The work done by stretching the spring to the given distance is:
[tex]W=\frac{kx_{0} }{3}[/tex]
The given parameters:
Applied force on the spring = F
Extension of the spring, = x₀
The work done by stretching the spring to the given distance is calculated as follows:
[tex]W=\int\limits^b_a {F} \, dx[/tex]
Where:
F - Force.
W- Work.
x- Distance.
If we know that F=k.x², then the work of the nonlinear spring is:
[tex]W=\int\limits^b_a {F} \, dx \\W=\int\limits^b_a {kx^2} \, dx \\W=k\int\limits^b_a {x^2} \, dx \\\\W=k[\frac{x^3}{3}]\\\\W=k[\frac{x_b-x_a}{3} ]\\\\W=k[\frac{x_o-0}{3} ][/tex]
W=kx₀/3
Thus, the work done by stretching the spring to the given distance is: W=kx₀/3
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find the charge q(t) that crosses the dashed line going from left to right in the time interval (0, t) [s].
The charge q(t) that crosses the dashed line going from left to right in the time interval (0, t) [s] is given by q(t) = q0 + (t)I,
The charge q(t) that crosses the dashed line going from left to right in the time interval (0, t) [s] is calculated by the formula q(t) = q0 + (t)I, where q0 is the initial charge and I is the current.
The charge q(t) that crosses the dashed line going from left to right in the time interval (0, t) [s] is given by q(t) = q0 + (t)I, where q0 is the initial charge and I is the current. To calculate q(t), we need to first calculate the initial charge q0 and then calculate the current I. To calculate q0, we need to integrate the current I over the time interval (0, t). To calculate q(t), we need to know q0 and I. Then, we can calculate q(t) by multiplying t by I and adding it to q0. For example, if q0 = 3 C and I = 2 A, then q(t) = 3 + (t)(2) = 3 + 2t C.
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Find the Unknown value of the triangle: Trigonometry
The unknown values in the triangle is calculated to be
PL = 14.3 cm
CY = 7.74
NY = 9.9 mm
LJ = 8.3
How to find the unknown values in the triangleThe unknown values in the triangle is worked using SOH CAH TOA
Sin = opposite / hypotenuse - SOH
Cos = adjacent / hypotenuse - CAH
Tan = opposite / adjacent - TOA
1. using Sin = opposite / hypotenuse - SOH
sin x = PL / PX
sin 54.25 = PL / 17.62
PL = sin 54.25 * 17.62
PL = 14.3 cm
2. using Tan = opposite / adjacent - TOA
tan 52.67 = CY / 5.9
tan 52.67 * 5.9 = CY
CY = 7.74
3. using Sin = opposite / hypotenuse - SOH
sin 33.06 = YC / NY
NY = 5.4 / sin 33.06
NY = 9.9 mm
4. using Tan = opposite / adjacent - TOA
tan 55.05 = LJ / VJ
tan 55.05 * 5.8 = LJ
LJ = 8.3
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About 13% of the students at high school have a part time job what is the probability that they’re will be between 33 and 45 with part time jobs in a randomly selected group of 300
A:.173
B.736
C.694
D.172
The probability that they’re will be between 33 and 45 with part time jobs in a randomly selected group of 300 is given as follows:
B. 0.736.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].The parameters for the binomial distribution in this problem are given as follows:
p = 0.13, n = 300.
Hence the mean and the standard deviation are given as follows:
[tex]\mu = 300 \times 0.13 = 39[/tex][tex]\sigma = \sqrt{300 \times 0.13 \times 0.87} = 5.82[/tex]Using continuity correction, the probability of X between 33 and 45 is (32.5 < X < 45.5), which is the p-value of Z when X = 45.5 subtracted by the p-value of Z when X = 32.5, hence:
X = 45.5:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (45.5 - 39)/5.82
Z = 1.12
Z = 1.12 has a p-value of 0.8686.
X = 32.5:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (32.5 - 39)/5.82
Z = -1.12
Z = -1.12 has a p-value of 0.1314
Hence:
0.8686 - 0.1314 = 0.736 (approximately).
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Use the properties of exponents to generate an expression equivalent to each expression.
[tex]2^(3t+4)[/tex]
The expression equivalent to the expression 2(3t + 4) is 6t + 8
How to determine the expression equivalent to each expression.From the question, we have the following parameters that can be used in our computation:
2(3t + 4)
To start with, we need to open the bracket of the expression
so, we have the following representation
2 * 3t + 2* 4
Evaluate the products
6t + 8
hence, the equivalent expression is 6t + 8
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Answer both part a and b the problem
a. Equation for Bert to figure out Ernie's original number is x = 5(2x - 6) + (-6).
b. Ernie's original number/integer was 4.
c. On checking and verifying the original integer is obtained as 4.
What is integers?
The Latin term "Integer," which implies entire or intact, is where the word "integer" first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as integers.
Ernie picks an integer from 1 to 10.
Let the value of the integer be x.
First the number is doubled, so the value of integer becomes = 2x.
Then 6 is subtracted from the number, so the value of integer becomes = 2x - 6.
Then the entire difference is multiplied by 5, so the value of integer becomes = 5(2x - 6).
Lastly -6 is added to the product, so the value of integer becomes = 5(2x - 6) + (-6).
So, the equation to find the integer is -
x = 5(2x - 6) + (-6)
Now, simplify this expression to obtain the number.
x = 5(2x - 6) + (-6)
x = 10x - 30 - 6
x - 10x = -36
-9x = -36
x = -36/-9
x = 4
Check if 4 is the correct number -
First the number is doubled = 4 × 2 = 8
Subtract 6 from the number = 8 - 6 = 2
Then multiply the difference by 5 = 5 × 2 = 10
Then subtract -6 from the number = 10 - 6 = 4
Therefore, the original number was 4.
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use scenario 3-8. which of the following statements can be made on the basis of the computer output? group of answer choices 69.7% of the variation in fish length can be accounted for by the linear regression of egg production on fish length. 83.5% of the variation in egg production can be accounted for by the linear regression of egg production on fish length. 83.5% of the variation in fish length can be accounted for by the linear regression of egg production on fish length. 69.7% of the variation in egg production can be accounted for by the linear regression of egg production on fish length. 68.4% of the variation in fish length can be accounted for by the linear regression of egg production on fish length.
The computer output for scenario 3-8 indicates that 83.5% of the variation in egg production can be accounted for by the linear regression of egg production on fish length, while 69.7% of the variation in fish length can be accounted for by the same linear regression.
This indicates that there is a strong and significant correlation between fish length and egg production, with egg production increasing as fish length increases. This suggests that egg production is a good predictor of fish length and vice versa, and that changes in one can be used to predict changes in the other.
However, it is worth noting that this correlation is not perfect and that there is still a considerable amount of variation in each that cannot be explained by the linear regression.
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Correct question should be:
Use Scenario 3-8. Which of the following statements can be made on the basis of the computer output?
A. 83.5% of the variation in egg production can be accounted for by the linear regression of
egg production on fish length.
B. 69.7% of the variation in egg production can be accounted for by the linear regression of
egg production on fish length.
C. 83.5% of the variation in fish length can be accounted for by the linear regression of egg
production on fish length.
D. 69.7% of the variation in fish length can be accounted for by the linear regression of egg
production on fish length.
E. 68.4% of the variation in fish length can be accounted for by the linear regression of egg
production on fish length.
Find JK.
12
2
I
J
K
30°
Write your answer in simplified, rationalized form. Do not round.
JK =
If the series JK has 12 2 I J K 30, the value of JK is 720
How do we determine the value?In order to find the value of JK, multiply the given values all through and then generate the answer in a simplified manner
JK=(12)(2)(30)
Multiply all the figures
JK = 12 X 2 X 30
JK = 720
Therefore, the correct answer is as given above. It can be concluded that the value of JK is 720 as a result of the multiplication.
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write an arm assembly language program to calculate the factorial of any reasonable small integer that is in register r0 using loop constructs. the final result, .e., the factorial should be in register r7. as an example, you can calculate 5! (ta may ask you to calculate factorial for another numbers, to make sure the program will work for numbers besides 5).
The result of the calculation, 120, will be stored in register r7.
Factorial is a mathematical operation where a number is multiplied by all its preceding numbers until 1. For example, 5! is calculated by multiplying 5 by 4, then 4 by 3, then 3 by 2, then 2 by 1. In mathematical notation, this is expressed as:
5! = 5 * 4 * 3 * 2 * 1
= 120
In arm assembly language, we can create a loop that multiplies the number that is stored in register r0 with all of its preceding numbers until 1 is reached. The result of the calculation will be stored in register r7. For example, to calculate 5!, we can set the initial value of r0 to 5 and the initial value of r7 to 1. Then, we can create a loop that multiplies the value of r0 by the value of r7 and then decrements the value of r0. This loop will run until r0 is equal to 1. At that point, the loop will end and the result of the calculation will be stored in r7. The following example shows the code for a program to calculate 5!:
MOV R0, #5
MOV R7, #1
LOOP:
MUL R7, R0
SUB R0, #1
CMP R0, #1
BNE LOOP
The result of the calculation, 120, will be stored in register r7.
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what angles are supplementary
Answer:
∠MKN and ∠OKN
Step-by-step explanation:
∠MKN and ∠OKN are supplementary because they add up to 180°
Based on a random sample of 50 students, the 90 percent confidence interval for the mean amount of money students spend on lunch at a certain high school is found to be ($3.45, $4.15). Which of the following statements is true?
answer choices
90% of the time, the mean amount of money that all students spend on lunch at this high school will be between $3.45 and $4.15.
90% of all students spend between $3.45 and $4.15 on lunch at this high school.
90% of all random samples of 50 students obtained at this high school would result in a sample mean amount of money students spend on lunch between $3.45 and $4.15.
90% of all random samples of 50 students obtained at this high school would result in a 90% confidence interval that contains the true mean amount of money students spend on lunch.
Approximately 45 of the 50 students in the random sample will spend between $3.45 and $4.15 on lunch at this high school.
The right answer is that a 90% confidence interval containing the actual mean amount of money kids spend on lunch would be produced by 90% of all random samples of 50 students collected at this high school.
It is given that the 90% confidence interval for the mean amount of money students spend on lunch time at a high school is ($3.45,$4.15)
A 90% confidence interval interprets that, 90% of the random samples will result in a 90% confidence interval that contain the true value of the population parameter.
In other words, 90% of the random samples of 50 students obtained at this high school would result in a 90% confidence interval that contains the true mean amount of money students spend on lunch.
Thus, the correct answer is option (D) or The right answer is that a 90% confidence interval containing the actual mean amount of money kids spend on lunch would be produced by 90% of all random samples of 50 students collected at this high school.
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Write an explicit formula for a(n), the n(th) term of the sequence 11,4,-3, ....
The explicit formula that represents the nth term of the sequence will be aₙ = 18 - 7n.
What is an arithmetic sequence?A series of integers called an arithmetic succession or arithmetic chain of events has a fixed difference between the terms.
Let a₁ be the first term and d be a common difference.
Then the nth term of the arithmetic sequence is given as,
aₙ = a₁ + (n - 1)d
The arithmetic sequence is given below.
11, 4, -3, .....
The first term is 11. And the common difference is given as,
d = 4 - 11
d = - 7
The explicit formula that represents the nth term of the sequence is given as,
aₙ = a₁ + (n - 1)d
aₙ = 11 + (n - 1)(-7)
aₙ = 11 - 7n + 7
aₙ = 18 - 7n
The explicit formula that represents the nth term of the sequence will be aₙ = 18 - 7n.
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Harry places a ladder against his house to fix the roof, leaving 5 feet between the edge of the house and the ladder. The roof is 18 feet off the ground. Using the Triangle Inequality Theorem, determine the range of heights his ladder should have.
Using the Triangle Inequality Theorem, the range of heights his ladder should have would be greater than or = 23 feet.
What is Triangle Inequality Theorem?The Triangle Inequality Theorem states that the sum of any two sides of a triangle is greater than or equal to the third side.
That is c >/= a + b
a = 5 feet
b = 18 feet
C >/ = 5+18 = 23 feet
Therefore, the range of heights his ladder should have would be greater than or = 23 feet.
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A grocery store is placing an order for dairy products. there are 8 varieties of dairy products sold in cases, where each case contains all the same variety of dairy products. the store will order 50 cases in total. 1) How many ways are there to place the order? 2)one of the varieties of dairy products is milk. how many ways are there to place the order if they order at most 20 cases of milk
We have 50 cases to distribute among 8 varieties of dairy products, with a restriction of 20 cases of milk.
How find the total number?To find the total number of ways to place the order, you can use the binomial coefficient formula, which is often used in combinatorics to count the number of ways to choose k items from a set of n items without regard to order. In this case, you have 8 varieties of dairy products and you want to choose 50 cases in total, with no regard to the order in which you choose the cases. So, the number of ways to place the order is:
(n k) = (8 50) = 8! / (50! (8-50)!) = 8! / (50! (-42)!) = 8! / 50! = 87654321 / (50494847464544434241)
This is a very large number, and it's impractical to calculate it exactly.
To find the number of ways to place the order if they order at most 20 cases of milk, you can use a similar approach, but with a restriction on the number of cases of milk. One way to do this is to use the stars and bars method, which is a combinatorial method for counting the number of ways to distribute n identical items into k distinguishable containers, with a restriction on the number of items in some of the containers.
In this case, we have 50 cases to distribute among 8 varieties of dairy products, with a restriction of 20 cases of milk. So, the number of ways to place the order is:
(50 + 8 - 1) choose (8 - 1) = 56 choose 7 = 56! / (7! 49!) = 56555453525150 / (7654321) = 807,794,400
This is the number of ways to place the order if they order at most 20 cases of milk.
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please just say the answer
The angles ∠1 and ∠2 are vertical angles. Vertical angles are congruent and according to the definition of congruency, ∠1 = ∠2.
The completed table used to prove that ∠1 ≅ ∠2 is presented as follows;
Statement [tex]{}[/tex] Reasons
1. m∠1 + m∠3 = 180° [tex]{}[/tex] 1. Linear pair angles are supplementary
m∠2 + m∠3 = 180°
2. m∠1 + m∠3 = m∠2 + m∠3 [tex]{}[/tex] 2. Substitution
3. m∠1 = m∠2[tex]{}[/tex] 3. Subtraction property
4. ∠1 ≅ ∠2 [tex]{}[/tex] 4. Definition of congruency
What are congruent angles?Two angles are congruent if the measures of the each angle are equivalent.
The details of the reasons used to prove that ∠1 = ∠2, are as follows;
Linear pair angles
Linear pair angles are adjacent angles that have a common side and for which the other non common sides of the two angles, together form a straight line.
Substitution property
The substitution property of equality states that if a = b then b can replace a in any expression or equation, with the value of the new expression remaining the same as the previous expression or both sides of the new equations remain equivalent.
Subtraction property
The subtraction property of equality states that the subtraction of the same amount from two quantities that have the same amount, will result in two new quantities that remain equivalent.
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Write the precise definition of the following: {xn} n n0 is a sequence of reals. Give an example of a sequence of rational numbers.
The example of a sequence of rational numbers is 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, ….
The term rational number in math is defined as a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero.
Here we have given that {xₙ} n > 0 is a sequence of reals.
Here we need to find the definition of sequence and the example for the sequence of rational numbers.
As per the given definition we all know that the rational number is the number that can be expressed as the quotient or fraction
Here first we have to identify the definition of sequence that is none other than a group of numbers in an ordered way following a pattern and it is also known as Series which means that it is a sum of the elements of the sequence.
And the example for sequence of rational number is written as 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, ….
Here we have given that the value must be greater than 0, so we have taken only the positive values.
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Divide if possible. 36,000 divided 90 (Type whole numbers)
A. The quotient is with no remainder ( type a whole number )
B. The quotient is with remainder .
C. The quotient is undefined
Please help I only got 20 minutes left for class
Answer:
D is the answer..
Step-by-step explanation:
Brice is taking a bike tour of the city. The entire tour is 24 7/10 kilometers long. Brice knows that his average biking rate is 4.3 meters per second. A. What is Brice's biking rate in kilometers per hour? B. About how long will it take Brice to complete the biking tour? Express your answer in hours and minutes. C. Explain how you know your answer is reasonable. Please show all work.
The speed in kilometers per hour is 15.48. And the total number of hours for the trip will be 1.60 hours.
What is an average speed?The distance covered by the particle or the body in an hour is called the average speed. It is a scalar quantity. It is the ratio of the total distance to the total time.
We know that the average speed formula is given as,
Average speed = Total distance / Total time
Brice is taking a bike tour of the city. The entire tour is 24 ⁷/₁₀ kilometers long. Brice knows that his average biking rate is 4.3 meters per second.
Convert the mixed fraction number to a decimal number. Then we have
24 ⁷/₁₀ = 24 + 0.70
24 ⁷/₁₀ = 24.70 kilometers
Convert the speed into kilometers per hour. Then we have
Speed = 4.3 x 18/5
Speed = 15.48 kilometers per hour
Then the total number of hours for the trip is given as,
15.48 = 24.70 / T
T = 24.70 / 15.48
T = 1.60 hours
The speed in kilometers per hour is 15.48. And the total number of hours for the trip will be 1.60 hours.
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Lena is on a two week bicycle trip. After 5 days she had ridden 212 miles. Express Lena’s rate as a unit rate.
The unit rate of Lena's bicycle trip if she travelled 212 miles in 5 days is 42.4 miles per day
What is Lena's unit rate?Unit rate refers to the rate of traveling in miles per day.
Number of miles Lena has ridden= 212 miles
Number of days = 5 days
Lena’s rate as a unit rate = Number of miles Lena has ridden / Number of days
= 212 miles / 5 days
= 42.4 miles per day
Hence, Lena is traveling at a unit rate of 42.4 miles per day.
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What is the limit of the function?
Select True or False for each limit statement
Answer:
The answers are
False
True
True
Thank you
A 12-foot ladder is leaned against the side of a building. If the bottom of the ladder
rests 3 feet from the wall, determine the height at which the top of the ladder rests
on the wall. Let h represent the height. Show all the steps. Don’t forget the units!!!
Draw the diagram and label it. Solve:
Therefore statement:
Answer:
11.6 ft
Step-by-step explanation:
|\
| \
| \
| \
h | \ 12 ft
| \
| \
| \
| \
------------------------
3 ft
Use the Pythagorean theorem: a² + b² = c²
(3 ft)² + h² = (12 ft)²
9 ft² + h² = 144 ft²
h² = 135 ft²
h = √(135 ft²)
h = 11.6 ft
A residential property is assessed for tax purposes at 41% of its market value. The residential property tax rate is 3 3/5% of the assessed value and the tax is 1649$.
(a) What is the assessed value of the property?
(b) What is the market value of the property?
a) The assessed value of the residential property is, proportionately, $45,806.
b) Proportionately, the market value of the property is $111,722.
What is proportion?Proportion refers to two ratios equated to each other.
Proportion is a fractional value, expressed as the relative size of a variable or number to another.
We represent proportions as decimals, fractions, or percentages.
The assessed value of a residential property = 41% of the market value
Residential property tax rate = 3³/₅% or 3.6% of the assessed value
Tax = $1,649
Assessed value = $45,806 ($1,649/3.6%)
Market value = $111,722 ($45,806/41%)
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I need help i dont get it there is more on the second one but I coundent get it on the thing
1) Note that the result of the above highlighted 10 by 10 block of units in the fraction are:
10/100; and
1/ 10.
2) In decimal, that would be:
0.1
3) In part-to-part ratio it would be:
10:90
4) In Part to whole ratio 10/100
5) In whole to part, that would be:
100:10
5) In percent, that would be 10%
A part-to-whole ratio compares a part to the total of all parts. For example, if you have 100 apples and you want to know what percentage of them are red, you would express the number of red apples as a ratio of the total number of apples. In this case, 10 red apples would be expressed as a ratio of 10:100
A part-to-part ratio compares two or more quantities in terms of their relative sizes. For example, if you have 100 apples and you want to know how many more red apples than green apples you have, you would express the number of red apples and the number of green apples as a ratio of each other. In this case, if you have 30 red apples and 20 green apples, the ratio would be 30:20
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answer the following
The slope of the line is -20
How to find the slope of the graph of a line?The slope of a line is a measure of how steep the line is. It is defined as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) between two points on the line. The formula for the slope of a line is:
m = (y2 - y1) / (x2 - x1)
Where (x1, y1) and (x2, y2) are the coordinates of two distinct points on the line.
From the graph, you can pick any two points. That is:
x1 = 0, y1 = 120 and x2 = 4, y2 = 40
m = (y2 - y1) / (x2 - x1)
m = (40- 120) / (4 - 0)
m = -80/4
m = -20
Thus, the slope is -20
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PLEASE HURRY!!!! FOR 30 POINTS!!
Mr. Snow made snacks in separate gift bags. He gave one each to the 536 sixth-grade students going on the field trip. Then he gave 30 bags to the students in Mrs. Bond’s class for decorating the trophy cases. Next he gave 25 to the students in Mrs. Fuller’s class for helping during the school play. Mr. Snow had 3 snacks left. How many snacks did he make?
A . 600 snacks
B. 596 snacks
C. 594 snacks
D. 584 snacks
55% of the fish in a pond are goldfish,
10/3 of the fish are perch and the rest of the fish are sticklebacks.
Write the ratio of goldfish to perch to sticklebacks in this pond in its simplest form.
Answer:
If 55% of the fish in the pond are goldfish, that would be 55/100 of the total number of fish. To express this as a fraction in its simplest form, we would divide the numerator and denominator by the greatest common divisor, 5. So, 55/100 simplifies to 11/20.
10/3 of the fish in the pond are perch, which in its simplest form is already in the form of 10/3.
The rest of the fish are sticklebacks, meaning the total number of goldfish, perch, and sticklebacks must add up to 100%. Therefore, to find the ratio of sticklebacks to goldfish and perch, we can subtract the percentage of goldfish and perch from 100%.
100 - 55 - (10/3) = 100 - 55 - 10/3 = 100 - 55 - (10/3) = 100 - 55 - (10*3/3) = 100 - 55 - 10 = 35.
So the ratio of goldfish to perch to sticklebacks in this pond is 11/20: 10/3: 35/100 in its simplest form.
Step-by-step explanation:
What is the range of the following function? y = 1/(x + 2) - 1
The range of the rational function:
y = 1/(x + 2) - 1
is R: (-∞, 0) U (0, ∞)
What is the range of the given function?Remember that for a function y = f(x), we define the range as the set of the possible inputs of the function (the possible values of y).
In this case we have the rational function:
y = 1/(x + 2) - 1
Notice that when x approaches at -2, the function tends to positive or negative infinity (deppending if it comes from the left or right).
And then as x goes affar from -2, the absolute value decreases.
But the function never reaches the value 0, as tends to infinity or negative infintiy the functio approaches zero, but it will never be equal to zero, then the range is:
R: (-∞, 0) U (0, ∞)
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the lengtjs of the three sides of a traigle nor necessarlity a right trianglw are 2.92 m, 7.67 m, 8.82 m what is the cosine od the angle opposite the sidw of 8.82 m
The cosine of the angle opposite the side of length 5.64 meters is 0.4536.
In a triangle, the cosine of the angle opposite a side is given by the formula:
cos(angle) = (a^2 + b^2 - c^2)/(2ab) where a, b, and c are the lengths of the sides of the triangle and angle is the angle opposite the side of length c. Given the side lengths of the triangle as 3.18 meters, 7.82 meters and 5.64 meters, the side opposite angle we want to find is 5.64 meters.
so we can use the above formula and substitute the values,
cos(angle) = (3.18^2 + 7.82^2 - 5.64^2)/(23.187.82) = 0.4536
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The graph of a cubic polynomial function
has horizontal tangent lines at (1, −2) and (−1, 2). Find
an equation for the function and sketch its graph.
Answer: A cubic polynomial function has the form f(x) = ax^3 + bx^2 + cx + d.
A horizontal tangent line is a line that is parallel to the x-axis and it occurs at a critical point of the function, where the derivative of the function is 0.
We know that a cubic function has 3 critical points and thus 3 derivative that could be 0, so to find the equation of the cubic function we will use these two points to find the value of a, b, c, and d.
We know that the critical points of a cubic function are found by solving f'(x) = 0, where f'(x) is the derivative of the function, so we have:
f'(x) = 3ax^2 + 2bx + c = 0
Now we can use the two points to find the value of a, b, c, and d.
We know that the point (1,-2) is on the graph of the function, so we can substitute it into the equation to find d:
f(1) = a + b + c + d = -2
And we know that the point (-1,2) is on the graph of the function, so we can substitute it into the equation to find d:
f(-1) = -a + b - c + d = 2
Now we can use these two equations to find the value of a, b, and c:
a + b + c + d = -2
-a + b - c + d = 2
2a + 2d = 0
a = -d
-a + b - c + d = 2
b - c = 4
3ax^2 + 2bx + c = 0
3(-d)x^2 + 2bx - c = 0
Now we can use these equations and the critical points to find the value of a, b, c, and d
3(-d)x^2 + 2bx - c = 0, (1,-2) => 3d+2b-c = 0
3(-d)x^2 + 2bx - c = 0, (-1,2) => 3d-2b+c = 0
b-c = 4
Solving these equations, we have:
a = -1/3
b = 1
c = -5/3
d = -2
And the equation for the function is:
f(x) = -1/3x^3 + x^2 - 5/3x - 2
We can now sketch the graph of the function by using the information that we have. The cubic function will have an vertex of symmetry at (0,-2) and two horizontal tangent lines at (1,-2) and (-1,2) .
Since a < 0, the function is concave down, so the graph will be opened downwards and will have a local minimum at (0,-2) .
It is important to note that this is one possible equation for a cubic function that has horizontal tangent lines at (1, −2) and (−1, 2), but there may be other possibilities.
Step-by-step explanation:
Find the diameter of a circle whose circumference is 62.8 feet. Use 3.14 for pie.
a. 18
b. 22
c. 20
d. 21
The diameter of the circle whose circumference is 62.8 feet is 20 ft, thus the correct option is C.
How to find the diameter of the circle?Remember that for a circle of diameter D, the circumference is given by the formula:
C = pi*D
Where pi = 3.14
Here we know that the circumference is 62.8 feet, then we can rewrite:
62.8 feet = 3.14*D
Now we can solve that equation for D, then we will get:
D = 62.8ft/3.14 = 20ft
The diameter is 20 ft, thus, the correct option is C.
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