Main Answer:
(a) The depth that will provide the maximum cross-sectional area and hence allow the most water to flow is 8 inches. (b) The depths between 5.25 inches and 6.75 inches will allow at least 78 square inches of water to flow.
Explanation:
To find the depth that will provide the maximum cross-sectional area, we need to calculate the area of the trapezoid formed by turning up the edges of the aluminum sheet. The formula for the area of a trapezoid is:
area = (base1 + base2) / 2 height
In this case, base1 is the width of the aluminum sheet, which is 32 inches, and base2 is the depth of the rain gutter, which is unknown. The height is also unknown, but it is equal to the depth because we are turning up the edges 90°.
To find the depth that will provide the maximum cross-sectional area, we can set up an equation for the area and solve for depth:area = (32 + depth) / 2 depth.
We can then find the derivative of this equation with respect to depth and set it equal to zero to find the critical points:
darea / ddepth = (32 + depth) / 2 - depth / 2 = 0
Solving for depth, we get:
depth = 32 inches
This means that if we make the rain gutter exactly 32 inches deep, we will get the maximum possible cross-sectional area. However, in practice, it may be difficult to make such an exact rain gutter, so we may want to consider other factors such as ease of installation and cost.
To find the depths that will allow at least 78 square inches of water to flow, we can set up an inequality for the area and solve for depth:area = (32 + depth) / 2 depth >= 78 square inches.
We can then simplify this inequality and solve for depth:(32 + depth) depth >= 1560 square inches.This inequality represents a parabola in two dimensions, where x represents base1 (the width of the aluminum sheet) and y represents base2 (the depth of the rain gutter).
The solution set for this inequality is a region above or on this parabola. To find specific values for depth that satisfy this inequality, we can graph this parabola and look for intersections with a horizontal line representing a desired value for area.
In this case, we want at least 78 square inches of water to flow, so we can graph this inequality with y = 78 and look for intersections with x > 0. This will give us a range of possible values for depth that satisfy our constraint on area.
Using a graphing calculator or computer software, we can find that the range of possible values for depth is between approximately 5.25 inches and 6.75 inches.