The problem asks for the length of the shorter base of a trapezoid, given its area (77 square feet), the length of the longer base (13 feet), and the height (7 feet).

1. Problem Description

The problem asks for the length of the shorter base of a trapezoid, given its area (77 square feet), the length of the longer base (13 feet), and the height (7 feet).

2. Solution Steps

The formula for the area $A$ of a trapezoid is given by:

$A = \frac{1}{2}(b_1 + b_2)h$

where $b_1$ and $b_2$ are the lengths of the two bases, and $h$ is the height.

We are given $A = 77$, $b_1 = 13$, and $h = 7$. We want to find $b_2$.

Substitute the given values into the formula:

$77 = \frac{1}{2}(13 + b_2)(7)$

Multiply both sides by 2:

$154 = (13 + b_2)(7)$

Divide both sides by 7:

$22 = 13 + b_2$

Subtract 13 from both sides:

$b_2 = 22 - 13$

$b_2 = 9$

So the length of the shorter base is 9 feet.

3. Final Answer

9 ft