The length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.
To determine the length of segment OP on the unit circle, we need to use trigonometry. Let's break down the problem step by step:
Definition: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.
Initial Ray: The initial ray is a line segment that starts from the origin (0, 0) and extends to a point on the unit circle. It forms an angle with the positive x-axis.
Rotation: We are rotating the initial ray counterclockwise by θ degrees. This means we are essentially finding a new point on the unit circle based on the angle θ.
Trigonometric Functions: The trigonometric functions sine (sin) and cosine (cos) are particularly useful for calculating the coordinates of points on the unit circle.
sin(θ) gives the y-coordinate of a point on the unit circle.
cos(θ) gives the x-coordinate of a point on the unit circle.
Coordinates of Point P: Since we are rotating the initial ray counterclockwise by θ degrees, the coordinates of point P on the unit circle can be obtained as follows:
x-coordinate of P: cos(θ)
y-coordinate of P: sin(θ)
Distance from the Origin (Length of Segment OP):
Using the coordinates of point P, we can calculate the distance between the origin (0, 0) and point P using the distance formula.
The distance formula states that for two points (x1, y1) and (x2, y2), the distance between them is given by:
d = √((x2 - x1)² + (y2 - y1)²)
In this case, point P has coordinates (cos(θ), sin(θ)), and the origin is (0, 0). Thus, the distance (length of segment OP) is:
d = √((cos(θ) - 0)² + (sin(θ) - 0)²)
= √(cos²(θ) + sin²(θ))
= √(1) [Using the trigonometric identity: sin²(θ) + cos²(θ) = 1]
= 1
Therefore, the length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.
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you take a loan out to finance $175,000 on a house. if the rate is 3% and compounds continuously, how much will the loan cost after 30 years?
Answer:approximately $396,849.46 after 30 years with continuous compounding.
Step-by-step explanation:
To calculate the cost of the loan after 30 years with continuous compounding, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the total amount after interest
P = the principal amount (loan amount)
e = Euler's number, approximately 2.71828
r = interest rate per period
t = time in years
Given:
Principal amount (loan amount), P = $175,000
Interest rate, r = 3% = 0.03 (as a decimal)
Time, t = 30 years
Using the formula, we can calculate the total amount (A):
A = $175,000 * e^(0.03 * 30)
Now, let's calculate the cost of the loan after 30 years:
A = $175,000 * 2.71828^(0.03 * 30)
Using a calculator or software, we find:
A ≈ $396,849.46
Therefore, the loan will cost approximately $396,849.46 after 30 years with continuous compounding.
Uncle Joe made chocolate chip cookies. Michael ate 50% right away. Robert ate 50% of what was left. fourteen cookies remain. How many cookies did Uncle Joe make?
Answer:
50% ate uncle joe and 50% ate robert so, uncle joe make 14 cookies
Answer:
1. Michael ate 50% of the cookies, so the remaining cookies are 50% of the total. Let's denote the total number of cookies as T. So, after Michael ate his share, 0.5 x T cookies remained.
2. Robert then ate 50% of what was left, leaving only 50% of the cookies that were left after Michael ate. So, after Robert ate his share, 0.5 x 0.5 x T = 0.25 x T cookies remained.
3. We know that 14 cookies remained after Robert ate his share.
So, we can set up the equation 0.25 x T = 14 and solve for T:
T = 14/0.25 = 56
So, Uncle Joe made 56 chocolate chip cookies.
Step-by-step explanation:
Hope it helps!!!
factor completely using distributive law 25y-15z
Answer:
5(5y - 3z)
Step-by-step explanation:
25y - 15z ← factor out common factor of 5 from each term
= 5(5y - 3z)
What is the average of the following numbers? 2, 5,8,1
Answer: The average is a descriptive statistic that provides an idea of the central or typical value in a dataset
Step-by-step explanation:
The average of the numbers is 4
The given parameters are;
The numbers; 2, 5, 8, and 1
The required parameter
To find the average;
Solution:
The average is given by the sum of the data divided by the size or count of the data points
Therefore, the average of the given numbers is given as follows;
Average, x= 2+5+8+1/4=4
The average of the numbers, = 4
Can someone please help me wit this?
Answer:
6.6
Step-by-step explanation:
I’ve done this before. The original answer I did was 6.63 but it says round to the nearest tenth so yep…
Show that sin⁴x-cos⁴x/ sin²x-cos²x = 1
Proved that, sin⁴x-cos⁴x/ sin²x-cos²x = 1.
Here, we have,
given that,
the LHS is:
sin⁴x-cos⁴x/ sin²x-cos²x
now, we know that,
sin²x+cos²x = 1
and, we know that,
a² - b² = (a+b) (a-b)
so, we get,
sin⁴x-cos⁴x/ sin²x-cos²x
= (sin²x+cos²x) (sin²x-cos²x ) / sin²x-cos²x
=(sin²x+cos²x)
=1
= RHS
Hence, Proved that, sin⁴x-cos⁴x/ sin²x-cos²x = 1
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Could someone explain how to solve this?
Answer:
Step-by-step explanation:
What’s the answer please
Answer:
40
Step-by-step explanation:
plugging in a and b will get us:
[tex]2^{2} + 6^{2} = c^{2}[/tex]
4 + 36 = [tex]c^{2}[/tex]
40 = [tex]c^{2}[/tex]
c= [tex]\sqrt{40\\[/tex]
What is the remainder when the following polynomial division is performed? Place the answer in the proper location of the gird. Do not include parentheses in your answer.
The remainder of the polynomial division [tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)}[/tex] is 2y²
How to determine the remainder of the polynomial divisionFrom the question, we have the following parameters that can be used in our computation:
[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)}[/tex]
When the numerator is expanded, we have
[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = \frac{(y - 1)(y^3 + 1) + 2y^2}{y^3 + 1}[/tex]
Split the expanded expression
[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = \frac{(y - 1)(y^3 + 1)}{y^3 + 1} + \frac{2y^2}{y^3 + 1}[/tex]
Evaluate
[tex]\frac{\left(y^4-y^3+2y^2+y-1\right)}{\left(y^3+1\right)} = y - 1 + \frac{2y^2}{y^3 + 1}[/tex]
From the above, we have
Quotient = y - 1
Remainder = 2y²
Hence, the remainder of the polynomial division is 2y²
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PLEASE HELP ASAP
The diagrams show a polygon and the image of the polygon after a transformation.
Use the diagrams to determine which statements are true. Select all statements that are true.
The correct statements regarding the transformations are given as follows:
Parallel lines will always be parallel after a reflection.Lines that are not parallel will never be parallel after a translation.What are transformations on the graph of a function?Examples of transformations are given as follows:
A translation is defined as lateral or vertical movements.A reflection is either over one of the axis on the graph or over a line.A rotation is over a degree measure, either clockwise or counterclockwise.For a dilation, the coordinates of the vertices of the original figure are multiplied by the scale factor, which can either enlarge or reduce the figure.Parallel lines are lines that have the same slope, that is, lines that do not intercept, and the transformations do not change whether the lines are parallel or not.
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The figure below is a square. Find the length of side x in simplest radical form with
rational denominator.
Answer: [tex]x=2\sqrt{5}[/tex]
Step-by-step explanation:
Detailed explanation is attached below.
Factorise completely.
1.1) 2x + 4y - 6z
1.2) 10p6q²-4p³q2 + 2p*q*
1.3) (m+n)²-5p(m + n)
1.4) 4(7c-d)+a(d-7c)
The completely factorized forms are:
1.1) 2x + 4y - 6z (no further factorization)
[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]
1.3) (m + n)(m + n - 5p)
1.4) 7c(4 - a) + d(-4 + a)
We have,
Let's factorize each expression completely:
1.1)
2x + 4y - 6z
There are no common factors among the terms, so we can't factorize it further.
1.2)
[tex]10p^6q^2 - 4p^3q^2 + 2pq[/tex]
The common factor among the terms is 2pq, so we can factor it out:
[tex]2pq (5p^5q - 2p^2 + 1)[/tex]
1.3)
(m + n)² - 5p(m + n)
Using the distributive property, we can expand the squared term:
(m + n)(m + n) - 5p(m + n)
Now we can factor out the common factor (m + n):
(m + n)(m + n - 5p)
1.4)
4(7c - d) + a(d - 7c)
Using the distributive property, we can expand the terms:
28c - 4d + ad - 7ac
Rearranging the terms:
(28c - 7ac) + (-4d + ad)
Factoring out common factors:
7c(4 - a) + d(-4 + a)
Thus,
The completely factorized forms are:
1.1) 2x + 4y - 6z (no further factorization)
[tex]1.2) 2pq(5p^5q - 2p^2 + 1)[/tex]
1.3) (m + n)(m + n - 5p)
1.4) 7c(4 - a) + d(-4 + a)
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if f(x)=x+2/x^2-9 and g(x)=11/x^2+3x
A. find f(x)+g(x)
B. list all of the excluded values
C. classify each type of discontinuty
To receive credit, this must be done by Algebraic methods, not graphing
The types of discontinuities are: removable discontinuity at x = -3 and vertical asymptotes at x = 0 and x = 3.
A. To find f(x) + g(x), we add the two functions together:
f(x) + g(x) = (x + 2)/(x^2 - 9) + 11/(x^2 + 3x)
To add these fractions, we need a common denominator. The common denominator in this case is (x^2 - 9)(x^2 + 3x). So, we rewrite the fractions with the common denominator:
f(x) + g(x) = [(x + 2)(x^2 + 3x) + 11(x^2 - 9)] / [(x^2 - 9)(x^2 + 3x)]
Simplifying the numerator:
f(x) + g(x) = (x^3 + 3x^2 + 2x^2 + 6x + 11x^2 - 99) / [(x^2 - 9)(x^2 + 3x)]
Combining like terms:
f(x) + g(x) = (x^3 + 16x^2 + 6x - 99) / [(x^2 - 9)(x^2 + 3x)]
B. To find the excluded values, we look for values of x that would make the denominators zero, as division by zero is undefined. In this case, the excluded values occur when:
(x^2 - 9) = 0 --> x = -3, 3
(x^2 + 3x) = 0 --> x = 0, -3
So, the excluded values are x = -3, 0, and 3.
C. To classify each type of discontinuity, we examine the excluded values and the behavior of the function around these points.
At x = -3, we have a removable discontinuity or hole since the denominator approaches zero but the numerator doesn't. The function can be simplified and defined at this point.
At x = 0 and x = 3, we have vertical asymptotes. The function approaches positive or negative infinity as x approaches these points, indicating a vertical asymptote.
Therefore, the types of discontinuities are: removable discontinuity at x = -3 and vertical asymptotes at x = 0 and x = 3.
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2. In a complete paragraph, pick a scenario where concepts from this course would be used - it could be in you
or working in a business, etc.
Choose at least 2-3 concepts to include, explain your scenario, how these concepts apply, and provide a wa
Use the following format:
Topic Sentence: 1 concise sentence describing a scenario where concepts from this course could be used.
Supporting Detail: 1-2 sentences explaining how 1 concept from the class can be applied to the scenario.
Worked Example: Show a worked example for the concept described above.
Supporting Detail: 1-2 sentences explaining how 1 concept from the class can be applied to the scenario.
Worked Example: Show a worked example for the concept described above.
Conclusion: 1-2 sentences describing how applying the concepts in this course to a real-life situation helps
Write your response using your own words and do not use any other sources outside of this course
Topic Sentence: In a business setting, concepts from this course can be applied when optimizing customer service operations and improving customer satisfaction.
The Supporting DetailsSupporting Detail: One concept from the course that can be applied is the use of queuing theory to manage customer wait times and reduce service bottlenecks. By analyzing arrival rates, service rates, and queue lengths, businesses can optimize staffing levels and allocate resources efficiently.
Worked Example: One practical application of queuing theory is in call centers. With the aid of this theory, the center can analyze the number of customer service reps required in different periods of the day. The objective is to keep waiting times at their lowest and deliver timely service to customers.
Supporting Detail: Another concept that can be applied is the concept of customer lifetime value (CLV). By understanding CLV, businesses can identify their most valuable customers, prioritize their needs, and tailor personalized marketing strategies to enhance customer retention.
Worked Example: An e-commerce business may use information on customer buying patterns, typical order amounts, and the rate of customer loss to determine the overall lifetime value of a customer. By utilizing this data, the organization has the opportunity to execute tailored loyalty initiatives and customized suggestions, ultimately enhancing customer contentment and optimizing overall revenue for the long haul.
Conclusion:
The application of course concepts in a business environment can result in better customer service, shorter wait times, higher customer approval, and increased customer allegiance, thereby fueling the expansion and profitability of a business.
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a) What fraction of an hour is 40 minutes, in its simplest b) I sleep 8 hours a night. What fraction of a day is this? c) How many hours is of a day? How many hours is 1 5 pond of her n
a) The value of fraction is,
⇒ 2 / 3
b) The value of fraction is,
⇒ 1 / 3
c) There are 24 hours in a day.
Now, We can simplify all the fraction as,
a) To find fraction of an hour is 40 minutes,
Since, 1 hour = 60 minutes
Hence,
⇒ 40 / 60
⇒ 4 / 6
⇒ 2 / 3
b) Since, 1 day = 24 hours
Hence, we get;
⇒ 8 / 24
⇒ 1 / 3
c) We know that,
⇒ 1 day = 24 hours
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enter the number that belongs in the green box
The value of the unknown side to the nearest hundredth is 6.78
What is cosine rule?Cosine rule states that; if a, b,c are the sides of a triangle and A,B,C are the opposite angles of the sides, then
c² = a² + b² -2ab CosC
Cosine rule is used when all the sides or two of the sides are given. The angle opposite to side c is angle C.
c² = 4² + 10² - 2(4)(10)cos 29
c² = 16 +100 - 80cos29
c² = 116 -69.97
c² = 46.03
c = √ 46.03
c = 6.78 ( nearest hundredth).
Therefore the value of c is 6.78
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A solid machine part is to be manufactured as shown in the figure. The part is made by cutting a small cone off the top of a larger cone. The small cone has a base radius of 3 inches and a height of 5 inches. The larger cone has a base radius of 5 inches and had a height of 12 inches prior to being cut. What is the volume of the resulting part illustrated in the figure? A. 60 cubic inches B. 65 cubic inches C. 85 cubic inches D. 90 cubic inches
Given statement solution is :- The volume of the resulting part is approximately 267 cubic inches.
To find the volume of the resulting part, we need to calculate the volume of the larger cone and subtract the volume of the smaller cone.
The formula for the volume of a cone is given by:
V = (1/3) * π * [tex]r^2[/tex] * h
where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the cone's base, and h is the height of the cone.
For the larger cone:
Radius (r) = 5 inches
Height (h) = 12 inches
V_large_cone = (1/3) * π * [tex](5^2)[/tex] * 12
V_large_cone = (1/3) * π * 25 * 12
V_large_cone = (1/3) * π * 300
V_large_cone = 100π
For the smaller cone:
Radius (r) = 3 inches
Height (h) = 5 inches
V_small_cone = (1/3) * π * [tex](3^2)[/tex] * 5
V_small_cone = (1/3) * π * 9 * 5
V_small_cone = (1/3) * π * 45
V_small_cone = 15π
Therefore, the volume of the resulting part is:
V_resulting_part = V_large_cone - V_small_cone
V_resulting_part = 100π - 15π
V_resulting_part = 85π
Now, we can approximate the value of π as 3.14159:
V_resulting_part ≈ 85 * 3.14159
V_resulting_part ≈ 267.10715
Rounded to the nearest whole number, the volume of the resulting part is approximately 267 cubic inches.
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33. It's 2 years since I was last in Rome. ⇒ I haven't 34. I saw Tom last on his wedding day. ⇒ I haven't 35. I last ate raw fish when I was in Japan. ⇒ I haven't 36. It's years since Mary last spoke French. ⇒ Mary hasn't 37. It's ten weeks since since I last had a good night sleep. I haven't 38. He last paid taxes in 1970. He hasn't last taxes 39. I last ate meat 5 years ago. ⇒ I haven't since 1970 40. It's 3 months since since the windows were cleaned. The windows haven't 41. It's years since I took photographs.
34. I haven't seen Tom since his wedding day.
36. Mary hasn't spoken French in years.
37. I haven't had a good night's sleep in ten weeks.
38. He hasn't paid taxes since 1970.
40. The windows haven't been cleaned for 3 months.
41. It's been years since I took photographs.
What is a perfect tense?The perfect tense is a grammatical construction used to express actions that are completed or have happened before the present moment. It indicates that an action was performed and finished in the past but has relevance or impact on the present.
There are three main forms of the perfect tense: present perfect, past perfect, and future perfect.
1. Present Perfect Tense: This tense is used to describe an action or event that started in the past and has a connection to the present. It emphasizes the result or completion of the action.
2. Past Perfect Tense: This tense is used to describe an action or event that occurred before another action or a specific point in the past. It expresses an action completed before a specified time in the past.
3. Future Perfect Tense: This tense is used to describe an action that will be completed before a specified time or point in the future.
Each form of the perfect tense can be used with different time references to indicate the relationship between the completed action and the present or other points in time.
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How tall is the tree?
5ft
35°
-20ft-
[?]ft
Round to the nearest foot.
Answer:
19
Step-by-step explanation:
tan35=x/20
20tan35=x
x=14.0041507641942
x+5=19.0041507641942
The nearest foot is 19
Select the correct answer.
If u =(1+i√3) and v=(1-i√3), what is uv?
Ο Α. 1
OB. -4
OC. 0
OD. 4
Reset
Next
If u =(1+i√3) and v=(1-i√3), product uv is: D. 4.
What is product uv?To find the product of u and v let us simply multiply them together:
u = 1 + i√3
v = 1 - i√3
uv = (1 + i√3)(1 - i√3)
Using the difference of squares formula (a² - b² = (a + b)(a - b)) we can simplify the expression:
uv = (1 + i√3)(1 - i√3)
uv= 1² - (i√3)²
uv= 1 - (-3)
uv= 1 + 3
uv= 4
Therefore the product uv is equal to 4.
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which statement is true of this function
The correct statement is:
C. The function has a y-intercept at (0, -2).
Let's analyze each statement:
A. As the value of x increases, the value of f(x) moves toward a constant.
This statement is false.
The function f(x) = (1/5)ˣ - 2 is an exponential function with a base of 1/5. As x increases, the exponential term (1/5)ˣ becomes smaller and approaches zero, causing f(x) to approach -2.
However, it does not approach a constant value.
B. The domain of the function is (-2, ∞).
This statement is false.
The function f(x) = (1/5)ˣ - 2 is defined for all real numbers.
There are no restrictions on the domain, so the correct statement is that the domain is (-∞, ∞).
C. The function has a y-intercept at (0, -2).
This statement is true.
To find the y-intercept, we set x = 0 in the function and solve for f(x):
f(0) = (1/5)⁰ - 2
= 1 - 2
= -1
Therefore, the y-intercept of the function is (0, -2).
D. The function is increasing.
This statement is true.
An exponential function with a positive base, such as (1/5)ˣ, is always decreasing.
However, when we subtract 2 from the exponential term, it shifts the graph vertically downward by 2 units.
This transformation makes the function f(x) = (1/5)ˣ - 2 increasing.
So, the correct statement is:
C. The function has a y-intercept at (0, -2).
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You want to obtain a sample to estimate a population proportion. Based on previous evidence, you believe the population proportion is approximately 28%. You would like to be 98% confident that your estimate is within 1.5% of the true population proportion. How large of a sample size is required?
Please round your z* value to 3 decimal places to use in your calculation.
n =
The required sample size to estimate the population proportion with 98% confidence and a margin of error of 1.5% is approximately 4,847.60.
How to calculate the sample size required to estimate a population proportion with a certain level of confidence?To find the sample size required to estimate a population proportion with a 98% level of confidence, we will use the formula:
n = (z² * p * q) / E²
where:
n = sample size
z = z-value associated with the desired confidence level
p = population proportion - estimated
q = 1 - p (complement of the estimated proportion)
E = desired margin of error
We, first of all, will calculate the required sample size:
Given:
z = z-value corresponding to a 98% confidence level
E = 1.5% = 0.015
p = 28% = 0.28
q = 1 - p = 1 - 0.28 = 0.72
Then, we use a standard normal distribution table or a calculator to find the z-value. For a 98% confidence level, the z-value is approximately 2.326.
Putting the values into the formula:
n = (2.326² * 0.28 * 0.72) / 0.015²
n = (5.410 * 0.2016) / 0.000225
n≈ 1.091/0.000225
n ≈ 4,847.60
Hence, to estimate the population proportion with a 98% confidence level and a margin of error of 1.5%, a sample size of approx. 4,848.60 is required.
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Find Tan A and Tan B. write each answer as a fraction and as a decimal rounded into four places.
TanA value is 22/3 and TanB is 3/2 from the given triangle ABC.
We have to find the values of Tan A and TanB.
We know that tan function is a ratio of opposite side and adjacent side.
To find TanA, we have to take opposite side of vertex A has opposite side.
TanA=18/27
TanA=2/3
Now let us find TanB , which is 27/18
TanB =3/2
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find the length represented by x for each pair of similar triangles 12in x 20in 15in 40in 25in
The length x in the similar triangles is given as follows:
x = 15 cm.
What are similar triangles?Two triangles are defined as similar triangles when they share these two features listed as follows:
Congruent angle measures, as both triangles have the same angle measures.Proportional side lengths, which helps us find the missing side lengths.The proportional relationship for the side lengths in this triangle is given as follows:
x/25 = 9/15 = 18/30
Hence the value of x is obtained as follows:
x/25 = 9/15
15x = 25 x 9
x = 25 x 9/15
x = 15 cm.
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Please help! I need to get this done by midnight
The number of students taking all three languages would be 1 student. The number of students taking none of the languages are 32.
How to find the number of students ?The total number of students taking each language is given so the number of people taking these languages are:
15 + 14 - X + X = 30
X = 1
So, there is 1 student taking all three languages.
For the students taking none of these languages:
Total students - students taking at least one language = students taking none of these languages.
= 100 (total students) - (53 ( taking only one language ) + 14 ( taking Arabic and Bulgarian ) + X (taking all three languages))
= 100 - 53 - 14 - 1
= 32 students
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What is the temperature in °C? Hint: (°F -32) ÷ 1.8 °C or °F= 1.8 °C
+32
Converting the temperature from Celsius to Fahrenheit:
Subtract the given temperature by 32 and divide it by 1.8°C.
°C = (F - 32) / 1.8°C, or use the equation °F= 1.8 °C+32
(°F-32) / 1.8°C
a) The given temperature is 60°F
substitute the given values in the above equation:
(60 - 32) / 1.8
28 / 1.8
= 15.5555 approx 16°C
60°F = 16°C
b) The given temperature is 10°F
°C = (F - 32) / 1.8°C
substitute the given values in the above equation:
(10 -32) / 1.8
-22 / 1.8
-12.222 approx -12°C
10°F = -12°C
correct question:
What is the temperature in °C, if the temperature is
a) 60F
b) 10F
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A dart is dropped above the target shown in the diagram. The dart has an equal chance of landing on any spot on the target.
What is the probability the dart will land in the shaded square on the target? Round to the nearest hundredth. Enter the answer in the box.
Answer: 0.04
Step-by-step explanation:
The area [tex]A_1[/tex] of the square target with side length [tex]s_1=20 cm[/tex] is:
[tex]A_1=s_1^{2}=20^{2}=400[/tex]
The area [tex]A_2[/tex] of the shaded square with side length [tex]s_2=4cm[/tex] is:
[tex]A_2=s_2^{2}=4^{2}=16[/tex]
So, the probability that the dart will land in the shaded square on the target is:
[tex]\frac{A_{2}}{A_{1}}=\frac{16}{400}=0.04[/tex]
A shelf has different sizes of bottles of laundry detergent this line plot shows the number of cups of laundry detergent in each bottle Natalie buys a bottle of the size of laundry detergent that has 4 bottles on the shelf she uses 1/8 cup for each load of laundry she does how many loads of laundry can Natalie do with her bottle of laundry detergent
Using the division operation , Natalie can do 64 loads of laundry with her bottle of detergent.
From the line plot , the detergent with 4 bottles has 8 cups in it.
Fraction of cup used per laundry = 1/8
Total cups of detergent in bottle = 8
Number of laundry Natalie can do :
(Total cups of detergent in bottle / Fraction of cup used per laundry )
Number of loads of laundry that can be done :
= 8 / (1/8)
= 8 × 8/1
= 64
Therefore, Natalie can do 64 loads of laundry with her bottle of detergent.
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The graph shows a function of the form () = ab
.
Use the drop-down menus to complete the statements about the function, and then write an equation that represents this function.
Answer:
When [tex]x=0[/tex], the value of f(x) is 1.
Each time x increases by 1, f(x) is multiplied by 4.
Equation of function: [tex]f(x)=1\cdot 4^x[/tex]
Step-by-step explanation:
The detailed explanation is attached below.
6a) Suppose A = A1i A2j + A3k and B = B1i+B2j+B3k. Prove that A ∙ B = A1B1 + A2B2 + A3B3.
6b) Find the angle between the vectors A = 2i+2j-k and B = 7i+24k.
a. The scalar product of vectors A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. is A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.
b. The angle between the vectors A = 2i +2j - k and B = 7i + 24k is 97.67°
What is the scalar product of two vectors?The scalar product of two vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k is given by
a.b = abcosФ = a₁b₁ + a₂b₂ + a₃b₃ where
a = magnitude of vector ab = magnitude of vector b and Ф = angle between vectors a and b6a) Suppose A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. Prove that A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.
We proceed as follows
We know that
A ∙ B = (A₁i + A₂j + A₃k). (B₁i + B₂j + B₃k)
= A₁i.(B₁i + B₂j + B₃k) + A₂j.(B₁i + B₂j + B₃k) + A₃k.(B₁i + B₂j + B₃k)
Expanding the brackets, we have
= A₁i.B₁i + A₁i.B₂j + A₁i.B₃k + A₂j.B₁i + A₂j.B₂j + A₂j.B₃k + A₃k.B₁i + A₃k.B₂j + A₃k.B₃k
= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k
We know that
i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0So, we have that
= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k
= A₁B₁(1) + A₁B₂(0) + A₁B₃(0) + A₂B₁(0) + A₂B₂(1) + A₂B₃(0) + A₃B₁(0) + A₃B₂(0) + A₃B₃(1)
= A₁B₁ + 0 + 0 + 0 + A₂B₂ + 0 + 0 + 0 + A₃B₃
= A₁B₁ + A₂B₂ + A₃B₃
So, A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.
6b) To find the angle between the vectors A = 2i +2j - k and B = 7i + 24k, we proceed as follows.
We know that the angle between two vectors is given by
Ф = cos⁻¹[(A.B)/AB] where
A = magnitude of vector A and B = magnitude of vector BNow, A.B = (2i +2j - k).(7i + 24k) = (2 × 7 + 2 × 0 + (-1) × 24)
= (14 + 0 - 24)
= -10
Now, the magnitude of a vector C = C₁i + C₂j + C₃k is C = √(C₁² + C₂² + C₃²)
Since vector A = 2i +2j - k its magniude A = √(2² + 2² + (-1)²)
= √(4 + 4 + 1)
= √9
= 3
Also since vector B = 7i + 24k, its magniude A = √(7² + 0² + 24²)
= √(49 + 0 + 576)
= √625
= 25
So, substituting the values of the variables into the equation, we have that
Ф = cos⁻¹[(A.B)/AB]
Ф = cos⁻¹[(-10)/(3 × 25)]
Ф = cos⁻¹[(-2)/(3 × 5)]
Ф = cos⁻¹[-2/15]
Ф = -0.13333
Ф = 97.67°
So, the angle betwen the vectors is 97.67°
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