The average rate of change of f(x) on the interval [4, 8] is 21.
How to find the rate of change?
For a function f(x), the rate of change on an interval [a, b] is given by:
R = ( f(b) - f(a))/(b - a)
In this case the function is:
f(x) = x² + 9x
And the interval is [4, 8]
Then the rate of change is
R = ( f(8) - f(4))/(8 - 4)
we will get:
f(8) = 8² + 9*8 = 136
f(4) = 4² + 9*4 + 16 + 36 = 52
Replacing that we get:
R = (136 - 52)/4 = 21
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You have maxed out your $800 credit card from Utopian One Bank. The interest on the card is 21%. Find the interest and the final cost you pay on this bill. (Hint this problem is computed like sales tax)
Given the question
$800 dollars
21% interest
Caclulate the question how you would calculate the sales tax.
B1 = 800 dollars
B2 = the final cost
[tex]\frac{21\times800}{100}=168[/tex]Interest rate = 168
800 + 168 = 968
968 is the final cost.
Interest rate = $168
Final cost = $968
why is 5.1 bigger than 5.099
5.1 is greater than 5.099, because the value of the 1 in 5.1 is more than the value of the 99 in 5.099.
[tex]\begin{gathered} 5.1=5+0.1=5+\frac{1}{10}=5+\frac{100}{1000} \\ 5.099=5+0.099=5+\frac{99}{1000} \end{gathered}[/tex]The value of 1 in 5.1 is 0.1, while the value of the 99 in 5.099 is 0.099.
Since 0.1 is bigger than 0.099, then 5.1 is bigger than 5.099.
Also, 5.1 is greater than 5.099 because a bigger number on 5.1 (which is 1) is closer to the decimal point compared to 5.099 (1 is bigger than 0). the closer a decimal is to the decimal point the higher its value.
-10x - 2y = 28
y = -5x - 14
Answer:
parallel
(0, 0)
Step-by-step explanation:
I have no clue how to answer this so I'll do my best.
-10x - 2y = 28 y = -5x - 14
-2y = 10x = 28
÷-2 ÷-2 ÷-2
--------------------
y = -5x - 14
The lines are parallel
----------------------------------------------------------------------------------------------------------
I hope this helps!
What is the slope of the line that goes through the points (1,-5) and (4,1)?OA-4/3OB.-3/4Ос. 1/2OD. 2
We can calculate the slope using the next formula
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where
(1,-5)=(x1,y1)
(4,1)=(x2,y2)
We substitute the values into the formula above
[tex]m=\frac{1+5}{4-1}=\frac{6}{3}=2[/tex]the slope of the line is m=2, the answer is D
Suppose that the annual rate of return for a common biotechnology stock is normally distributed with a mean of 5% and a standard deviation of 6%. Find the probability that the one-year return of this stock will be negative. Round to four decimal places.
===================================================
Work Shown:
Compute the z score for x = 0.
z = (x - mu)/sigma
z = (0 - 0.05)/(0.06)
z = -0.83333 approximately
Then use a calculator to find that P(Z < -0.83333) = 0.2023
There's about a 20.23% chance of getting negative returns, i.e. the person will lose money on the investment.
Find the measure of the missing angle. *Don't worry about the degree symbol
this is an ange formed by two intersecting chords
then the missing angle is given by
[tex]\theta=\frac{1}{2}(arcCB+arcSD)[/tex][tex]\theta=\frac{1}{2}(191+55)[/tex][tex]\theta=\frac{1}{2}(246)[/tex][tex]\theta=123\degree[/tex]the missing angle is 123°
Which ones are considered functions
Answer:
The answer is be because the x dose not repeat
Step-by-step explanation:
the x dose not repeat just look for the non repeating x
A box has length 4 ft, width 5 ft, and height 6 ft. What is the volume?
The volume of box will be 120 ft.³
What is volume ?
Volume is a three dimensional space occupy by the body of particular shape such as here :
Volume of cuboidal box = lbh
where, length "l" = 4 ft.
width "b" = 5 ft.
height "h" = 6 ft.
now, the volume of box will be :
V = lbh
V = 4 x 5 x 6
V = 120 ft.³
Therefore, the volume of box will be 120 ft.³
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A manufacturing process produces a critical part of average length 90 millimeters, with a standard deviation 2 of millimeters. All parts deviating by more than 5 millimeters from the mean must be rejected. What percentage of the parts must be rejected, on average? Assume a normal distribution.
We have that
[tex]X\sim N(\mu=90,\sigma^2=4^{})[/tex]The parts the will be rejected when it's above 95 or when it's under 85, if we plot the normal distribution it would be
Then, the percentage of the parts that will be rejected corresponds to the area in blue, then, we must calculate the area under the normal distribution for
[tex]P(X<85)+P(X>95)[/tex]The normal distribution is symmetrical, then calculate P(X < 85) is the same as P(X > 95), then we write it as
[tex]2\cdot P(X>95)[/tex]Calculate that integral is very hard, then, we must transform that in a standard normal X ~ N(0, 1) and use a table to find the result, to do that we must write a value z, it's a transformation to take a value on our normal and leads it to the standard normal, it's
[tex]Z=\frac{X-\mu}{\sigma}[/tex]We have X = 95, μ = 90 and σ = 2
[tex]Z=\frac{95-90}{2}=2.5[/tex]Then 2.5 is the value we are going to search in our table, using the complementary cumulative table for 2.5 we get 0.00621, which means
[tex]P(X>95)=0.00621[/tex]And the total percentage will be
[tex]P(X<85)+P(X>95)=0.01242[/tex]We can write it in percentage
[tex]0.01242=1.242\%[/tex]Therefore, only 1.24% will be rejected.
It's a very low value, but it's expected because it's more than 2 standard deviations (95%).
The table shows the amount of water Joel had in his bathtub to wash his dog and his desired water level. If the water drains at a rate of 14 gallons per minute, how many minutes will it take the tub to drain to his desired level?
Starting Water Level = 42 gallons
Desired Water Level = 28 gallons
It will take 1 minute to tub to drain to his desired level, by Rate of change.
What is rate of change?
Rate of change is used to mathematically describe the percentage change in value over a defined period of time.
Given, starting water value = 42 and desired water level = 28
Rate of change = 14 gallons.
Let x be the time,
According to question,
42-14x=28
-14x=-14
x=1
Hence, it will take one minute.
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Enter the answer in the space provided.Consider the functionWhat is the average rate of change of f(z) from z = 6toz=6?
To find the average rate of ch
In earn contains two white marbles, three green marbles, and 5 red marbles. A marble is drawn and then replaced. Then the second marble is drawn. What is the probability that the first marbel was white and the second was Green?
total number of outcomes = 2 + 3 + 5 = 10
The probability of getting a white marble is:
[tex]P(white)=\frac{2}{10}=\frac{1}{5}[/tex]The probability of getting a green marble is:
[tex]P(green)=\frac{3}{10}[/tex]The events: getting a white marble and getting a green marble are independent since there is a replacement after each drawing. Then, the probability that the first marble was white and the second was Green is:
[tex]\text{ P(white and gr}een\text{) =}P(white)\cdot P(green)=\frac{1}{5}\cdot\frac{3}{10}=\frac{3}{50}[/tex]Write the correct equation for the following statement.
The product of x and nine is three
Answer:
x*9 = 3
that's all
=)
Answer:
x*9=3
Step-by-step explanation:
Atleast I think so
Given f(x)=cosxf(x)=cosx, which function below doubles the amplitude and has a period of 3π3π?g(x)=3cos2xg of x is equal to 3 cosine 2 xg(x)=12cos2xg of x is equal to 1 half cosine 2 xg(x)=2cos2x3g of x is equal to 2 cosine 2 x over 3g(x)=3cos3x2g of x is equal to 3 cosine 3 x over 2
Answer:
[tex]g(x)=2\cos \frac{2x}{3}[/tex]Explanation:
A cosine function is generally given as;
[tex]\begin{gathered} y=a\cos (b) \\ \text{where Amplitude }=|a| \\ \text{ Period }=\frac{2\pi}{|b|} \end{gathered}[/tex]Given the below function;
[tex]f(x)=\cos x[/tex]If we compare both functions, we'll see that a = 1 and b = 1.
If we need another function with double the amplitude, then the value of a in that function will be (a = 2 x 1 = 2).
If we're to have another function g(x), with a period of 3 pi, let's go ahead and determine the value of b in the second function;
[tex]\begin{gathered} \frac{2\pi}{b}=3\pi \\ 3\pi\cdot b=2\pi \\ b=\frac{2\pi}{3\pi} \\ b=\frac{2}{3} \end{gathered}[/tex]Since we now have that for the second function g(x), a = 2 and b = 2/3, therefore g(x) can be written as below;
[tex]g(x)=2\cos \frac{2x}{3}[/tex]
Answer:
The derivation that correctly uses the cosine sum identity to prove the cosine double angle identity is A. A 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = cosine (x) cosine (x) minus sine (x) sine (x), = cosine squared (x) minus sine squared (x)
Step-by-step explanation:
It should be noted that the cosine difference identity is found by simplifying the equation by first squaring both sides.
Therefore, the derivation that correctly uses the cosine sum identity to prove the cosine double angle identity is that a 2-column table with 3 rows. Column 1 has entries 1, 2, 3. Column 2 is labeled Step with entries cosine (2 x) = cosine (x + x), = cosine (x) cosine (x) minus sine (x) sine (x), = cosine squared (x) minus sine squared (x).
In conclusion, the correct option is A.
LESSON Sine and Cosine Ratios 13-2 Practice and Problem Solving: A/B After verifying that the triangle to the right is a right triangle, use a calculator to find the given measures. Give ratios to the nearest hundredth and angles to the nearest degree. 1. Use the Pythagorean Theorem to confirm that the triangle is a right triangle. Show your work.
I saw this
is this correct? to start solving your question?
1.- Pythagorean theorem
6^2 = (5.2)^2 + (3)^2
36 = 27.04 + 9
36 = 36 Yes, it is a right triangle
2.-
[tex]\begin{gathered} \sin \text{ 1 = }\frac{5.2}{6} \\ \sin 2\text{ = }\frac{3}{6}\text{ = }\frac{1}{2} \end{gathered}[/tex]An object is launched at 18.4 meters per second (m/s) from a 36.8-meter tall platform. The equation for the object's heights at time t seconds after launch is s(t) = -4.912 + 18.4t + 36.8, where s is in meters. • When does the object strike the ground? (Select ] How long did it take the object to get to its maximum height? Select ] What was the height of the object at 3.32 seconds? | Select ]
1) To find when the object strikes the ground, we need to find the roots of the equation. Using quadratic formula:
[tex]\begin{gathered} t_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ t_{1,2}=\frac{-18.4\pm\sqrt[]{18.4^2-4(-4.912)(36.8)}}{2(-4.912)} \\ t_{1,2}=\frac{-18.4\pm\sqrt[]{1061.6064}}{-9.824} \\ \\ t_1=\frac{-18.4+32.582}{-9.824}=-1.44 \\ t_2=\frac{-18.4-32.582}{-9.824}=5.2 \end{gathered}[/tex]t can't be negative, then the object strikes the ground after 5.2 seconds
2) The maximum height is the vertex of the parabola. The t-coordinate is computed as follows:
[tex]t=\frac{-b}{2a}=\frac{-18.4}{2(-4.912)}=1.87[/tex]It takes 1.87 seconds for the object to get to its maximum height
3) To find the height after 3.32 seconds, we have to replace t = 3.32 int the equation:
[tex]\begin{gathered} s(3.32)=-4.192(3.32)^2+18.4(3.32)+36.8 \\ s(3.32)=-54.142+61.088+36.8 \\ s(3.32)=43.746 \end{gathered}[/tex]The height was 43.746 meters
What does it mean to take a derivative? I know how it's done, but not why.
Finding the derivative of the function is just a way for us to discuss how the function changes. For example, if we want to get the derivative of function y, with respect to x (dy/dx), then it is a formal way of discussing, how y changes when x changes.
4.
The Freshman Class treasury has 30
ten- and twenty-dollar bills that have
a total value of $430. How many of
each bill do they have?
There are 13 $20 bills and 17 $10 bills, respectively.
A linear equation is what?Constants and variables are used in conjunction to create linear equations. A linear equation with one variable is shown in the following standard form: Where a 0 and x is the variable, ax + b = 0.
Due to that,
There are 30 bills in all.
Total = $430
Let,
x = the number of $20 bills.
Amount in $10 banknotes = (30-x)
20x+10(30-x) = 430
20x+300-10x = 430
10x = 430-300
10x = 130
x = 13
$20 bills: x = 20; y = 13.
30 x = 30 13 = 17 = number of $10 banknotes
Therefore, there are 13 $20 bills and 17 $10 banknotes.
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5. GMAT scores are approximately normally distributed with a mean of 547 and a standard deviation of 95. Estimate the percentage of scores that were(a) between 357 and 737. %(b) above 737. %(c) below 452. %(d) between 452 and 737. %
Problem Statement
The question tells us that the GMAT scores are approximately normally distributed with a mean of 547 and a standard deviation of 95.
We are asked to find the percentage of scores that were:
a) between 357 and 737.
b) above 737
c) below 452
d) between 452 and 737.
Solution
a) Between 357 and 737:
[tex]\begin{gathered} 357\text{ is 2 standard deviations less than the mean of 547. That is,} \\ 547-2(95)=357 \\ \text{This means that 357 is }\frac{95}{2}\text{ \% from the mean}=47.5\text{ \% from 547.} \\ \\ 737\text{ is 2 standard deviations greater than the mean of 547. That is,} \\ 737-2(95)=547. \\ \text{This means that 737 is }\frac{95}{2}\text{ \% from the mean }=47.5\text{ \% from 547} \\ \\ \text{Thus the range 'Between 357 and 737' is:} \\ (47.5+47.5)\text{ \%}=95\text{ \%} \end{gathered}[/tex]b) Above 737
[tex]\begin{gathered} 737\text{ is 2 standard deviations away from the mean as shown in question A.} \\ \text{Thus, the percentage of scores above 737 must be:} \\ 100\text{ \% - (50 + 47.5)\% }=2.5\text{ \%} \end{gathered}[/tex]c) Below 452:
[tex]\begin{gathered} 452\text{ is 1 standard deviation from the mean.} \\ \text{Thus the percentage of scores below 452 must be:} \\ 50\text{ \% - 34\% = 16\%} \end{gathered}[/tex]d) Between 452 and 737:
[tex]\begin{gathered} 452\text{ is 1 standard deviation lower than the mean 547. Thus, the percentage from 452 to 547 is 34\%} \\ 737\text{ is 2 standard deviations higher than the mean of 547. Thus the percentage from 547 to 737 is: 47.5\%} \\ \\ \text{Thus the percentage between 452 and 737 is: (34 + 47.5)\%= 81.5\%} \end{gathered}[/tex]1. Squares with side lengths 6, 8, and 10 meters?2. Squares with areas 64 in?, 100 in?, 144 in2? 3. Two squares with side length 5 feet and a square with area 50 square feet?4. Explain how you know whether three squares will join at their corners to form a right triangle.
In order to know if 3 squares will form a right triangle,
a. The sum of the length of two of the squares must be greater than the length of the last square.
b. The lengths of the squares (if they are integers) must form a Pythagorean triple.
Pythagorean triples are:
3, 4, 5
5, 12, 13
8, 15, 17
9, 40, 41
there are more triples but we only need these for this question
c. They must conform to the Pythagoras Theorem.
[tex]\begin{gathered} \text{Pythagoras theorem is:} \\ c^2=a^2+b^2 \\ \text{where c is the largest side of the right angled triangle or hypothenus} \\ \text{while a and b are adjacent and opposite of the right angled triangle} \end{gathered}[/tex]Now we can proceed with these points at hand.
1. Squares with side lengths 6, 8, 10 can be written as:
2(3), 2(4), 2(5).
Ignoring the "2", we can see that this follows the Pythagorean triple.
therefore, 6, 8, 10 can form a right-angled triangle
2. 64, 100, 144 can be written as:
4(16), 4(25), 4(36)
Ignoring the "4", we can see that this does not follow the Pythagorean triple.
If we input the values into the Pythagoras theorem, we shall have:
[tex]\begin{gathered} 64^2+100^2\text{ = 4096 + 10000 = 14096} \\ 144^2=\text{ 20736} \\ \text{Therefore, we can s}ee\text{ that:} \\ 64^2+100^2\text{ }\ne\text{ }144^2 \end{gathered}[/tex]Therefore, 64, 100, 144 cannot form a right-angled triangle
3. Two squares with lengths 5 and a Square with an area of 50 square feet:
We need to find the length of the square with an area of 50 square feet.
[tex]\begin{gathered} \text{Area of square = l}^2 \\ \text{where l is the length of the side} \\ 50=l^2 \\ \text{square root both sides} \\ l\text{ = }\sqrt[]{50\text{ }}\text{ = 5}\sqrt[]{2} \end{gathered}[/tex]Now that we know the length of the 3rd and largest side of this triangle, we can now determine whether it is a right-angled triangle.
This case has a non-integer as part of the sides of the triangle, thus, condition b does not apply.
We must check via Pythagoras theorem:
[tex]\begin{gathered} By\text{ pythagoras:} \\ 5^2+5^2=25+25=50 \\ \text{while,} \\ (5\sqrt[]{2})^2=5^2\times(\sqrt[]{2})^2=25\times2=50 \\ \text{Thus we can s}ee\text{ that:} \\ 5,5,5\sqrt[]{2}\text{ can form a right-angled triangle} \end{gathered}[/tex]Therefore, the final answer: 1 and 3 can form a right-angled triangle but 2 cannot
4. I have given the reasons why they form a right-angled triangle above. But let me restate them:
In order to know if 3 squares will form a right triangle,
a. The sum of the length of two of the squares must be greater than the length of the last square.
b. The lengths of the squares (if they are integers) must form a Pythagorean triple.
Pythagorean triples are:
3, 4, 5
5, 12, 13
8, 15, 17
9, 40, 41
there are more triples but we only need these for this question
c. They must conform to the Pythagoras Theorem.
[tex]\begin{gathered} \text{Pythagoras theorem is:} \\ c^2=a^2+b^2 \\ \text{where c is the largest side of the right angled triangle} \\ \text{while a and b are adjacent and opposite of the right angled triangle} \end{gathered}[/tex]
Aurora raised money for a white water rafting trip Jacy made the first donation Gillermo’s donation was twice Jacy’s donation Rosa’s mother tripled what Aurora had raised so far now Aurora has $120. how much did Jacy donate?
Help please!!!
The amount of money donated by Jacy is $10.
Given that, Jacy made the first donation Gillermo’s donation was twice Jacy’s donation Rosa’s mother tripled what Aurora had raised so far now Aurora has $120.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Let the donation given by Jacy be $x.
Divide that amount by 4. One part is Guillermo's and Jacy's donation and three parts is the amount donated by Rosa's mother = 120 ÷ 4 = $30
Divide that amount by 3. One part is for Jacy's donation ($x) and two part is the amount Guillermo donated x = 30/3 = $10
Jacy was the first to donate. So, Jacy donated is $10
Therefore, the amount of money donated by Jacy is $10.
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find the exact value of cos90°.
cos90° is equal zero
Pls I need help with 2 problems as quick as possible thank you
To get the surface area of the prism given, we have to find the total area of the nets
We first split the nets into three and then find the areas
Lets us start with B
[tex]\text{Area of B=Area of a rectangle = length }\times Breadth[/tex]Area of A and C are equal
so each of the areas is
[tex]\frac{1}{2}\times base\text{ }\times height[/tex]But we can use a general formula for an equilateral traingular prism
[tex]\begin{gathered} =\frac{\sqrt[]{3^{}}\text{ }\times a^2}{2}+3(a\times h) \\ \text{where a=7} \\ h=18 \end{gathered}[/tex][tex]\text{Surface Area =}\frac{\sqrt[]{3^{}}\text{ }\times7^2}{2}+3(7\times18)[/tex]Thus we have the total surface area to be approximately
[tex]\text{Surface area=}420.44ft^2[/tex]A hammock is suspended between two trees. The curve the hammock makes can bemodelled by the equation y = 0.2x² - 0.4x - 0.6, where x and y are measured inmetres.a) Find the x interceptsb) Find the vertex.c) What is the minimum height of the hammock?
We have the function that relates x and y expressed as:
[tex]y=0.2x^2-0.4x-0.6[/tex]a) We have to find the x-intercepts.
To do that we can use the quadratic equation:
[tex]\begin{gathered} x=\frac{-(-0.4)\pm\sqrt{(-0.4)^2-4(0.2)(-0.6)}}{2(0.2)} \\ x=\frac{0.4\pm\sqrt{0.16+0.48}}{0.4} \\ x=\frac{0.4\pm\sqrt{0.64}}{0.4} \\ x=\frac{0.4\pm0.8}{0.4} \\ x=1\pm2 \\ x_1=1-2=-1 \\ x_2=1+2=3 \end{gathered}[/tex]Then, we have x-intercepts at x = -1 and x = 3.
b) We have to find the vertex.
We can find the x-coordinate of the vertex using the linear coefficient b = -0.4 and the quadratic coefficient a = 0.2:
[tex]x_v=\frac{-b}{2a}=\frac{-(-0.4)}{2(0.2)}=\frac{0.4}{0.4}=1[/tex]It can also be calculated as the average of the x-intercepts.
Knowing the x-coordinate of the vertex, we can find the y-coordinate of teh vertex using the formula applied to x = 1:
[tex]y=0.2(1)^2-0.4(1)-0.6=0.2-0.4-0.6=-0.8[/tex]Then, the vertex is (1, -0.8).
c) The minimum height will be given by the y-coordinate of the vertex.
Relative to the horizontal axis (y = 0), the minimum height will be -0.8 meters below that level.
Answer:
a) The x-intercepts are x = -1 and x = 3.
b) The vertex is (1,-0.8)
c) The minimum height is 0.8 units below the horizontal axis.
Casey deposited $1,550 in a bank account that earned simple interest at an interest rate of 4%. How much interest, in dollars, was earned in 6 years?
Answer:
$372
Explanation:
From the given problem, we have the following:
• The amount deposited, Principal = $1,550
,• The interest rate, r = 4%
,• Time = 6 years
To determine the amount of interest earned, at simple interest, we use the formula below:
[tex]$$Simple\: Interest=\frac{Principal\times Rate\times Time}{100}$$[/tex]Substitute the given values:
[tex]\begin{gathered} Simple\:Interest=\frac{1550\times4\times6}{100} \\ =\$372 \end{gathered}[/tex]The interest that was earned in 6 years is $372.
Fay is paid semimonthly. The net amount of each paycheck is $670.50.What is her net annual income?a. $17,433b. $4,023c. $16,092d. $8,046
Answer:
c. $16,092
Explanation:
• Fay is paid semimonthly, that is, ,twice a month,.
,• There are ,12 months in a year,.
Thus, the number of paychecks she receives annually is: 2 x 12 = 24.
The net amount of each paycheck is $670.50.
In order to get her net annual income, multiply the net amount on each paycheck by the number of payments.
[tex]\text{Net Annual Income}=24\times670.50=\$16,092[/tex]Fay's net annual income is $16,092.
Option C is correct.
What's the value of b ? See attached screenshot.
The value of b would be 25/4.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Since the equation of the line is 2y = 4.5, where c is a constant, the y-coordinate of the intersection point must be c.
The parabola has a equation y = -4x² + bx, where bis a positive constant.
The solution to this quadratic equation will gives the x-coordinate(s) of the point(s) of intersection
Since it’s given that the line and parabola intersect at exactly one point, the equation y = -4x² + bx has exactly one solution.
A quadratic equation in the form ax²+bx+c has exactly one solution when its discriminant b²−4ac is equal to 0.
Therefore, if the line y = 22.5 intersects the parabola defined by exactly one point, then by = 25/4 .
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what is the least multiple of 8
The least multiple of a number is itself.
Least multiple of 8 : 8
can you help me i have to see if it is a direct variation
a direct variation is a relationship between two variables that can be expressed by an equation in which one variable is equal to a constant times the other. For instance,
[tex]y=mx[/tex]in which variables x and y are related by a constant m. A numerical examples of a Direct variations are:
[tex]\begin{gathered} y=-5x \\ y=\frac{2}{3}x \\ y=-\frac{7}{6}x \end{gathered}[/tex]etc. In our case, the figure shows a line. The general equation of a line is in the form
[tex]y=mx+b[/tex]This is almost the same a direc variation, however it has the y-intercept b. In our case, form the graph, we can see that
b=4. In other words, b is the point in which the lines cross y-axis.
Hence, our line doesnt represent a Direct variation since there is a y-intercept b=4.
[tex] \rm\int_{0}^{ \frac{\pi}{2} } \frac{1}{ \sqrt{1 - {sin}^{2} ( \frac{1}2) {sin}^{2} \varphi } } d \varphi \\ [/tex]
This is an another elliptical integral, but of the first kind:
[tex]\displaystyle F(k) = \int_0^{\pi/2} \frac{dx}{\sqrt{1-k^2\sin^2(x)}}[/tex]
[tex]\implies \displaystyle \int_0^{\pi/2} \frac{d\varphi}{\sqrt{1-\sin^2\left(\frac12\right)\sin^2(\varphi)}} = \boxed{F\left(\sin\left(\frac12\right)\right)}[/tex]