Question

A trapezoid has bases of 8 and one third
yd and 10 and one fifth
​yd, and a height of 3 and three fourths
yd. Compose the trapezoid into a parallelogram. What is the area of this​ trapezoid?

Answer 1

To find the area of a trapezoid, you can use the formula:

\[
\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h
\]

where \(b_1\) and \(b_2\) are the lengths of the two bases, and \(h\) is the height.

1. **Convert the base lengths and height into improper fractions:**

- For the first base \(b_1 = 8 \frac{1}{3}\) yd:
\[
8 \frac{1}{3} = \frac{25}{3} \text{ yd}
\]

- For the second base \(b_2 = 10 \frac{1}{5}\) yd:
\[
10 \frac{1}{5} = \frac{51}{5} \text{ yd}
\]

- For the height \(h = 3 \frac{3}{4}\) yd:
\[
3 \frac{3}{4} = \frac{15}{4} \text{ yd}
\]

2. **Calculate the area of the trapezoid:**

Using the formula provided,
\[
\text{Area} = \frac{1}{2} \times \left(\frac{25}{3} + \frac{51}{5}\right) \times \frac{15}{4}
\]

First, find a common denominator for \( \frac{25}{3} \) and \( \frac{51}{5} \). The least common multiple of 3 and 5 is 15.

- Convert \( \frac{25}{3} \):
\[
\frac{25}{3} = \frac{25 \times 5}{3 \times 5} = \frac{125}{15}
\]

- Convert \( \frac{51}{5} \):
\[
\frac{51}{5} = \frac{51 \times 3}{5 \times 3} = \frac{153}{15}
\]

Now add the two fractions:
\[
\frac{125}{15} + \frac{153}{15} = \frac{278}{15}
\]

Substitute into the area formula:
\[
\text{Area} = \frac{1}{2} \times \frac{278}{15} \times \frac{15}{4}
\]

Simplifying:
\[
\text{Area} = \frac{1}{2} \times \frac{278 \times 15}{15 \times 4}
\]

The \(15\) cancels out:
\[
\text{Area} = \frac{1}{2} \times \frac{278}{4}
\]

Simplifying \(\frac{278}{4}\):
\[
\frac{278}{4} = 69.5
\]

Now calculate the area:
\[
\text{Area} = \frac{1}{2} \times 69.5 = 34.75 \text{ yd}^2
\]

Therefore, the area of the trapezoid is:

\[
\boxed{34.75} \text{ yd}^2
\]