Using partial derivatives, we know that in order to optimize profit, the company should produce 7,000 type A solar panels and 8,000 type B solar panels annually.
What are partial derivatives?A function of two or more variables can yield partial derivatives.
Writing the derivative with regard to x while treating all other variables as constants allows one to determine the partial derivative of a function with respect to one variable, let's say x.
The partial derivatives for each of the other
So, for a business making two different types of solar panels, the revenue and cost functions are provided by:
[tex]\begin{aligned}&R(x, y)=4 x+2 y \\&C(x, y)=x^2-4 x y+9 y^2+22 x-114 y-5\end{aligned}[/tex]
If revenue and cost are stated in millions of dollars annually, x represents the thousands of type A solar panels produced and sold annually, and y represents the thousands of type B solar panels produced and sold annually.
By deducting costs from revenues, the profit function is obtained as follows:
[tex]\begin{aligned}&P(x, y)=R(x, y)-C(x, y) \\&P(x, y)=4 x+2 y-\left(x^2-4 x y+9 y^2+22 x-114 y-5\right) \\&P(x, y)=-x^2-9 y^2-18 x+116 y+4 x y+5\end{aligned}[/tex]
The profit function's partial derivatives with regard to x and y are as follows:
[tex]\begin{aligned}&\frac{\partial P}{\partial x}=-2 x+4 y-18 \\&\frac{\partial P}{\partial y}=4 x-18 y+116\end{aligned}[/tex]
When both of these partial derivatives are equal to zero, the profit is maximized:
[tex]\begin{aligned}&0=-2 x+4 y-18 \\&0=4 x-18 y+116\end{aligned}[/tex]
The first equation above is multiplied by two, and the two equations are then added to yield:
[tex]\begin{aligned}&0=-10 y+80 \\&y=8\end{aligned}[/tex]
Inputting into the first equation results in:
[tex]\begin{aligned}&0=-2 x+4(8)-18 \\&2 x=14 \\&x=7\end{aligned}[/tex]
Therefore, using partial derivatives, we know that in order to optimize profit, the company should produce 7,000 type A solar panels and 8,000 type B solar panels annually.
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explain how the is related to the standard error of the estimated difference in average birth weight for smoking and nonsmoking mothers.
Babies born to nonsmokers typically weighed between 1,000 and 5,000 grams. Babies of smokers typically ranged in weight from 500 to 4500 grams.
Given,
What is the weight at birth?
Birth weight is the term used to describe a newborn's body weight. Typically, newborns of European heritage weigh between 2.5 and 4.5 kilos, on average 3.5 kilograms, at delivery. Babies from South Asia and China typically weigh 3.26 kg.
Here,
Babies of nonsmokers frequently ranged in weight from 1,000 to 5,000 grams. Babies born to smokers frequently weighed between 500 and 4500 grams.
The usual ranges of birth weights between the two groups do show some notable differences, though. Nearly half of neonates among nonsmokers (56 out of 115, 48.7%) have birth weights between 3,000 and 4,000 grams since there are fewer babies in the lower weight ranges. Fewer babies are in the bigger weight groups, with 40 of 74, or 54%, of the newborns delivered to smokers having birth weights between 2,000 and 3,000 grams.
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at a certain university in the u.s., all phone numbers are 7-digits long and start with either 824 or 825. 1) how many different phone numbers are possible?
There are 20,000 different phone numbers possible that is 7-digits long and start with either 824 or 825.
Fundamental Principle of Counting, also called the counting rule, is the method to count how many ways multiple independent events can occur.
Based on the given information, all phone numbers are 7-digits long and starts with either 824 or 825. Hence, the four last digits will be the combination of numbers from 0 to 9.
Thus, by Fundamental Principle of Counting, the number of possible outcomes is 20,000.
= (number of phone numbers starting with 824) + (number of phone numbers starting with 825)
= (10 x 10 x 10 x 10) + (10 x 10 x 10 x 10)
= 20,000
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14 on a certain sight-seeing tour, the ratio of the number of women to the number of children was 5 to 2. what was the number of men on the sight-seeing tour?
The number of men on the sight-seeing tour was 22.
What is the proportion?
A unique class of ratio called a percentage has the numerator and denominator combined. An illustration is the ratio of deaths to men, which is calculated by dividing deaths to men by deaths to men and women (i.e. the total population).
We have,
w/c = 5/2
w/c = 5/2
w = 5x
and c=2x
for some integer x. Q : m =?
(1) on the sight-seeing tour, the ratio of the number of children to the number of men was 5 to 11
c/m = 5/11cm
c=5y and m = 11y,
for some integer y. Not sufficient to calculate m.
(2) the number of women on the sight-seeing tour was less than 30 --> w<30. Not sufficient to calculate m.
(1) + (2)
w = 5x < 30w --> x<6x.
c=2x and as c=5y,
c is a multiple of 5 --> x=5x
c = 10 = 5y --> y = 2
m = 11y = 22.
Hence, the number of men on the sight-seeing tour was 22.
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A fence is to enclose a field and divide it into 3 equal areas. If there is 1600m of fencing available. Find the dimensions of the field that will allow for the maximum area.
The dimensions of the field that will allow for the maximum area are 200 meters by 400/3 meters
How to determine the dimensions of the field that will allow for the maximum area.From the question, we have
Fencing available, Perimeter = 1600 meters
Number of sections, n = 3
Represent the dimensions with x and y
So, we have:
Area = Number of sections* x * y = 3xy
Perimeter = (Number of sections * 2x + 2(Number of sections) * y) = 6x + 4y
Recall that
Perimeter = 1600 meters
So, we have
6x + 4y = 1600
A = 3xy
Divide 6x + 4y = 1600 by 2
3x + 2y = 800
Make y the subject
y = (800 - 3x)/2
Substitute y = (800 - 3x)/2 in A = 3xy
A = 3x(800 - 3x)/2
So, we have
A = 1200x - 4.5x²
Differentiate and set to 0
1200 - 9x = 0
So, we have
9x = 1200
Divide by 9
x = 400/3
Recall that y = (800 - 3x)/2
So, we have
y = (800 - 3 x 400/3)/2
Evaluate
y = 200
Hence, the dimensions are 200 by 400/3
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a popular brand of pen is available in 15 colors and 3 writing tips. how many different choices of pens do you have with this brand?
45 different choices of pens you have with this band.
What is combination?
A combination is a choice made in mathematics from a group of different elements when the order of the choices is irrelevant (unlike permutations). For instance, if three fruits, such as an apple, an orange, and a pear, are supplied, there are three possible pairings of the two: an apple and a pear. Formally speaking, a k-combination of a set S is a subset of S's k unique components. In other words, two combinations are the same if and only if they have the same members. (It is not important how the individuals in each set are arranged.) The quantity of k-combinations for a set with n components
A popular brand of the pen is available in 15 colors and 3 writing tips.
Choosing from one of 15 colors and one of 3 writing tips.
Each color could have one of two different writing tips.
To find the total different pens available multiply the number of colors by the number of tips: 15*3=45
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TRUE/FALSE. you cannot use both the current value of an explanatory variable and lags of that same explanatory variable in a regression.
It is "false" to claim that you cannot use both the present value of an explanatory variable and its delays in a regression.
Define the term regression?A statistical method called regression links a dependent variable with one or much more independently (explanatory) variables.
A regression model can demonstrate whether changes in one or more several explanatory variables are related to changes in the dependent variable.Political scientists frequently use lag- ged explanatory variables as a solution to endogeneity issues with observational data.Although some research claims that using lag dependent variables (LDVs) in regressions results in negatively biased coefficient estimates, even when the LDV is a part of the data-generating process, LDVs have been used in regression analysis provide the robust estimates of the outcomes of independent variables.Thus, it is "wrong" to claim that you cannot use both the present value of an explanatory variable and its delays in a regression.
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There are 6 circles and 10 triangles. What is the simplest ratio of circles to total shapes?.
If there are 6 circles and 10 triangles, then the simples ratio of circles to total shapes is 3 : 8
Total number of circles = 6 circles
Total number of triangles = 10 triangles
Total number of shapes = Total number of circles + Total number of triangles
Substitute the values in the equation
Total number of shapes = 6 + 10
= 16 shapes
The ratio of circles to total shapes = Total number of circles : Total number of shapes
Substitute the values
= 6 : 16
Simplify the ratio
= 3 : 8
Therefore, the simplified ratio of circles to total shapes is 3 : 8
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13. A hardware store sells many different sizes of chains. The price of each chain is determined by
its length. The table below shows the price for different lengths of chain.
Price of Chains
Length
(ft)
1.8
3
$1.35
$2.25
$2.55
6.2
$4.65
Which equation best represents the relationship between x, the length of the chain in feet,
and y, the price of the chain?
Price
3.4
The equation y=0.75x is the relationship between x, the length of the chain in feet, and y, the price of the chain.
What is slope intercept form of line?The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
Given that hardware store sells many different sizes of chains.
The price of each chain is determined by its length.
From the table we need to find the equation best represents the relationship between x, the length of the chain in feet, and y, the price of the chain.
Let us find this by taking 0.75 as x and multiplying with lengths to get the price.
1.8×0.75=1.35
3×0.75=2.25
3.4×0.75=2.55
6.2×0.75=4.65
y=0.75x.
Hence, y=0.75x is equation best represents the relationship between x, the length of the chain in feet, and y, the price of the chain.
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2. what conditions must be met if we want to perform a hypothesis test and answer the question of the management? select all that apply:
1.The following is the 90% confidence interval for the mean difference between the park's tree heights in 2009 and 2019:
(11.59, 14.21)
2. The conditions for the confidence interval are of:
A. Large samples and no extreme outliers.
D. Independently sampled pairs.
Given,
What is a confidence interval for the t-distribution?
When the population's standard deviation is unknown, the t-distribution is utilized, and the boundaries of the confidence interval are provided using the following formula:
x ± t s/√n
When the equation's variables are laid down as follows:
The sample mean = x
The critical value = t
The sample size = n
The standard deviation for the sample = s
Using a t-distribution calculator, the critical value is t = 1.6766 for a two-tailed 90% confidence interval with 50 - 1 = 49 df.
The remaining parameters are given as follows:
x = 12.9, s = 5.52, n = 50
Then the lower bound of the interval is of:
12.9 - 1.6766 x 5.52/sqrt(50) = 11.59.
The upper bound of the interval is of:
12.9 + 1.6766 x 5.52/sqrt(50) = 14.21.
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Question is incomplete. Completed question is given below;
Forest rangers wanted to better understand the rate of growth for younger trees in the park. They took measurements of a random sample of 50 young trees in 2009 and again measured those same trees in 2019. The data below summarize their measurements, where the heights are in feet.
Year Mean SD n
2009 11.4 3.6 50
2019 24.3 8.8 50
Difference 12.9 5.52 50
Round all calculated values to 4 decimal places as appropriate.
1. Construct a 90 confidence interval for the mean difference between the height of trees in the park in 2009 and in 2019.
2. What conditions must be met if we want to perform a hypothesis test and answer the question of the management? Select all that apply:
A. Large samples and no extreme outliers.
B. There must be at least 3 levels of the categorical variable.
C. np^≥10 and n(1−p^)≥10
D. Independently sampled pairs.
Alr last 1 HELP ME OUT……..
Answer: 89.7
Step-by-step explanation:
find the equation of the line that is perpendicular to the line 5x-2y=12
The slope of two perpendicular lines is the negative inverse of each other thus the equation of the perpendicular line will be y = (-2/5)x + c.
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A linear function always has the same and constant slope.
The formula for a linear function is f(x) = ax + b, where a and b are real values.
As per the given line,
5x-2y=12
2y = 5x - 12
y = (5/2)x - 6
The slope of the above line is 5/2
Since the slope of two perpendicular lines is the negative inverse of each other.
Thus,slope of line = -1/(5/2) = -2/5
So equation will be y = (-2/5)x + c
Hence"The slope of two perpendicular lines is the negative inverse of each other thus the equation of the perpendicular line will be y = (-2/5)x + c".
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a pizza parlor has been experimenting with lowering the price of their large one-topping pizza to promote sales. the average revenues from the sale of large one-topping pizzas on a friday night (5 p.m. to midnight) at various prices are given below. revenue from the sale of pizzas at different prices price (dollars x) revenue (dollars r) 9.25 1202.50 10.50 1228.50 11.75 1210.25 13.00 1131.00 14.25 1054.50 (a) find the function for the quadratic model that gives the average revenue in dollars where x is the price in dollars of a large one-topping pizza, with data from 9.25 ≤ x ≤ 14.25. (round all numerical values to two decimal places.) r(x)
The function for the quadratic model that gives the average revenue in dollars where x is the price in dollars of a large one-topping pizza, with data from 9.25 ≤ x ≤ 14.25 is -105.6x² + 254.28x - 12.16.
Quadratic model
In regression, quadratic model means mathematical model represented by a quadratic equation such as Y = aX² + bX + c, or by a system of quadratic equations.
Given,
A pizza parlor has been experimenting with lowering the price of their large one-topping pizza to promote sales. the average revenues from the sale of large one-topping pizzas on a Friday night (5 p.m. to midnight) at various prices are given below.
Here we need to find the quadratic model function with data from 9.25 ≤ x ≤ 14.25.
While we plot the given table of values in the graph then we get the graph like the following.
Through the graph we have identified that the value of
mean of x 11.75
mean of y 1,165.35
correlation coefficient r 0.9924547615
A -105.6
B 254.28
C -12.16
So, through these values the quadratic model of the function is written as,
=> -105.6x² + 254.28x - 12.16.
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duke earned $250 washing cars last summer. he spent all of his earning on a new stereo and a new bike. if he paid $190 for the stereo, what percent of his summer earnings did he soend on his bike
Answer: 24%
Step-by-step explanation: First find the percentage of the stereo.
190/250 = 0.76 of 76%
Then subtract 76% from 100% or subtract 190 from 250.
250 - 190 = 60 then 60/250 = 0.24 or 24%
Duke spent 24% of his earnings on his bike.
What is the area of the triangle with vertices (1, 4), (3, 1) and (11, 1)?
Answer:
Step-by-step explanation:
go to school
Quick, please! Stephanie is making a gingerbread house similar to her real-life house. She is using a scale of 1 cm equals 5 feet. If Stephanie's house is 30 feet wide, how wide will the gingerbread house be?
Answer:
6cm
Step-by-step explanation:
30 divded by 5 times 1
Answer:
Step-by-step explanation:
5cm
hi pls help me pls
hello help me
Answer:
see pic
Step-by-step explanation:
see pic
Can someone Explain to me how to do this step by step
The diameter of IC 1101 is 50 times greater than the diameter of the Milky way.
What does diameter mean?Diameter is a measurement that goes all the way around from starting point to the end point.
Given, the largest galaxy IC 1101 has a diameter of 5 * 10²² and the Milky way has a diameter of 1 * 10²¹.
Diameter of IC 1101 = 5 * 10²²
Rearranging the above term;
Diameter of IC 1101 = 1 * 5 * 10²¹ * 10
Diameter of IC 1101 = 1 * 10²¹ * 50
Diameter of IC 1101 = 50 * diameter of Milky way
Therefore, the answer is 50 times.
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: a marketing manager for a phone company claims that the percentage of children aged 8-12 who have phones differs from 52%. in a survey of 832 children aged 8-12 by the national consumers league, 449 of them had phones. can you conclude that the manager's claim is true? use the
The percentage of children aged between the ages of 8 - 12 who own phones are, in fact, lower than the 52% industry average. Of the 832 children aged 8 to 12 who were questioned by the national consumer league, 449 had cell phones.
You must use survey data to calculate the percentage of youngsters between the ages of 8 as well as 12 who own phones to ascertain whether the commercial manager's assertion is accurate. To achieve this, divide the total number of children polled (832) by the total number of kids who own phones (449), multiply the outcome by 100 to get the percentage, and so on.
This estimate gives a percentage of 53.8%, which is slightly higher than the advertising manager's stated figure.
This implies that the claim made by the management could not be genuine, and more investigation would be required to ascertain if it is real or not.
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in a specific university, getting a full scholarship and getting an internship are conditionally independent given that you are a top student. it is known that 50% of top students get a full scholarship and 40% of top students get an internship. if a randomly selected student is known to be a top one, what is the probability they will get a full scholarship and an internship? assume event a: getting a full scholarship; event b: getting an internship; event c: being a top student.
The probability they will get a full scholarship and an internship Is 0.2
What is Probability?
Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can vary from 0 to 1, with 0 being an impossibility and 1 denoting a certainty, probability is a crucial subject since it teaches all the fundamental ideas of the subject. One is the probability of every event in a sample space.
However, if two coins are tossed, four outcomes might occur: (H, H), (H, T), (T, H), and (T, T).
For instance, when we flip a coin, there are just two potential results: Head OR Tail (H, T).
Given that these two events are independent. Hence,
P(Full scholarship and an internship)
= P(Full scholarship) * P(Internship)
= 0.50 * 0.40
= 0.20
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8(8 - k) = -72k
please help
Answer:
k=-1
Step-by-step explanation:
Please look at the attachment, hope this helps, have a great day! :D
Find the value of X
2x
3x+10
Answer:
-10 = x
Step-by-step explanation:
2x = 3x + 10 Is this what you meant to write? Subtract 2x from both sides
0 = x + 10 Subtract 10 from both sides
-10 = x
or are you combining like terms?
2x + 3x + 10
5x + 10
A cylinder has a base radius of 6ft and a height of 18ft. What is its volume in cubic ft, to the nearest tenths place?.
2034.72ft³ is the required cylinder volume.
What do we mean by the volume of the cylinder?The capacity of a cylinder determines how much material it can carry, or we may say that the volume of a cylinder is the total number of unit cubes that can fit inside of it.
The volume of a cylinder is r²h, and its surface area is 2πrh+2πr².
The volume of cylinder Formula: V = A × h
We will assume the cylinder is a right circular cylinder.
Then, we have the formula:
V = πr²h
We already know:
Radius = 6ft
Height = 18ft
Now, calculate the volume as follows:
(V) = π× 6² × 18
(V) = 2034.72ft³
Therefore, 2034.72ft³ is the required cylinder volume.
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1a) as the newly hired manager of a company that provides cell phone service, you want to determine the percentage of adults in your state who live in a household with cell phones and no land-line phones. how many adults must you survey? assume that you want to be 98% confident that the sample percentage is within three percentage points of the true population percentage. use statcrunch to calculate the following sample size. assume that nothing is known about the percentage of adults who live in a household with cell phones and no land-line phones.
In this instance, a sample size of 393 adults who live in homes without landlines and use cell phones would be required.
Picking the number of observations or replicates to include in a statistical sample is known as sample size determination. Any empirical study whose objective is to draw conclusions about a population from a sample must take into account the sample size. In reality, a study's sample size is typically chosen based on the expense, convenience, or time required to collect the data as well as the requirement that the sample size have a high enough statistical power. There could be a variety of sample sizes used in complex studies; for instance, there might be different sample sizes for each stratum in a stratified survey. The intended sample size for a census is equal to the population because the entire population's data is being sought.
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step 1 of 4 : state the null and alternative hypotheses in terms of the expected proportion for each day. write your answer as a fraction or a decimal rounded to six decimal places, if necessary.
By using the concept of testing of hypothesis, the null and alternative hypothesis is given by
[tex]H_0 :p_i = \frac{1}{5}[/tex]
[tex]H_a : p_i \neq \frac{1}{5}[/tex] for atleast one day
What is testing of hypothesis?
Suppose there is a hypothesis and there is a data at hand. Testing of Hypothesis determines whether a particular hypothesis is supported by the data at hand.
The null and alternative hypothesis can be written are as follows-
Null hypothesis: [tex]H_0[/tex] : There is no difference in the number of swimmers from day to day.
Alternative hypothesis: [tex]H_a[/tex] : There is a difference in the number of swimmers from day to day.
[tex]H_0 :p_i = \frac{1}{5}[/tex]
[tex]H_a : p_i \neq \frac{1}{5}[/tex] for atleast one day
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Complete Question
The manager of the city pool has scheduled extra lifeguards to be on staff for Saturdays. However, he suspects that Fridays may be more popular than the other weekdays as well. If so, he will hire extra lifeguards for Fridays, too. In order to test his theory that the daily number of swimmers varies on weekdays, he records the number of swimmers each day for the first week of summer. Test the manager’s theory at the 0.10 level of significance.
Swimmers at the City Pool
Monday Tuesday Wednesday Thursday Friday
Number 46 68 43 51 70
Step 1 of 4 :
State the null and alternative hypotheses in terms of the expected proportion for each day. Enter your answer as a fraction or a decimal rounded to six decimal places, if necessary.
H0: pi=⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Ha: There is a difference in the number of swimmers from day to day.
Use point-slope form to write the equation of a line that passes through the point
(-14, 12) with slope -7/8
Answer:
Step-by-step explanation:
Please Simplify
√81x2
Answer:
18
Step-by-step explanation:
1)since 9*9 = 81 the square root of 81 is 9
9*2
2)simplify
18
Let f(x) = -3x² - 1 and g(x) = -5x. Find f(g(x)).
-15x²-1
0-75x²-1
15x²+5
15x² + 5x
Answer:
[tex] \huge{ \boxed{f(g(x)) = - 75 {x}^{2} - 1}}[/tex]
Step-by-step explanation:
To find f(g(x)), substitute g(x) into f(x). That is for every x found in f(x), replace it by g(x) and solve.
[tex]f(x) = - 3 {x}^{2} - 1 \\ g(x) = - 5x[/tex]
[tex]f(g(x) )= - 3 ({ - 5x})^{2} - 1 \\ = - 3(25 {x}^{2} ) -1 \\ = - 75 {x}^{2} - 1 \: \: \: \: \: \: [/tex]
We have the final answer as
[tex] \bold{f(g(x)) = - 75 {x}^{2} - 1}[/tex]
find the perimeter.
10ft. 13ft. 17ft.
Answer: 40
Step-by-step explanation:
P = a+b+c = 10+17+13 = 40
an air plane is flying at a height of 6 miles on a flight path that will take it directly over a radar tracking station. the distance d between them is decreasing at a rate of 400 miles per hour when the distance between them is 10 miles. what is the speed of the plane?
500 mph is the speed of the plane.
This creates a right triangle, with the distance between the plane and the radar station serving as the hypotenuse (s). Let y represent the plane's height and x represent its flight route.
s2 = y2 + x2
Given: the constant value of y = 6
negative since decreasing, ds/dt = -400
Trying to locate dx/dt
Taking the implicit derivative of the above equation's two sides
0 + 2xdx/dt = 2sds/dt
(S/X) ds/dt = dx/dt
When x = 8
s = √(62+82) = √100 = 10
dx/dt = (10/8)(-400) = -500 mph, or 500 mph traveling toward the radar station at a distance of 6 miles.
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In each of the following equations, determine which variable represents the angle and which variable represents the value of the trigonometric function. (a) In the equation q = sin d, represents the angle and represents the value of the trigonometric function. (b) In the equation p = tan^1 c, represents the angle and represents the value of the trigonometric function. (c) In the equation arccos n = upsilon, represents the angle and represents the value of the trigonometric function.
a) q= sin d
Angle is represented by d, and value is represented by q.
b) p = tan1 c
Angle is represented by c, and value is represented by p.
c) arccos n = ν
Angle is represented by n, and value is represented by v.
Given that,
We have to find the variables that indicate the angle and the value of the trigonometric function in each of the following equations.
In the following three equations:
(a) q = sin d;
(b) p = tan1 c; and
(c) arccos n = ν,
We know that,
a) q= sin d
Angle is represented by d, and value is represented by q.
b) p = tan1 c
Angle is represented by c, and value is represented by p.
c) arccos n = ν
Angle is represented by n, and value is represented by v.
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