The object traveled a total distance of 500 meters.
To calculate the total distance traveled by the object, we can add up the individual distances traveled in each direction.
The distances traveled in each direction are as follows:
- 27 meters northwest
- 27 meters
- 405 meters northwest
- 21 meters
- 20 meters northwest
To calculate the total distance traveled, we add these distances together:
27 + 27 + 405 + 21 + 20 = 500 meters
Therefore, the object traveled a total distance of 500 meters.
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If the area of a circle is 821 what is the radius
Answer: r≈16.17
Step-by-step explanation: r=A
π=821
π≈16.16578
. Translate each of the following problem into mathematial sentence then solve. Write your answer in your notebook. (3)/(4) multiplied by (16)/(21) is what number? The product of 5(7)/(9) and (27)/(56) is what number? 4(2)/(5) times 7(1)/(3) is what number? Twice the product of (8
1. The product of (3/4) multiplied by (16/21) is 4/7.
2. The product of 5(7/9) and (27/56) is 189/100.
3. 4(2/5) times 7(1/3) is 484/15.
4. Twice the product of (8/11) and (9/10) is 72/55.
To solve the given problems, we will translate the mathematical sentences and perform the necessary calculations.
1. (3/4) multiplied by (16/21):
Mathematical sentence: (3/4) * (16/21)
Solution: (3/4) * (16/21) = (3 * 16) / (4 * 21) = 48/84 = 4/7
Therefore, the product of (3/4) multiplied by (16/21) is 4/7.
2. The product of 5(7/9) and (27/56):
Mathematical sentence: 5(7/9) * (27/56)
Solution: 5(7/9) * (27/56) = (35/9) * (27/56) = (35 * 27) / (9 * 56) = 945/504 = 189/100
Therefore, the product of 5(7/9) and (27/56) is 189/100.
3. 4(2/5) times 7(1/3):
Mathematical sentence: 4(2/5) * 7(1/3)
Solution: 4(2/5) * 7(1/3) = (22/5) * (22/3) = (22 * 22) / (5 * 3) = 484/15
Therefore, 4(2/5) times 7(1/3) is 484/15.
4. Twice the product of (8/11) and (9/10):
Mathematical sentence: 2 * (8/11) * (9/10)
Solution: 2 * (8/11) * (9/10) = (2 * 8 * 9) / (11 * 10) = 144/110 = 72/55
Therefore, twice the product of (8/11) and (9/10) is 72/55.
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Sales Determination An appliance store sells a 42 ′′
TV for $400 and a 55 ′′
TV of the same brand for $730. During a oneweek period, the store sold 5 more 55 ′′
TVs than 42 ′′
TVs and collected $26,250. What was the total number of TV sets sold?
The total number of TV sets sold is 20 + 25 = 45.
Let the number of 42′′ TV sold be x and the number of 55′′ TV sold be x + 5.
The cost of 42′′ TV is $400.The cost of 55′′ TV is $730.
So, the total amount collected = $26,250.
Therefore, by using the above-mentioned information we can write the equation:400x + 730(x + 5) = 26,250
Simplifying this equation, we get:
1130x + 3650 = 26,2501130x = 22,600x = 20
Thus, the number of 42′′ TV sold is 20 and the number of 55′′ TV sold is 25 (since x + 5 = 20 + 5 = 25).
Hence, the total number of TV sets sold is 20 + 25 = 45.
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An architect built a scale model of Cowboys Stadium using a scale in which 2 inches represents 40 feet. The height of Cowboys Stadium is 320 feet. What is the height of the scale model in inches?
If an architect built a scale model of Cowboys Stadium using a scale in which 2 inches represents 40 feet and the height of Cowboys Stadium is 320 feet, then the height of the scale model in inches is 16 inches.
To find the height in inches, follow these steps:
According to the scale, 40 feet corresponds to 2 inches. Hence, 1 foot corresponds to 2/40 = 1/20 inches.Then, the height of the Cowboys Stadium in inches can be written as 320 feet * (1/20 inches/feet) = 16 inches.Therefore, the height of the scale model in inches is 16 inches.
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The Moore family received 23 pieces of mail on July 28 . The mail consisted of letters, magazines, bills, and ads. How many letters did they receive if they received five more ads than magazines, thre
The Moore family received 12 letters in their mail on July 28.
Let the number of magazines received be x.
According to the question, the number of ads is 5 more than the number of magazines i.e., ads = x + 5.
Also, the number of bills is three times the number of magazines i.e., bills = 3x.
Therefore, the total number of pieces of mail can be represented as:
Total pieces of mail = letters + magazines + bills + ads
23 = letters + x + 3x + (x+5)
Simplifying the above equation:
23 = 5x + 5
18 = 5x
x = 3.6
Since x represents the number of magazines, it cannot be a decimal value. So, we take the closest integer value, which is 4.
Hence, the number of magazines received by the Moore family is 4.
Now, substituting the values of magazines, ads, and bills in the equation:
letters = 23 - magazines - ads - bills
letters = 23 - 4 - 9 - 12
letters = 12
Therefore, the number of letters received by the Moore family on July 28 is 12.
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The number sequence is 1, 2, 4, 8, 6, 1, 2, 4, 8, 6,. How many sixes are in the first 296 numbers of the sequence?
Given sequence is 1, 2, 4, 8, 6, 1, 2, 4, 8, 6,. The content loaded is that the sequence is repeated. We need to find out the number of sixes in the first 296 numbers of the sequence. Solution: Let us analyze the given sequence first.
Number sequence is 1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....On close observation, we can see that the sequence is a combination of 5 distinct digits 1, 2, 4, 8, 6, and is loaded. Let's repeat the sequence several times to see the pattern.1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....1, 2, 4, 8, 6, 1, 2, 4, 8, 6, ....We see that the sequence is formed by repeating the numbers {1, 2, 4, 8, 6}. The first number is 1 and the 5th number is 6, and the sequence repeats. We have to count the number of 6's in the first 296 terms of the sequence.So, to obtain the number of 6's in the first 296 terms of the sequence, we need to count the number of times 6 appears in the first 296 terms.296 can be written as 5 × 59 + 1.Therefore, the first 296 terms can be written as 59 complete cycles of the original sequence and 1 extra number, which is 1.The number of 6's in one complete cycle of the sequence is 1. To obtain the number of 6's in 59 cycles of the sequence, we have to multiply the number of 6's in one cycle of the sequence by 59, which is59 × 1 = 59.There is no 6 in the extra number 1.Therefore, there are 59 sixes in the first 296 numbers of the sequence.
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If 13x = 1989 ,then find the value of 7x.
Answer:
1071
Step-by-step explanation:
1989÷13=153
so x=153
153×7=1071
so 7x=1071
Answer:
1,071
Explanation:
If 13x = 1,989, then I can find x by dividing 1,989 by 13:
[tex]\sf{13x=1,989}[/tex]
[tex]\sf{x=153}[/tex]
Multiply 153 by 7:
[tex]\sf{7\times153=1,071}[/tex]
Hence, the value of 7x is 1,071.
1. Explain Sampling 2. Differentiate between probability and non-probability sampling techniques. 3. State and explain the various forms of sampling under probability sampling. 4. State and explain the various forms of sampling under non-probability sampling. 5. Write down the advantages and disadvantages of each of the forms listed above.
Sampling is a method in research that involves selecting a portion of a population that represents the entire group. There are two types of sampling techniques, including probability and non-probability sampling techniques.
Probability sampling techniques involve the random selection of samples that are representative of the population under study. They include stratified sampling, systematic sampling, and simple random sampling. On the other hand, non-probability sampling techniques do not involve random sampling of the population.
It can provide a more diverse sample, and it can be more efficient than other forms of non-probability sampling. Disadvantages: It may introduce bias into the sample, and it may not provide a representative sample of the population. - Convenience Sampling: Advantages: It is easy to use and can be less costly than other forms of non-probability sampling. Disadvantages: It may introduce bias into the sample, and it may not provide a representative sample of the population.
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The functions g(x) and h(x) are defined on the domain (-[infinity], [infinity]). Com- pute the following values given that
g(-1)= 2 and h(-1) = -10, and
g(x) and h(x) are inverse functions of each other (i.e., g(x) = h-¹(x) and h(x) = g(x)).
(a) (g+h)(-1)
(b) (g-h)(-1)
The g(h(-1)) = g(-10) = -1 ------------ (1)h(g(x)) = x, which means h(g(-1)) = -1, h(2) = -1 ------------ (2)(a) (g + h)(-1) = g(-1) + h(-1)= 2 + (-10)=-8(b) (g - h)(-1) = g(-1) - h(-1) = 2 - (-10) = 12. The required value are:
(a) -8 and (b) 12
Given: g(x) and h(x) are inverse functions of each other (i.e.,
g(x) = h-¹(x) and h(x) = g(x)).g(-1) = 2 and h(-1) = -10
We are to find:
(a) (g + h)(-1) (b) (g - h)(-1)
We know that g(x) = h⁻¹(x),
which means g(h(x)) = x.
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A firm manufactures a commodity at two different factories, Factory X and Factory Y. The total cost (in dollars) of manufacturing depends on the quantities, x and y produced at each factory, A firm manufactures a commodity at two different factories, Factory X and Factory Y. The total cost (in dollars) of manufacturing depends on the quantities, x and y produced at each factory, respectively, and is expressed by the joint cost function: C(x,y)=x 2
+xy+2y 2
+1500 A) If the company's objective is to produce 1,000 units per month while minimizing the total monthly cost of production, how many units should be produced at each factory? (Round your answer to whole units, i.e. no decimal places.) To minimize costs, the company should produce: units at Factory X and units at Factory Y B) For this combination of units, their minimal costs will be dollars.respectively, and is expressed by the joint cost function: C(x,y)=x2 +xy+2y2+1500 A) If the company's objective is to produce 1,000 units per month while minimizing the total monthly cost of production, how many units should be produced at each factory? (Round your answer to whole units, i.e. no decimal places.) To minimize costs, the company should produce: _________units at Factory X and __________units at Factory Y B) For this combination of units, their minimal costs will be ________dollars.
To minimize the total monthly cost of production, we need to minimize the joint cost function C(x,y) subject to the constraint that x + y = 1000 (since the objective is to produce 1000 units per month).
We can use the method of Lagrange multipliers to solve this problem. Let L(x,y,λ) be the Lagrangian function defined as:
L(x,y,λ) = x^2 + xy + 2y^2 + 1500 + λ(1000 - x - y)
Taking partial derivatives and setting them equal to zero, we get:
∂L/∂x = 2x + y - λ = 0
∂L/∂y = x + 4y - λ = 0
∂L/∂λ = 1000 - x - y = 0
Solving these equations simultaneously, we obtain:
x = 200 units at Factory X
y = 800 units at Factory Y
Therefore, to minimize costs, the company should produce 200 units at Factory X and 800 units at Factory Y.
Substituting these values into the joint cost function, we get:
C(200,800) = 200^2 + 200800 + 2(800^2) + 1500 = $1,622,500
So, for this combination of units, their minimal costs will be $1,622,500.
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Kenzie purchases a small popcorn for $3.25 and one ticket for $6.50 each time she goes to the movie theater. Write an equation that will find how 6.50+3.25x=25.00 many times she can visit the movie th
Kenzie can visit the movie theater approximately 5 times, given the prices of a ticket and a small popcorn.
To find how many times Kenzie can visit the movie theater given the prices of a ticket and a small popcorn, we can set up an equation.
Let's denote the number of times Kenzie visits the movie theater as "x".
The cost of one ticket is $6.50, and the cost of a small popcorn is $3.25. So, each time she goes to the movie theater, she spends $6.50 + $3.25 = $9.75.
The equation that represents this situation is:
6.50 + 3.25x = 25.00
This equation states that the total amount spent, which is the sum of $6.50 and $3.25 multiplied by the number of visits (x), is equal to $25.00.
To find the value of x, we can solve this equation:
3.25x = 25.00 - 6.50
3.25x = 18.50
x = 18.50 / 3.25
x ≈ 5.692
Since we cannot have a fraction of a visit, we need to round down to the nearest whole number.
Therefore, Kenzie can visit the movie theater approximately 5 times, given the prices of a ticket and a small popcorn.
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The area of a room is roughly 9×10^4 square inches. If a person needs a minimum of 2.4×10^3square inches of space, what is the maximum number of people who could fit in this room? Write your answer in standard form, rounded down to the nearest whole person. The solution is
Based on the given area of the room and the minimum space required per person, we have determined that a maximum of 37 people could fit in this room.
To find the maximum number of people who can fit in the room, we need to divide the total area of the room by the minimum space required per person.
Given that the area of the room is approximately 9×10^4 square inches, and each person needs a minimum of 2.4×10^3 square inches of space, we can calculate the maximum number of people using the formula:
Maximum number of people = (Area of the room) / (Minimum space required per person)
First, let's convert the given values to standard form:
Area of the room = 9×10^4 square inches = 9,0000 square inches
Minimum space required per person = 2.4×10^3 square inches = 2,400 square inches
Now, we can perform the calculation:
Maximum number of people = 9,0000 square inches / 2,400 square inches ≈ 37.5
Since we need to round down to the nearest whole person, the maximum number of people who could fit in the room is 37.
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Question 3 ABC needs money to buy a new car. His friend accepts to lend him the money so long as he agrees to pay him back within five years and he charges 7% as interest (compounded interest rate). a) ABC thinks that he will be able to pay him $5000 at the end of the first year, and then $8000 each year for the next four years. How much can ABC borrow from his friend at initial time. b) ABC thinks that he will be able to pay him $5000 at the end of the first year. Estimating that his salary will increase through and will be able to pay back more money (paid money growing at a rate of 0.75). How much can ABC borrow from his friend at initial time.
ABC needs money to buy a new car.
a) ABC can borrow approximately $20500.99 from his friend initially
b) Assuming a payment growth rate of 0.75, ABC can borrow approximately $50139.09
a) To calculate how much ABC can borrow from his friend initially, we can use the present value formula for an annuity:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where PV is the present value, PMT is the annual payment, r is the interest rate, and n is the number of years.
In this case, ABC will make annual payments of $5000 in the first year and $8000 for the next four years, with a 7% compounded interest rate.
Calculating the present value:
PV = 5000 * [(1 - (1 + 0.07)^(-5)) / 0.07]
PV ≈ $20500.99
Therefore, ABC can borrow approximately $20500.99 from his friend initially.
b) If ABC's salary is estimated to increase at a rate of 0.75, we need to adjust the annual payments accordingly. The new payment schedule will be $5000 in the first year, $5000 * 1.75 in the second year, $5000 * (1.75)^2 in the third year, and so on.
Using the adjusted payment schedule, we can calculate the present value:
PV = 5000 * [(1 - (1 + 0.07)^(-5)) / 0.07] + (5000 * 1.75) * [(1 - (1 + 0.07)^(-4)) / 0.07]
PV ≈ $50139.09
Therefore, ABC can borrow approximately $50139.09 from his friend initially, considering the estimated salary increase.
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A high school student volunteers to present a report to the administration about the types of lunches students prefer. He surveys members of his class and records their choices. What type of sampling did the student use?
The type of sampling the student used is known as convenience sampling.
How to determine What type of sampling the student usedConvenience sampling involves selecting individuals who are easily accessible or readily available for the study. In this case, the student surveyed members of his own class, which was likely a convenient and easily accessible group for him to gather data from.
However, convenience sampling may introduce bias and may not provide a representative sample of the entire student population.
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In a random sample, 10 students were asked to compute the distance they travel one way to school to the nearest tenth of a mile. The data is listed below. Compute the range, standard deviation and variance of the data.
1.1 5.2 3.6 5.0 4.8 1.8 2.2 5.2 1.5 0.8
The range of the given data is 4.4 miles, the variance of the given data is 2.99054 and the standard deviation of the given data is 1.728 (approx).
To compute the range, standard deviation and variance of the given data we have to use the following formulae:
Range = Maximum value - Minimum value
Variance = (Σ(X - μ)²) / n
Standard deviation = √Variance
Here, the data given is:
1.1 5.2 3.6 5.0 4.8 1.8 2.2 5.2 1.5 0.8
First we will find out the range:
Range = Maximum value - Minimum value= 5.2 - 0.8= 4.4
Now, we will find the mean of the data.
μ = (ΣX) / n= (1.1 + 5.2 + 3.6 + 5.0 + 4.8 + 1.8 + 2.2 + 5.2 + 1.5 + 0.8) / 10= 30.2 / 10= 3.02
Now, we will find out the variance:
Variance = (Σ(X - μ)²) / n= [(1.1 - 3.02)² + (5.2 - 3.02)² + (3.6 - 3.02)² + (5.0 - 3.02)² + (4.8 - 3.02)² + (1.8 - 3.02)² + (2.2 - 3.02)² + (5.2 - 3.02)² + (1.5 - 3.02)² + (0.8 - 3.02)²] / 10= [(-1.92)² + (2.18)² + (0.58)² + (1.98)² + (1.78)² + (-1.22)² + (-0.82)² + (2.18)² + (-1.52)² + (-2.22)²] / 10= (3.6864 + 4.7524 + 0.3364 + 3.9204 + 3.1684 + 1.4884 + 0.6724 + 4.7524 + 2.3104 + 4.9284) / 10= 29.9054 / 10= 2.99054
Now, we will find out the standard deviation:
Standard deviation = √Variance= √2.99054= 1.728 (approx)
Hence, the range of the given data is 4.4 miles, the variance of the given data is 2.99054 and the standard deviation of the given data is 1.728 (approx).
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The number of bacteria P(h) in a certain population increases according to the following function, where time (h) is measured in hours.
P(h)=1900 e^{0.18 h}
How many hours will it take for the number of bacteria to reach 2500 ?
Round your answer to the nearest tenth, and do not round any inteediate computations.
The number of bacteria in a certain population increases according to the function P(h) = 100(2.5)^h, where time (h) is measured in hours. we get h ≈ 5.6. Thus,by solving the equation t it will take approximately 5.6 hours of time for the population of bacteria to reach 2500.
The task is to determine how many hours it will take for the number of bacteria to reach 2500, rounded to the nearest tenth. The given function that models the population growth of bacteria is P(h) = 100(2.5)^h, where h is the number of hours. It can be observed that the initial population is 100 when h = 0, and the population doubles every hour as the base of 2.5 is greater than 1. The task is to find how many hours it will take for the population to reach 2500.
So, we have to solve the equation 100(2.5)^h = 2500 for h. Dividing both sides of the equation by 100, we get (2.5)^h = 25. Now, we can take the logarithm of both sides of the equation, with base 2.5 to obtain h.
log2.5(2.5^h) = log2.5(25)
h = log2.5(25)
Using a calculator, we get h ≈ 5.6. we get h ≈ 5.6. Thus, it will take approximately 5.6 hours for the population of bacteria to reach 2500.
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What is the general equation of an ellipse whose standard equation is left parenthesis x plus 3 right parenthesis squared over 4 plus left parenthesis y minus 5 right parenthesis squared over 16 equals 1 ?
The general equation of the given ellipse is [tex]((x + 3)^2 / 4) + ((y - 5)^2 / 16) = 1.[/tex]
The standard equation of an ellipse is given by:
[tex]((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1[/tex]
where (h, k) represents the coordinates of the center of the ellipse, and a and b are the lengths of the major and minor axes, respectively.
In the given equation, we have:
[tex]((x + 3)^2 / 4) + ((y - 5)^2 / 16) = 1[/tex]
Comparing this with the standard equation, we can deduce the following information:
The center of the ellipse is (-3, 5), which is obtained from the opposite signs of the x and y terms in the standard equation.
The length of the major axis is 2a, which is equal to 2 times the square root of 4, resulting in a value of 4.
Therefore, the major axis has a length of 8 units.
The length of the minor axis is 2b, which is equal to 2 times the square root of 16, resulting in a value of 8.
Therefore, the minor axis has a length of 16 units.
Using this information, we can conclude that the general equation of the ellipse is:
[tex]((x + 3)^2 / 4) + ((y - 5)^2 / 16) = 1[/tex]
This equation represents an ellipse with center (-3, 5), a major axis of length 8 units, and a minor axis of length 16 units.
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Find the equation of a line that is parallel to the line y=-7 and passes through the point (-1,9).
Hence, the equation of the line that is parallel to the line y = -7 and passes through the point (-1, 9) is y = 9.
Given that a line that is parallel to the line y = -7 and passes through the point (-1, 9) is to be determined.
To find the equation of the line that is parallel to the line y = -7 and passes through the point (-1, 9), we need to make use of the slope-intercept form of the equation of the line, which is given by y = mx + c, where m is the slope of the line and c is the y-intercept of the line.
In order to determine the slope of the line that is parallel to the line y = -7, we need to note that the slope of the line y = -7 is zero, since the line is a horizontal line.
Therefore, any line that is parallel to y = -7 would also have a slope of zero.
Therefore, the equation of the line that is parallel to the line y = -7 and passes through the point (-1, 9) would be given by y = 9, since the line would be a horizontal line passing through the y-coordinate of the given point (-1, 9).
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and sample standard deviation cost of repair. The range is $216. s 2
=9602 dollars 2
(Round to the nearest whole number as needed.) s=$ (Round to two decimal places as needed.)
Given,Range = $216s^2 = 9602 dollar^2Now, we are supposed to find the Sample Standard Deviation Cost of Repair.
Solution:Formula for the Sample standard deviation is:s = √[Σ(x-µ)²/(n-1)]Now, we have to find the value of ‘s’.Hence, by substituting the given values we get,s = √[Σ(x-µ)²/(n-1)]s = √[9602/(n-1)]Now, in order to solve the above equation, we need to find the value of n, mean and summation of x.Here, we can observe that the number of observations 'n' is not given. Hence, we can’t solve this problem. But, we can say that the value of sample standard deviation ‘s’ is directly proportional to the value of square root of range 'r'.i.e., s ∝ √rOn solving the given problem, the value of range is 216. Hence, the value of square root of range ‘r’ can be calculated as follows:r = 216 = 6 × 6 × 6Now, substituting the value of 'r' in the above expression, we get,s ∝ √r = √(6×6×6) = 6√6Thus, the sample standard deviation cost of repair is 6√6 dollar. Hence, the answer is s=6√6 dollars.
Sample standard deviation is an estimation of population standard deviation. It is a tool used for analyzing the spread of data in a dataset. It is used for measuring the amount of variation or dispersion of a set of values from its average or mean value. The formula for calculating sample standard deviation is s = √[Σ(x-µ)²/(n-1)]. The given problem is about calculating the sample standard deviation of the cost of repair. But, the problem lacks the number of observations 'n', mean and summation of x. Hence, the problem can't be solved directly.
But, we can say that the value of sample standard deviation ‘s’ is directly proportional to the value of square root of range 'r'.i.e., s ∝ √rOn solving the given problem, the value of range is 216. Hence, the value of square root of range ‘r’ can be calculated as follows:r = 216 = 6 × 6 × 6Now, substituting the value of 'r' in the above expression, we get,s ∝ √r = √(6×6×6) = 6√6Thus, the sample standard deviation cost of repair is 6√6 dollar. Therefore, the answer is s=6√6 dollars.
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Find an equation of the plane through the three points given: P=(4,0,0),Q=(3,4,−4),R=(5,−1,−4)=−80
The equation of the plane is -16x - 12y - 4z + 64 = 0.
Given three points P = (4, 0, 0), Q = (3, 4, -4), R = (5, -1, -4) and a plane equation through the three points. We need to find the equation of the plane.
Let's start with the vector PQ and PR will lie on the plane
PQ vector = Q - P = (3, 4, -4) - (4, 0, 0)
= (-1, 4, -4)
PR vector = R - P = (5, -1, -4) - (4, 0, 0)
= (1, -1, -4)
The normal vector of the plane will be perpendicular to both the above vectors.
N = PQ × PRN = (-1, 4, -4) x (1, -1, -4)
N = (-16, -12, -4)
The equation of the plane is of the form ax + by + cz = d. Now we will substitute any one of the three points to find the value of d. We use point P as P.
N + d = 0(-16)(4) + (-12)(0) + (-4)(0) + d = 0 +d = 64
The equation of the plane is -16x - 12y - 4z + 64 = 0. The plane is represented by the equation -16x - 12y - 4z + 64 = 0.
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The student council is hosting a drawing to raise money for scholarships. They are selling tickets for $7 each and will sell 700 tickets. There is one $2,000 grand prize, four $200 second prizes, and sixteen $10 third prizes. You just bought a ticket. Find the expected value for your profit. Round to the nearest cent.
Given Data: Price of a single ticket = $7Number of tickets sold = 700Amount of Grand Prize = $2,000Amount of Second Prize (4) = $200 x 4 = $800Amount of Third Prize (16) = $10 x 16 = $160
Expected Value can be defined as the average value of each ticket bought by each person.
Therefore, the expected value of the profit is the sum of the probabilities of each winning ticket multiplied by the amount won.
Calculation: Expected value for your profit = probability of winning × amount wonProbability of winning Grand Prize = 1/700
Therefore, the expected value of Grand Prize = (1/700) × 2,000 = $2.86
Probability of winning Second Prize = 4/700Therefore, the expected value of Second Prize = (4/700) × 200 = $1.14
Probability of winning Third Prize = 16/700Therefore, the expected value of Third Prize = (16/700) × 10 = $0.23
Expected value of profit = (2.86 + 1.14 + 0.23) - 7
Expected value of profit = $3.23 - $7
Expected value of profit = - $3.77
As the expected value of profit is negative, it means that on average you would lose $3.77 on each ticket you buy. Therefore, it is not a good investment.
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A sample of four 35-year-old males is asked about the average number of hours per week that he exercises, and is also given a blood cholesterol test. The data is recorded in the order pairs given below, in the form (Hours Exercising, Cholesterol Level):
(2.4,222), (3,208), (4.8, 196), (6,180)
Suppose that you know that the correlation coefficient r = -0.980337150474362.
Find the coefficient of determination for this sample.
r-squared =
Which of the following is a correct interpretation of the above value of 22
A. Spending more time exercising will make your muscles go big.
B. Spending more time exercising causes cholesterol levels to go down.
OC. 96.106% of the variance in hours spent exercising is explained by changes in cholesterol levels. D. 96.106% of the variance in cholesterol levels is explained by changes in hours spent exercising.
The coefficient of determination (r-squared) is calculated by squaring the correlation coefficient (r).
Given that r = -0.980337150474362, we can find r-squared as follows:
r-squared = (-0.980337150474362)^2 = 0.9609
Therefore, the coefficient of determination for this sample is 0.9609.
The correct interpretation of this value is:
D. 96.106% of the variance in cholesterol levels is explained by changes in hours spent exercising.
Note: The coefficient of determination represents the proportion of the variance in the dependent variable (cholesterol levels) that can be explained by the independent variable (hours spent exercising). In this case, approximately 96.106% of the variance in cholesterol levels can be explained by changes in hours spent exercising.
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Suppose elementary students are asked their favorite color, and these are the results: - 24% chose blue - 17% chose red - 16% chose yellow What percentage chose something other than red, blue, or yellow? (Each student was only allowed to choose one favorite color.) Your Answer:
The percentage of students who chose something other than red, blue, or yellow is 43%.
To find the percentage of students who chose something other than red, blue, or yellow, we need to subtract the percentages of students who chose red, blue, and yellow from 100%.
Given:
- 24% chose blue
- 17% chose red
- 16% chose yellow
Let's calculate the percentage of students who chose something other than red, blue, or yellow:
Percentage of students who chose something other than red, blue, or yellow = 100% - (percentage of students who chose red + percentage of students who chose blue + percentage of students who chose yellow)
= 100% - (17% + 24% + 16%)
= 100% - 57%
= 43%
43% of the students chose something other than red, blue, or yellow as their favorite color.
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Use implicit differentiation to find the derivatives dy/dx of the following functions. For (c) and (d), express dxdy in terms of x only. (a) x^3+y^3=4 (b) y=sin(3x+4y) (c) y=sin^−1x (Hint: y=sin^−1x⟹x=siny, and recall the identity sin^2y+cos^2y=1 ) 6 (d) y=tan^−1x (Hint: y=tan−1x⟹x=tany, and recall the identity tan^2y+1=sec^2y )
(a) The derivative of x^3+y^3=4 is given by 3x^2+3y^2(dy/dx)=0. Thus, dy/dx=-x^2/y^2.
(b) The derivative of y=sin(3x+4y) is given by dy/dx=3cos(3x+4y)/(1-4cos^2(3x+4y)).
(c) The derivative of y=sin^(-1)x is given by dy/dx=1/√(1-x^2).
(d) The derivative of y=tan^(-1)x is given by dy/dx=1/(1+x^2).
(a) To find dy/dx for the equation x^3 + y^3 = 4, we can differentiate both sides of the equation with respect to x using implicit differentiation:
d/dx (x^3 + y^3) = d/dx (4)
Differentiating x^3 with respect to x gives us 3x^2. To differentiate y^3 with respect to x, we use the chain rule. Let's express y as a function of x, y(x):
d/dx (y^3) = d/dx (y^3) * dy/dx
Applying the chain rule, we get:
3y^2 * dy/dx = 0
Now, let's solve for dy/dx:
dy/dx = 0 / (3y^2)
dy/dx = 0
Therefore, the derivative dy/dx for the equation x^3 + y^3 = 4 is 0.
(b) For the equation y = sin(3x + 4y), let's differentiate both sides of the equation with respect to x using implicit differentiation:
d/dx (sin(3x + 4y)) = d/dx (y)
Using the chain rule, we have:
cos(3x + 4y) * (3 + 4(dy/dx)) = dy/dx
Rearranging the equation, we can solve for dy/dx:
4(dy/dx) - dy/dx = -cos(3x + 4y)
Combining like terms:
3(dy/dx) = -cos(3x + 4y)
Finally, we can express dy/dx in terms of x only:
dy/dx = (-cos(3x + 4y)) / 3
(c) For the equation y = sin^(-1)(x), we can rewrite it as x = sin(y). Let's differentiate both sides with respect to x using implicit differentiation:
d/dx (x) = d/dx (sin(y))
The left side is simply 1. To differentiate sin(y) with respect to x, we use the chain rule:
cos(y) * dy/dx = 1
Now, we can solve for dy/dx:
dy/dx = 1 / cos(y)
Using the Pythagorean identity sin^2(y) + cos^2(y) = 1, we can express cos(y) in terms of x:
cos(y) = sqrt(1 - sin^2(y))= sqrt(1 - x^2) (substituting x = sin(y))
Therefore, the derivative dy/dx for the equation y = sin^(-1)(x) is:
dy/dx = 1 / sqrt(1 - x^2)
(d) For the equation y = tan^(-1)(x), we can rewrite it as x = tan(y). Let's differentiate both sides with respect to x using implicit differentiation:
d/dx (x) = d/dx (tan(y))
The left side is simply 1. To differentiate tan(y) with respect to x, we use the chain rule:
sec^2(y) * dy/dx = 1
Now, we can solve for dy/dx:
dy/dx = 1 / sec^2(y)
Using the identity tan^2(y) + 1 = sec^2(y), we can express sec^2(y) in terms of x:
sec^2(y) = tan^2(y) + 1= x^2 + 1 (substituting x = tan(y))
Therefore, the derivative dy/dx for the equation y = tan^(-1)(x) is:
dy/dx = 1 / (x^2 + 1)
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The Flemings secured a bank Ioan of $320,000 to help finance the purchase of a house. The bank charges interest at a rate of 3%/year on the unpaid balance, and interest computations are made at the end of each month. The Flemings have agreed to repay the in equal monthly installments over 25 years. What should be the size of each repayment if the loan is to be amortized at the end of the term? (Round your answer to the nearest cent.)
The size of each repayment should be $1,746.38 if the loan is to be amortized at the end of the term.
Given: Loan amount = $320,000
Annual interest rate = 3%
Tenure = 25 years = 25 × 12 = 300 months
Annuity pay = Monthly payment amount to repay the loan each month
Formula used: The formula to calculate the monthly payment amount (Annuity pay) to repay a loan amount with interest over a period of time is given below.
P = (Pr) / [1 – (1 + r)-n]
where P is the monthly payment,
r is the monthly interest rate (annual interest rate / 12),
n is the total number of payments (number of years × 12), and
P is the principal or the loan amount.
The interest rate of 3% per year is charged on the unpaid balance. So, the monthly interest rate, r is given by;
r = (3 / 100) / 12 = 0.0025 And the total number of payments, n is given by n = 25 × 12 = 300
Substituting the given values of P, r, and n in the formula to calculate the monthly payment amount to repay the loan each month.
320000 = (P * (0.0025 * (1 + 0.0025)^300)) / ((1 + 0.0025)^300 - 1)
320000 = (P * 0.0025 * 1.0025^300) / (1.0025^300 - 1)
(320000 * (1.0025^300 - 1)) / (0.0025 * 1.0025^300) = P
Monthly payment amount to repay the loan each month = $1,746.38
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A cylindrical object is 3.13 cm in diameter and 8.94 cm long and
weighs 60.0 g. What is its density in g/cm^3
A cylindrical object is 3.13 cm in diameter and 8.94 cm long and weighs 60.0 g. The density of the cylindrical object is 0.849 g/cm^3.
To calculate the density, we first need to find the volume of the cylindrical object. The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius (half of the diameter) and h is the height (length) of the cylinder.
Given that the diameter is 3.13 cm, the radius is half of that, which is 3.13/2 = 1.565 cm. The length of the cylinder is 8.94 cm.
Using the values obtained, we can calculate the volume: V = π * (1.565 cm)^2 * 8.94 cm = 70.672 cm^3.
The density is calculated by dividing the weight (mass) of the object by its volume. In this case, the weight is given as 60.0 g. Therefore, the density is: Density = 60.0 g / 70.672 cm^3 = 0.849 g/cm^3.
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Write the equation of a line with the slope, (3)/(2) ,which passes through the point (0,-4). Write the answer in slope -intercept form.
The equation of the line with a slope of 3/2, passing through the point (0, -4), in slope-intercept form is y = (3/2)x - 4.
To write the equation of a line in slope-intercept form, we need two key pieces of information: the slope of the line and a point it passes through. Given that the slope is 3/2 and the line passes through the point (0, -4), we can proceed to write the equation.
The slope-intercept form of a line is given by the equation y = mx + b, where m represents the slope and b represents the y-intercept.
Substituting the given slope, m = 3/2, into the equation, we have y = (3/2)x + b.
To find the value of b, we substitute the coordinates of the given point (0, -4) into the equation. This gives us -4 = (3/2)(0) + b.
Simplifying the equation, we have -4 = 0 + b, which further reduces to -4 = b.
Therefore, the value of the y-intercept, b, is -4.
Substituting the values of m and b into the slope-intercept form equation, we have the final equation:
y = (3/2)x - 4.
This equation represents a line with a slope of 3/2, meaning that for every 2 units of horizontal change (x), the line rises by 3 units (y). The y-intercept of -4 indicates that the line intersects the y-axis at the point (0, -4).
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Convert each individual dato value to a standardized z.score. a-1. Ages of airline passengers: x=81,μ=49,σ=9 (Round your answer to 3 decimal places.) a-2. Is it an outlier? Yes, this is an outlier. No, this is an unusual observation. No, this is not an outlier nor is it unusual. b-1. FiCO credit scores: x=569,μ=738,σ=74 (Round your answer to 3 decimal places. Negative amount should be indicated by a minus sign.) b-2. Is it an outier? No, this is an unusual observation. No, this is not an outlier nor is it unusual. Yes, this is an outlier. c-1. Condo rental vacancy days: x=21,μ=20,σ=6 (Round your answer to 3 decimal places.) c-2. Is it an outlier? No, this is not an outlier nor is it unusual. Yes, this is an outlier. No, this is an unusual observation.
a-1: The standardized z-score for the age of the airline passenger is approximately 3.556.
a-2. The statement provided does not indicate whether the given age value (81) is considered an outlier or unusual observation.
To convert the age of an airline passenger (x=81) to a standardized z-score, use the formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
Plugging in the values,
z = (81 - 49) / 9 =3.556
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Determine the equation of the circle that passes through point (-3, -2) whose center is at (-3, 5)
The equation of the circle is [tex](x + 3)^2 + (y - 5)^2 = 49[/tex].
The equation of the circle that passes through point (-3, -2) and whose center is at (-3, 5) can be determined as follows:
Center of the circle (h, k) = (-3, 5)
And the point (-3, -2) lies on the circle.
We can find the radius of the circle using the distance formula between two points in a plane. The formula is:
[tex]r = \sqrt[2]{(x2 - x1)^2 + (y2 - y1)}[/tex]
where (x1, y1) and (x2, y2) are the coordinates of the center and the given point on the circle respectively.
So, substituting the values, we get:
[tex]r = \sqrt[2]{((-3 - (-3))^2 + (5 - (-2)))}[/tex]
= [tex]\sqrt{(0^2 + 7^2)}[/tex]
= 7 units.
Now, the equation of the circle can be obtained using the standard equation of the circle:
[tex](x - h)^2 + (y - k)^2 = r^2[/tex]
Substituting the values of (h, k) and r, we get the equation of the circle as:
[tex](x - (-3))^2 + (y - 5)^2 = 7^2 or(x + 3)^2 + (y - 5)^2[/tex]
= 49
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Factor the following function by finding all rational and other zeros first: P(x)=x^(3)+2x^(2)+x+2.
The complete factorization of the function P(x) is [tex]P(x) = (x + 1)(x - [-1 + i*\sqrt{ (7)/ 2} (x - [-1 - i*\sqrt{(7)] / 2}.[/tex]
The function given to us is: P(x) = x³ + 2x² + x + 2
To find all the rational and other zeros of the given function, we can use the rational root theorem. According to the rational root theorem, if a polynomial function has a rational zero, then it must be of the form: p/q where p is a factor of the constant term of the function and q is a factor of the leading coefficient of the function.
Here, the constant term is 2 and the leading coefficient is 1, so the possible rational roots of the function P(x) are: ±1, ±2.
Next, we can test these possible rational roots using synthetic division:
Let's start with the root x = -1, we have the following synthetic division:
x | 1 2 1 2-1 |___|_______|_______|______|1 1 2 | 0
Since we get a zero remainder, x = -1 is a root of the function P(x).Using the factor theorem, we can write:
P(x) = (x + 1)(x² + x + 2)
Now, we need to find the roots of the quadratic factor x² + x + 2. Since there are no real roots of this quadratic, we can use the quadratic formula to find the complex roots:
x = [-b ± sqrt(b² - 4ac)] / 2a
Here, a = 1, b = 1, c = 2, so we have:
[tex]x = [-1 ± sqrt(1 - 4(1)(2))] / 2[/tex]
[tex]= [-1 ± sqrt(-7)] / 2[/tex]
[tex]= [-1 ± i*sqrt(7)] / 2[/tex]
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