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?

Answer 1

Area= width*height

Lets just do one side.
area= x*2cosX
darea/dx= 2x*2cosx- 2x^2sinx=0

4xcosx=2x^2 sinX

2= x tanx

I get a graphical solution very near PI/3, -PI/3, so sthe base is 2PI/3

I cant find an exact solution. Check my work.

Answer 2

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 3

Certainly! I can help you solve this problem step by step.

Step 1: Understand the problem
We want to find the length of the base of a rectangle that can be inscribed under the curve y = 2cos(x) between x = -π and x = π, such that it has the maximum area.

Step 2: Find the area of the rectangle
Let's start by understanding the properties of the rectangle. The rectangle is inscribed under the curve, meaning that its top side is the curve itself. The height of the rectangle, h, will be the maximum value of the curve, which is 2.

To find the length of the base, we need to find the width of the rectangle. Since the curve is symmetrical about the y-axis, the width will be two times the distance from the y-axis to the point where the curve intersects the x-axis. Let's find this point.

Step 3: Find the x-coordinate where the curve intersects the x-axis
To find the x-coordinate where the curve intersects the x-axis, we need to solve the equation y = 0. In this case, the equation is 2cos(x) = 0. Solving this equation, we get cos(x) = 0.

The solutions to cos(x) = 0 are x = ±π/2, ±3π/2, ±5π/2, and so on. However, we are only interested in the solutions between x = -π and x = π. Therefore, the only solution within this range is x = π/2.

Step 4: Find the width of the rectangle
Now that we know the x-coordinate where the curve intersects the x-axis, which is x = π/2, we can find the width of the rectangle. The width will be two times the distance from the y-axis to this x-coordinate.

The distance from the y-axis to the x-coordinate π/2 is π/2 - (-π/2) = π.

So, the width of the rectangle, w, is equal to 2 times the distance, which is 2π.

Step 5: Calculate the area of the rectangle
Now that we have the height, h = 2, and the width, w = 2π, we can calculate the area of the rectangle, A, using the formula A = h * w.

A = 2 * 2π = 4π

So, the area of the rectangle is 4π.

Step 6: Answer the question
The question asks for the length of the base of the rectangle with maximum area. In our case, the base length is equal to the width, which is 2π.

Therefore, the length of the base of the rectangle with maximum area is 2π.

I hope this step-by-step explanation helps you understand how to solve this problem. Feel free to ask if you have any further questions or need help with similar problems!