To determine the height of the mast, given that the lookout can see an island five miles away, we can use the formula [tex]\( d = \frac{5}{6} \sqrt{h} \)[/tex], where [tex]\( d \)[/tex] is the distance in miles that the lookout can see and [tex]\( h \)[/tex] is the height of the mast in feet.
We know that [tex]\( d = 5 \)[/tex] miles in this problem. Let's follow these steps to solve for [tex]\( h \)[/tex]:
1. Start with the formula:
[tex]\[
d = \frac{5}{6} \sqrt{h}
\][/tex]
2. Substitute the given distance:
[tex]\[
5 = \frac{5}{6} \sqrt{h}
\][/tex]
3. Isolate [tex]\(\sqrt{h}\)[/tex]:
To isolate [tex]\(\sqrt{h}\)[/tex], multiply both sides of the equation by [tex]\(\frac{6}{5}\)[/tex]:
[tex]\[
5 \times \frac{6}{5} = \sqrt{h}
\][/tex]
Simplifying this, we get:
[tex]\[
6 = \sqrt{h}
\][/tex]
4. Solve for [tex]\(h\)[/tex]:
To solve for [tex]\(h\)[/tex], square both sides of the equation:
[tex]\[
(6)^2 = h
\][/tex]
[tex]\[
36 = h
\][/tex]
5. Round if necessary:
Since [tex]\(h = 36\)[/tex] is already a whole number, rounding is not needed.
Therefore, the height of the mast is [tex]\( 36 \)[/tex] feet.
The correct answer is:
D. 36 feet
