A rain gutter is made from sheets of aluminum that are 16 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross- sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?

Depth of the rain gutter is 8 inches

Step-by-step explanation:

Let’s assume ‘x’ is the depth of the rain gutter

Then the width of the rain gutter can be written as 16 - 2x

Cross sectional area

A = depth x width

Substitute values

A = x*(16 - 2x)

A = 16x – 2x^2

Now according to axis of symmetry for maximum area x = -b/2a

x = -16/2*(-2)

x = 4 inches depth of rain gutter, substitute the value of x to get

Width of rain gutter 16 – 2(4) = 8 inches

Area of the rain gutter for maximum water flow

A = 4 * 8

A = 32 square inch.

Part A:
y=∛x satisfies the table values for x and y. This can also be written y³=x, but y=f(x)=∛x is the way it’s expressed as a function.
Part B:
f(x)=925x+1000, so f(24)=925×24+1000=22200+1000=23200. This represents the value in dollars of renting a house for 24 months (2 years). The coefficient 925 is the rent per month, and 1000 is a downpayment.