A long rectangle sheet of metal, 15 inches wide is to be made into a rain gutter by turning up equal amounts on each side so that they are perpendicular to the sheet. How many inches should be turned up in order to give the gutter its greatest capacity (i.e. maximize the costal-sectional area).
height = x
width = (15-2x)
A = x(15-2x) = 15 x - 2 x^2
at max or min dA/dx = 0
= 15 - 4 x
so
x = 15/4 (wish they had given us 16 :)
To find how many inches should be turned up to maximize the gutter's capacity, we need to consider the costal-sectional area. Let's break down the problem step by step:
1. Visualize the rectangle sheet of metal: Draw a rectangle representing the metal sheet, with a width of 15 inches and an unknown length.
2. Identify the sides to be turned up: The problem states that equal amounts on each side of the sheet should be turned up perpendicular to it. Let's label the length of each turned-up side as "x" inches.
3. Calculate the dimensions of the gutter: When the sides are turned up, the width of the gutter would remain the same, i.e., 15 inches. However, the length of the gutter would be the original length minus the length of the turned-up sides (2x), which we can label as "L."
4. Determine the coastal-sectional area: The coastal-sectional area of the gutter is found by multiplying the width by the length, i.e., A = width * length. In this case, the coastal-sectional area would be A = 15 * L.
5. Express the coastal-sectional area in terms of x: Since we know the length of the gutter is "L = original length - 2x," we can substitute L in the equation for the coastal-sectional area. This gives us A = 15 * (original length - 2x).
6. Maximize the coastal-sectional area: To find the maximum coastal-sectional area, we need to differentiate the equation for A with respect to x and set it equal to zero. By doing this, we can solve for the value of x that maximizes A.
7. Solve for x: Differentiate A = 15 * (original length - 2x) with respect to x, and set it equal to zero to find the critical point. Solve for x in this equation to determine the value that maximizes the coastal-sectional area.
8. Substitute the value of x into the equation for A to find the maximum coastal-sectional area.
By following these steps, you can determine the value of x that should be turned up to give the gutter its greatest capacity or maximum coastal-sectional area.