To solve for the height of a trapezoid given the area and the lengths of the two bases, we can use the formula for the area of a trapezoid. The formula for the area [tex]\(A\)[/tex] of a trapezoid is given by:
[tex]\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\][/tex]
where:
- [tex]\(A\)[/tex] is the area,
- [tex]\(b_1\)[/tex] is the length of the first base,
- [tex]\(b_2\)[/tex] is the length of the second base,
- [tex]\(h\)[/tex] is the height.
We are given:
- [tex]\(A = 70.55 \ \text{square feet}\)[/tex],
- [tex]\(b_1 = 11.4 \ \text{feet}\)[/tex],
- [tex]\(b_2 = 5.6 \ \text{feet}\)[/tex].
We need to find [tex]\(h\)[/tex], the height of the trapezoid. First, let's rearrange the area formula to solve for [tex]\(h\)[/tex]:
[tex]\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\][/tex]
Multiply both sides by 2 to isolate the term with [tex]\(h\)[/tex]:
[tex]\[
2A = (b_1 + b_2) \times h
\][/tex]
Next, divide both sides by [tex]\((b_1 + b_2)\)[/tex]:
[tex]\[
h = \frac{2A}{b_1 + b_2}
\][/tex]
Now, substitute the given values into the formula:
[tex]\[
h = \frac{2 \times 70.55}{11.4 + 5.6}
\][/tex]
Calculate the denominator:
[tex]\[
11.4 + 5.6 = 17 \ \text{feet}
\][/tex]
Then, compute the height:
[tex]\[
h = \frac{2 \times 70.55}{17} = \frac{141.1}{17} \approx 8.3 \ \text{feet}
\][/tex]
Therefore, the height of the trapezoid is approximately [tex]\(8.3\)[/tex] feet.
Of the given options:
- [tex]\(4.15 \ \text{ft}\)[/tex]
- [tex]\(4.15 \ \text{ft}^2\)[/tex]
- [tex]\(8.3 \ \text{ft}\)[/tex]
- [tex]\(8.3 \ \text{t}^2\)[/tex]
The correct choice is:
[tex]\[
\boxed{8.3 \ \text{ft}}
\][/tex]
