a rectangle is to be inscribed under the arch of the curve y = 2cosx from x=-pi to x=pi. What is the length of the base of the rectangle with maximum area?

Question

a rectangle is to be inscribed under the arch of the curve y = 2cosx from x=-pi to x=pi. What is the length of the base of the rectangle with maximum area?

can someone help me with this....i have a lot of problems to do just like this but i don't understand them. can someone show me how to do this problem step by step, please?

my teacher taught me how to do it like this but i cant seem to get an answer.
try to follow my work and please help

so the area for both sides is:

A= 2x(2cosx)

then i have to find the deriviative, which is:

a= 4cosx-4xsinx (is this deriviative right?)

then i have to set it equal to zero and solve but this is where i get stuck. can you help me with this part?

4cosx-4xsinx=0

Answer 1

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Answer 2

To find the length of the base of the rectangle with maximum area, we need to find the value of x that maximizes the area function A = 2x(2cosx).

Let's continue from where you left off. You correctly found the derivative of the area function:

A' = 4cos(x) - 4xsin(x)

To find the value of x that maximizes the area, we need to set the derivative equal to zero and solve for x:

4cos(x) - 4xsin(x) = 0

Now, let's solve this equation step by step:

1. Factor out 4:

4(cos(x) - xsin(x)) = 0

2. Divide both sides by 4:

cos(x) - xsin(x) = 0

3. Rearrange the equation:

cos(x) = xsin(x)

4. Divide both sides by sin(x) (assuming sin(x) is not equal to 0):

cot(x) = x

Now, we have a transcendental equation. To solve this type of equation, we can use numerical methods such as graphing or approximation techniques.

One way to solve this equation is by graphing. Plot the graphs of y = cot(x) and y = x on the same coordinate system. The points where these two graphs intersect will represent the values of x that satisfy the equation.

Alternatively, you can use approximation methods like Newton's method or the bisection method to estimate the value of x that satisfies the equation.

Once you find the value(s) of x, substitute it back into the original area function to find the corresponding length of the base of the rectangle.