Find the dimensions of the rectangle with the most area that will fit above the x-axis and below the graph of y=e^-x^2.
The curve looks sort of bell-shaped.
Let the point of contact in quadrant I be (x,y)
then the base of the rectangle is 2x and its height is y
A = 2xy
= 2x(e^(-x^2))
d(A)/dx = (2x)(-2x)(e^(-x^2)) + 2(e^(-x^2))
= 0 for a max/min of A
skipping some steps
e^(-x^2)) ( -2x + 1) = 0
x = 1/2
then y = e^(-1/4) = 1/e^(1/4)
so the rectangle has a length of 2x or 1
and a height of 1/e^(1/4)
Can you show me the steps you skipped?
Thanks
from
(2x)(-2x)(e^(-x^2)) + 2(e^(-x^2)) = 0
-2e^(-x^2) ( -2x^2 + 1) = 0
ahhh, just noticed an error from earlier, (that's what I get by skipping steps)
-2e^(-x^2) = 0 ----> no solution
or
-2x^2 + 1 = 0
x^2 = 1/2
x = 1/√2
then y = e^(-1/2) = 1/√e
so the rectangle has a length of 2x or 2/√2
and a height of 1/√e
To find the dimensions of the rectangle with the maximum area that fits above the x-axis and below the graph of y=e^(-x^2), we can use calculus to solve the problem.
Step 1: Define the problem
We want to find the dimensions of a rectangle that has the maximum area and fits above the x-axis and below the graph of y=e^(-x^2).
Step 2: Formulate the equation for the area of the rectangle
The area of a rectangle is the product of its length and width. Let's assume the length of the rectangle is 2x, and the width is 2y. Therefore, the area of the rectangle can be expressed as A = 4xy.
Step 3: Express y in terms of x
We know that the rectangle should be above the x-axis and below the graph of y=e^(-x^2). So, the lower boundary is y=0, and the upper boundary is y=e^(-x^2). We can express y in terms of x by taking the square root of both sides: y = sqrt(e^(-x^2)).
Step 4: Express the area function in terms of x
Substitute y with sqrt(e^(-x^2)) in the area equation: A = 4x(sqrt(e^(-x^2))). Simplify this expression.
Step 5: Find the derivative of the area function
Differentiate the area function with respect to x to find the derivative. This step helps us identify critical points where the derivative equals zero or does not exist.
Step 6: Find the critical points
Set the derivative equal to zero and solve for x to find the critical points.
Step 7: Determine the maximum area
Evaluate the area function at the critical points and endpoints to find the maximum area.
By following these steps, you should be able to find the dimensions of the rectangle with the maximum area that fits above the x-axis and below the graph of y=e^(-x^2).