Question

A piece of rectangular sheet metal is 20 inches wide. Its is to be made into a rain gutter by turning up the edges to form parallel sides. let x represent the length of each of the parallel sides. a) Give the restrictions on x b) Determine a function A that gives the area of the cross section of the gutter. C) for what value of x will A be a maximum (and thus maximize the amount of water that the gutter will hold) What is the max area?

Answer 1

The restrictions on x are x ≤ 10. The function A that gives the area of the cross section of the gutter is A = 20x square inches. The function A does not have a maximum value.

Step-by-step explanation:

To determine the restrictions on x, we need to consider the width of the sheet metal and how it will be folded to form the gutter.

Since the sheet metal is 20 inches wide, the sum of the widths of the parallel sides (2x) must be less than or equal to 20 inches.

This gives the inequality 2x ≤ 20. We can divide both sides of the inequality by 2 to solve for x: x ≤ 10. Therefore, the restriction on x is x ≤ 10.

To determine the function A that gives the area of the cross section of the gutter, we need to consider the dimensions of the gutter. The width of the gutter is the same as the width of the sheet metal, which is 20 inches.

The length of the parallel sides is represented by x.

The area of the cross section of the gutter is equal to the product of the width and the length of the parallel sides, which is A = 20x square inches.

To find the value of x that will maximize the area of the cross section (and thus maximize the amount of water the gutter can hold), we can take the derivative of the function A with respect to x and set it equal to zero. The derivative of A with respect to x is dA/dx = 20.

Setting this equal to zero gives 20 = 0, which is not possible.

Therefore, the function A does not have a maximum value.

Answer 2

Turning up the edges of the sheet metal will result in the 20 inches of metal being divided into x inches of the first side, then the unknown width of the gutter (let's call it "y") and then again x inches for the second side.

As a picture: x|______|x/y

As a formula: 20 = x + y + x = 2x + y

Resolving this to y we get: 20 - 2x = y

and switching it around: y = 20 - 2x,

Now for part b) of your question:
The area of the gutter's cross-section is its width (y) multiplied with its height (x).

A(x,y) = x * y
If we use our result from a) to eliminate y then we can see that
A(x) = x * y = x * (20-2x) = 20x-2x^2
So I'd say that the answer to part b) should be: A(x) = 20x - 2x^2