Can someone help me in this trig question, please? thanks
A person is on the outer edge of a carousel with a radius of 20 feet that is rotating counterclockwise around a point that is centered at the origin. What is the exact value of the position of the rider after the carousel rotates 5pi/12
Answers
[tex]\bf \textit{Sum and Difference Identities} \\\\ sin(\alpha + \beta)=sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta) \\\\ cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\ -------------------------------\\\\ cos\left( \frac{\pi }{6}+\frac{\pi }{4} \right)=cos\left( \frac{\pi }{6}\right)cos\left(\frac{\pi }{4} \right)-sin\left( \frac{\pi }{6}\right)sin\left(\frac{\pi }{4} \right)[/tex]
[tex]\bf cos\left( \frac{\pi }{6}+\frac{\pi }{4} \right)=\cfrac{\sqrt{3}}{2}\cdot \cfrac{\sqrt{2}}{2}-\cfrac{1}{2}\cdot \cfrac{\sqrt{2}}{2}\implies \cfrac{\sqrt{6}}{4}-\cfrac{\sqrt{2}}{4}\implies \boxed{\cfrac{\sqrt{6}-\sqrt{2}}{4}} \\\\\\ sin\left( \frac{\pi }{6}+\frac{\pi }{4} \right)=sin\left( \frac{\pi }{6}\right)cos\left( \frac{\pi }{4} \right)+cos\left( \frac{\pi }{6}\right)sin\left(\frac{\pi }{4} \right)[/tex]
[tex]\bf sin\left( \frac{\pi }{6}+\frac{\pi }{4} \right)=\cfrac{1}{2}\cdot \cfrac{\sqrt{2}}{2}+\cfrac{\sqrt{3}}{2}\cdot \cfrac{\sqrt{2}}{2}\implies \cfrac{\sqrt{2}}{4}+\cfrac{\sqrt{6}}{4}\implies \boxed{\cfrac{\sqrt{2}+\sqrt{6}}{4}}\\\\ -------------------------------\\\\ 20\left( \cfrac{\sqrt{6}-\sqrt{2}}{4} \right)\implies 5(-\sqrt{2}+\sqrt{6}) \\\\\\ 20\left( \cfrac{\sqrt{2}+\sqrt{6}}{4} \right)\implies 5(\sqrt{2}+\sqrt{6})[/tex]
The exact value of the position of the rider after the carousel rotates 5π/12 is 5 (-√2 + √6), 5(√2 + √6).
The positionSince the position of the carousel is (x, y) = (20cosθ, 20sinθ) and we need to find the position when θ = 5π/12 = 5π/12 × 180 = 75°
So, substituting the value of θ into the positions, we have
(20cos75°, 20sin75°)
The value of 20cos75°
20cos75° = 20cos(45 + 30)
Using the compound angle formula
cos(A + B) = cosAcosB - sinAsinB
With A = 45 and B = 30
cos(45 + 30) = cos45cos30 - sin45sin30
= 1/√2 × √3/2 - 1/√2 × 1/2
= 1/2√2(√3 - 1)
= 1/2√2(√3 - 1) × √2/√2
= √2(√3 - 1)/4
= (√6 - √2)/4
= (-√2 + √6)/4
So, 20cos75° = 20 × (-√2 + √6)/4
= 5 (-√2 + √6)
The value of 20sin75°
20sin75° = sin(45 + 30)
Using the compound angle formula
sin(A + B) = sinAcosB + cosAsinB
With A = 45 and B = 30
sin(45 + 30) = sin45cos30 + cos45sin30
= 1/√2 × √3/2 + 1/√2 × 1/2
= 1/2√2(√3 + 1)
= 1/2√2(√3 + 1) × √2/√2
= √2(√3 + 1)/4
= (√6 + √2)/4
= (√2 + √6)/4
So, 20sin75° = 20 × (√2 + √6)/4
= 5(√2 + √6)
Thus, (20cos75°, 20sin75°) = 5 (-√2 + √6), 5(√2 + √6).
So, the exact value of the position of the rider after the carousel rotates 5π/12 is 5 (-√2 + √6), 5(√2 + √6).
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Related Questions
Daniela is a stain glass artist; she created the window below for her living room. What is
the area of the window she created?
Answers
We know the diameter d = 12 ft.
The area of a (complete) circle is given by the formula:
A = π × r²
The radius can be found by halving the diameter:
r = d ÷ 2
= 12 ÷ 2
= 6 ft
Therefore the area of the circle will be:
A = π × r²
= 3.14 × 6²
= 3.14 × 36
= 113.04 ft²
And the area of the semi-circle:
A = 113.04 ÷ 2
= 56.52 ft²
Hence, the area of the window will be 56.52 ft².
If a recipe calls for 3 servings calls for 2 grams of saffron . how many mililgrams of saffron are needed for 6 servings
Answers
Which of the following gives all of the coefficients of the algebraic expression -2x^2 +17-15x+7xy
Answers
The coefficients of the algebraic expression -2x² + 17 -15x + 7xy are -2, 17, -15 and 7
How to determine the coefficient of the expression
From the question, we have the following parameters that can be used in our computation:
-2x² + 17 -15x + 7xy
Consider an expression represented as
Ax
Where A is a constant and x is the variable
The coefficient of the expression is A
Using the above as a guide, we have the following:
The coefficients of the algebraic expression are -2, 17, -15 and 7
Question
Which of the following gives all of the coefficients of the algebraic expression -2x^2 +17-15x+7xy
a. 2, 17, 15 and 7
b. -2, 17, -15 and 7
c. -2, -17, -15 and 7
d. -2, -17, 15 and 7
What is the equation of the line that passes through the points (0, -4) and (2, 8)?
A. y = 6x - 4
B. y = 6x + 4
C. y = 4x + 6
D. y = -4x + 6
Answers
m = Y₂ - Y₁ / X₂ - X₁
m = 8 - (- 4) / 2 - 0
m = 12 / 2
m = 6
So now your equation is y = 6x + b. This eliminates options C and D. Plug in any of your ordered pairs to find b. I picked the first one.
- 4 = 6(0) + b
- 4 = 0 + b
- 4 = b
Your equation is y = 6x - 4.
If you add the lengths of the focal radii of an ellipse, what other value will you produce?
2a
2b
2c
The eccentricity
Answers
Equation of an ellipse
→having center (0,0) , vertex ([tex](\pm a ,0)[/tex] and covertex [tex](\pm b ,0)[/tex] and focus [tex](\pm c ,0)[/tex] is given by:
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2}=1[/tex]
As definition of an ellipse is that locus of all the points in a plane such that it's distance from two fixed points called focii remains constant.
Consider two points (a,0) and (-a,0) on Horizontal axis of an ellipse:
Distance from (a,0) to (c,0) is = a-c = [tex]F_{1}[/tex]
Distance from (-a,0) to (c,0) is = a + c = [tex]F_{2}[/tex]
[tex]F_{1} + F_{2} =[/tex] a -c + a +c
= a + a
= 2 a →(Option A )
PLEASE ANSWER + BRAINLIEST!!!
What is the value of 3x^2 - 5xy + 5y^2 for x = -6 and y = 2?
A. -68
B. -4
C. 188
D. 208
Answers
Simplifying all the numbers, it's 3(36) + 30(2) + 5(4), and then 108+60+20. Going along, you get 188
Determine whether the given coordinates are the vertices of a triangle. Explain.
L(–24, –19), M(–22 ,20), N(–5, –7)
A:
yes:
LM + MN > LN,
LM + LN > MN,
LN + MN > LM
B:
no:
LM + MN = LN
C:
no:
LM + MN < LN
D:
yes:
LM + MN < LN,
LM + LN < MN,
LN + MN < LM
Answers
yes
LM + MN > LN,
LM + LN > MN,
LN + MN > LM
Explanation:
In any triangle, the sum of any two sides MUST BE GREATER than the third side.
Therefore, to decide whether the given coordinates can form a triangle or not, we will simply get the length of each side and then check the above condition.
The length can be calculated using the distance formula attached in the image.
LM = [tex] \sqrt{(-24--22)^2+(20--19)^2} [/tex] = 39.05 units
MN = [tex] \sqrt{(-5--22)^2+(-7-20)^2} [/tex] = 31.906 units
NL = [tex] \sqrt{(-5--24)^2+(-7--19)^2} [/tex] = 22.4722 units
Now, let's check:
LM + MN = 39.05 + 31.906 = 70.956 units > NL
MN + NL = 31.906 + 22.4722 = 54.3782 > LM
LM + NL = 39.05 + 22.4722 = 61.5222 > MN
Therefore,, the given coordinates can form a triangle
Hope this helps :)
Final answer:
After computing the distances between the points L, M, and N using the distance formula and comparing them, it is clear that the sum of the lengths of any two sides is greater than the length of the third side, satisfying the conditions of a triangle.
Explanation:
To determine whether the given coordinates are the vertices of a triangle, we can check if the sum of the lengths of any two sides is greater than the length of the third side, which is a principle derived from the triangular inequality theorem. First, we calculate the distances between the points L(–24, –19), M(–22, 20), and N(–5, –7) using the distance formula: √((x2-x1)² + (y2-y1)²).
For LM: √((-22 + 24)² + (20 + 19)²) = √(4 + 1521) = √(1525)
For MN: √((-5 + 22)² + (-7 - 20)²) = √(289 + 729) = √(1018)
For LN: √((-5 + 24)² + (-7 + 19)²) = √(361 + 144) = √(505)
Now we compare the lengths:
LM + MN > LN
LM + LN > MN
LN + MN > LM
Since all three conditions are satisfied, the points L, M, and N form a triangle.
A typist charges $0.90 a page and averages 45 pages per work day. If she works 3 days a week, how much does she earn in a week?
Answers
Final answer:
The typist earns $121.50 per week by multiplying the charge per page ($0.90) with the daily page average (45 pages) and the number of days worked per week (3 days).
Explanation:
The question asks us to calculate the weekly earnings of a typist who charges $0.90 per page and averages 45 pages per day, working 3 days a week. To find the total weekly earnings, we can multiply the number of pages typed per day by the charge per page, and then multiply that result by the number of days worked in a week.
Multiply the number of pages (45) by the charge per page ($0.90): 45 pages × $0.90 = $40.50 per day.
Multiply the daily earnings by the number of days worked in a week (3): $40.50 × 3 days = $121.50 per week.
Therefore, the typist earns $121.50 per week.
Complete the equation of the line through (-8,8) and (1,-10). Use exact numbers.
Answers
y = mx + b
-10 = m(1) + b
m + b = -10
b = -10 - m
Now that we have a value for b, we can plug this into the other equation:
8 = m(-8) + b
8 = -8m + (-10 - m)
8 = -9m -10
18 = -9m
m = -2
Now that we know what m is equal to, we can plug this into our first equation to get an answer for b:
b = -10 - m
b = -10 -(-2)
b = -10 + 2
b = -8
Our final equation is y = -2x - 8
The equation of the line through (-8,8) and (1,-10) is y = -2x - 8
Using the slope intercept form equation.
y = mx + b
where
m = slope
b = y-intercept
Therefore,
(-8,8) (1,-10)
m = -10 - 8 / 1 + 8 = - 18 / 9 = -2
Therefore,
let's find b as follows:
(1, -10)
y = -2x + b
-10 = -2(1) + b
b = -10 + 2
b= -8
Therefore,
y = -2x - 8
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What is the trigonometric ratio for sin C?
Answers
Park rangers notice a negative correlation between the number of available campsites in a park and the amount of litter on the park's trails. Which variable is most likely the lurking variable that explains the correlation? Question 3 options: age of trees in the park number of eagles spotted in the park number of visitors to the park density of trees in the park
Answers
We have been given that a park rangers notice a negative correlation between the number of available campsites in a park and the amount of litter on the park's trails.
We need to figure out the lurking variable that describes the given correlation.
Out of the given options, the most appropriate variable that describes that negative correlation between number of available campsites in the part and the amount of litter on the park's trails is number of visitors to the park.
The reason, why number of visitors to the park is the lurking variable for the given negative correlation, is that more the number of visitors, more the litter they produce and consequently, lesser the number of available campsites.
The Candle Emporium is pouring cylindrical candles. Each candle has a radius of 5 inches and a height of 18 inches. Which formula represents the correct way to find the volume of a candle?
a. Volume= 5x18
b. Volume= pi (5) (18)
c. Volume= pi (18)squared (5)
d. Volume= pi (5)squared (18)
Answers
The formula for finding the volume of a cylinder is [tex]V = \pi r ^{2}h[/tex]. The answer choice that matches this is answer choice D. Hope I helped! :3
Answer:
Option D is correct.
Step-by-step explanation:
We are given,
Height of the candles, h = 18 inches
Radius of the candles, r = 5 inches
Candles are in shape of cylinder.
We have to find correct representation of volume of candle.
Volume of cylinder = [tex]\pi r^2h[/tex]
Volume of candle = ]tex]\pi (5)^2\times18[/tex]
Therefore, Option D is correct.
You wish to invest $500. You are going to earn 5% simple interest. How much interest will you earn?
Answers
Answer:You would earn .05 x $500 or $25 over that time period.
Step-by-step explanation:
Which system of inequalities is represented by the graph
Answers
y = x
y = -x
We then have to evaluate points that belong to the shaded region:
For (0, 1) we have:
1> = 0
1> = - 0
We observe that both equalities are fulfilled, therefore, we have that the inequalities are:
y> = x
y> = - x
Answer:
y> = x
y> = - x
option D
Simplify 11/12 - 6/4
Answers
Short answer: -7/12
Why?
So, here are the steps on how to solve this. It takes a bit of work. (First step: 6/4 can be simplified down to 3/2, so do that.) - (Second step: Find the least common denominator. The least common denominator is 12. Convert 3/2 to a 12 denominator by multiplying both sides by 6. {3 * 6} / {2 * 6} ==> 18/12) - (Third step: Subtract the numerators. {11}/12 - {18}/12 ==> -7/12
Hope this helped you! ♥
Rico saved $30 to buy office supplies. He needed a binder that cost $9.99 , including tax,and three pens that cost $1.99 each,including tax. How much money will Rico have left after he buys the pens and the binder?
Answers
Which expressions are equivalent to the one below? Check all that apply.
21^x/7x
A. 3^x-7
B. 3
C. 3^x
D. (21-7)^x
E. 7^x * 3^x/ 7^x
F. (21/7)^x
Answers
After simplifying the original expression, the equivalent expressions to [tex]21^x/7x[/tex] are C. [tex]3^x[/tex] and F. [tex](21/7)^x.[/tex]
Explanation:The original expression in the question is [tex]21^x/7x[/tex]. To find equivalent expressions, we first simplify the original expression. The expression [tex]21^x[/tex]is the same as [tex](7*3)^x,[/tex] which, by property of exponents, can be written as [tex]7^x * 3^x.[/tex]Then, given the expression is divided by 7x, the x in the denominator cancels out with the x from [tex]7^x.[/tex] Hence, the simplified form will be [tex]3^x.[/tex] Now, we compare it with the given options.
Option C is directly equal to the simplified form. While options A,D, E, and B are not equivalent to the original expression. Considering option F, 21/7 simplifies to 3 and so, [tex](21/7)^x[/tex] can be rewritten as [tex]3^x[/tex] which is equal to the simplified form. Therefore, the equivalent expressions are C. [tex]3^x[/tex]and F. [tex](21/7)^x.[/tex]
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The equivalent expressions to [tex]21^x/7x[/tex] are 3^x and [tex]7^x * 3^x/ 7^x[/tex], and[tex](21/7)^x,[/tex]which can all be simplified to result in 3^x.
Explanation:The expressions equivalent to 21^x/7x are checked by determining if they simplify down to the same value as the original expression. Taking the original expression 21^x/7x, it's clear that 21 can be rewritten as 7*p, so 21^x/7x simplifies to [tex](7*3)^x/7x = 7^x*3^x/7x = 3^x[/tex] (because [tex]7^x/7^x = 1).[/tex]
From the given options, Option C: 3^x, and Option E: [tex]7^x * 3^x/ 7^x[/tex]satisfy this criterion, as they also simplify to 3^x. Option F: (21/7)^x also simplifies as explained earlier to 3^x. So, options C, E, and F are equivalent to the original expression.
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22 POINTS, NEED URGENT HELP
Write the equation of each circle:
1. R with center R(-1, 8) and radius 5
2. K with center K(4,0) and that passes through (2,0)
Answers
(x+1)^2 + (y-8)^2 = 5^2 (answer)
Find the amount in a continuously compiunded account for the following condition. Principal, $4000; Annual interest rate, 5.1%; time, 2 years
Answers
Using the given principal of $4000, an annual interest of 5.1%, and a time period of 2 years, the amount in a continuously compounded account will be approximately $4483.48 using the formula A = P*e^(rt).
Explanation:In this question, we are asked to find the amount in a continuously compounded account. The formula to calculate this is A = P*e^(rt), where A is the amount, P is the principal, r is the rate, t is the time, and e is the base of the natural logarithm, approximately equal to 2.71828.
In this case, P = $4000, r = 0.051 (5.1% expressed as a decimal), and t = 2.
Substituting these values into the formula, we get: A = 4000 * e^(0.051*2). Using a calculator, we can find this to be approximately $4483.48.
So, after 2 years with an annual interest rate of 5.1%, compounded continuously, the account will contain about $4483.48.
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asmine works at the public library after school. She makes $7.00 per hour. She cannot work more than 15 hours per week. If y = 7x represents Jasmine's earning, what are the values in the domain?
Answers
Q1 :Let f(x)=2x^2−8
The quadratic function g(x) is f(x) translated 2 units down
What is the equation for g(x)
Q2 Let f(x)=[tex] \frac{3}{4} [/tex]x2−1.
The function g(x) is a vertical stretch of f(x) by a factor of 8.
What is the equation of g(x)?
Q3 The graph of the function g(x) is a transformation of the parent function f(x)=x2 .
Which equation describes the function g?
(refer to picture attached)
A. g(x)=x2+2
B. g(x)=(x−2)2
C. g(x)=(x+2)2
D. g(x)=x2−2
Answers
[tex]f(x) = 2 x^{2} -8 \\[/tex]
g(x) is the translation of f(x) 2 units down.
Vertical translation can be expressed as addition or subtraction of a number from the function value. Subtraction indicates that the function is translated down. So, translation of f(x) by 2 units down can be expressed as:
g(x) = f(x) - 2
Using the value of f(x), we can write:
[tex]g(x)= 2x^{2} -8-2 \\ \\ g(x)=2 x^{2} -10[/tex]
Question # 2
[tex]f(x)= \frac{3}{4} x^{2} -1 [/tex]
A function can be vertically stretched by multiplying it with a number having magnitude greater than 1. f(x) is to be stretched by a factor of 8 to get g(x), so we can write:
g(x) = 8 * f(x)
Using the value of f(x) we can write:
[tex]g(x)=8* (\frac{3}{4} x^{2} -1) \\ \\ g(x)=6 x^{2} -8[/tex]
If you want to put a 4x8 piece of plywood through a 3-foot square opening in your ceiling by turning it diagonally, is the opening big enough? Use 45-45-90 triangle since it is a square
Answers
The first step is to calculate the length of the diagonal of the square opening.
The length of a diagonal of square = [tex] \sqrt{2} [/tex] * side
Given side = 3 feet
Length of the square hole = [tex] \sqrt{2} [/tex] *3 = 4.24 feet
since the diagonal is 4.24 feet and the shorter side of the plywood piece is only 4 feet, we can slide it through the square hole diagonally
Answer is YES since the diagonal 4.24 feet is greater than the shorter side of the plywood piece *4 feet
When is periodic data useful? Give examples to support your answer.
Answers
Periodic data is useful for analyzing trends over time, helping to forecast future events, and measure frequencies of occurrences. It is applicable in various fields including business, environmental science, and health for planning strategies and understanding patterns.
Periodic data is useful when we are interested in analyzing and understanding trends over regular intervals of time. It helps us to observe patterns and forecast future occurrences based on historical information. For example, in business, we may look at quarterly sales figures to identify seasonal effects on sales and plan inventory accordingly. In environmental science, climate data recorded over decades can be critical for studying changes in weather patterns and predicting future climate events. In health, collecting epidemiological data periodically can enable public health officials to track the spread of diseases and the effectiveness of interventions.
Frequency, relative frequency, and cumulative relative frequency are measures that can describe how often certain events occur within periodic data. These measures help answer questions like 'How many students study five hours or more for an exam?' or 'What percent of families own two pets?' They are crucial for understanding the distribution of data and for making informed decisions based on that data.
The resonance frequency f in an electronic circuit containing inductance L and capacitance C in series is given by f=1/2π√LC. Determine the inductance L in an electric circuit if the resonance frequency is 6.2 and the capacitance is 0.0001. Round your answer to the nearest tenth.
10270.1
6.6
256.8
0
Answers
f = 1 / (2π√LC)
Clearing L we have:
√LC = 1 / (2πf)
LC = (1 / (2πf)) ^ 2
L = (1 / C) * (1 / (2πf)) ^ 2
Substituting values:
L = (1 / 0.0001) * (1 / (2 * 3.14 * 6.2)) ^ 2
L = 6.596253438
Round to the nearest tenth:
L = 6.6
Answer:
L = 6.6
f = 1 / (2π√LC)
To find L
√LC = 1 / (2πf)
LC = (1 / (2πf)) ^ 2
L = (1 / C) * (1 / (2πf)) ^ 2
Substitute the values:
L = (1 / 0.0001) * (1 / (2 * 3.14 * 6.2)) ^ 2
L = 6.596253438
L = 6.6
So the Answer is this:
L = 6.6
What is the solution to the linear equation ? 2/5+p=4/5+3/5p
Answers
Answer:
p=1
Step-by-step explanation:
2/5+p=4/5+3/5p
[tex]\frac{2}{5} +p= \frac{4}{5} +\frac{3}{5} p[/tex]
p or p/1 are same
LCD is 5, multiply 5 with each term. the equation becomes
[tex]\frac{2}{5} \cdot 5+p \cdot 5= \frac{4}{5} \cdot 5 +\frac{3}{5}p \cdot 5[/tex]
[tex]2+5p = 4 +3p[/tex]
Subtract 3p from both sides
[tex]2+2p = 4[/tex]
Subtract 2 on both sides
[tex]2p= 2[/tex]
Divide by 2 on both sides
p= 1
So the value of p =1 for the given equation
A data set consists of the following data points:(3, 5), (5, 8), (6, 13)The line of best fit has the equation y = 2.5x – 3. What does this equation predict for a value of x = 3?
Answers
y = 2.5x - 3
For x = 3 we have:
y = 2.5 (3) - 3.
Rewriting:
y = 7.5 - 3.
y = 4.5
Therefore, the ordered pair that predicts the line is:
(x, y) = (3, 4.5)
Answer:
this equation predict for a value of x = 3:
(x, y) = (3, 4.5)
Sam is driving from Chicago to San Francisco, a distance of 2220 miles. He covers the first 192 miles in 4 hrs. At this rate, how long will the entire trip take?
Answers
[tex] \frac{192}{4} = \frac{2220}{x} [/tex]
This states that Sam can drive 192 miles in 4 hours and 2220 miles in x hours. To solve, we can use cross-multiplication:
[tex]192x = 2220*4 \\ 192x = 8880 \\ x = \frac{8880}{192} \\ x = 46.25[/tex]
It will take Sam 46.25 hours to drive the whole trip, or 46 hours and 15 minutes.
solve for z (attachment)
Answers
z / 4 = (16 + 4)/z
z^2 / 4 = 20
z^2 = 80
z = 4√5
Therefore the answer to this question is z = 4√5.
[tex]\Delta ABD\ and\ \Delta BCD\ are\ similar,\ therefore:\\\\\dfrac{y}{4}=\dfrac{16}{y}\ \ \ \ |cross\ multiply\\\\y\cdot y=4\cdot16\\\\y^2=64\to y=\sqrt{64}\to y=8[/tex]
Use Pythagorean theory:
[tex]z^2=4^2+8^2\\\\z^2=16+64\\\\z^2=80\to z=\sqrt{80}\\\\z=\sqrt{16\cdot5}\\\\z=\sqrt{16}\cdot\sqrt5\\\\\boxed{z=4\sqrt5}[/tex]
Write the equation of a circle, given the center is at point a(4,6) and the radius = 3.
Answers
[tex](x - a)^{2} + (y - b)^{2} = r^{2} [/tex]
Answer:
[tex](x - 4)^{2} + (y - 6)^{2} = 9[/tex]
A rectangular plot of land that contains 1500 square meters will be fenced and divided into two equal portions by an additional fence parallel to two sides. find the dimensions of the land that require the least amount of fencing.
Answers
Let w= the width
Two sections are required, hence 2w of fence required
2x+3w=500
this can be written as:
3w=1500-2x
w=(1500-2x)/3
Area=x*w
replacing w with our expression:
A=x(1500-2x)/3
A=(500x-2x^2)/3
This is a quadratic equation, if we find the axis of symmetry we will know what value of x gives maximum area:
Axis of symmetry: x=-b/2a
From our equation we get:
a=-2/3; b=1500/3
thus
x=(1500/3)/(-(-2/3))
x=750
thus the maximum area will be given by length of 750