To determine the height of the mast given that the lookout can see an island five miles away, we use the formula provided:
[tex]\[ d = \frac{5}{6} \sqrt{h} \][/tex]
where:
- [tex]\( d \)[/tex] is the distance the lookout can see (5 miles),
- [tex]\( h \)[/tex] is the height of the mast which we need to determine.
Here are the steps to find the height [tex]\( h \)[/tex]:
1. Substitute the given distance [tex]\( d \)[/tex] into the formula:
Since [tex]\( d = 5 \)[/tex] miles, we get:
[tex]\[ 5 = \frac{5}{6} \sqrt{h} \][/tex]
2. Solve for [tex]\( \sqrt{h} \)[/tex]:
To isolate [tex]\( \sqrt{h} \)[/tex], multiply both sides of the equation by [tex]\( \frac{6}{5} \)[/tex]:
[tex]\[ 5 \times \frac{6}{5} = \left( \frac{5}{6} \sqrt{h} \right) \times \frac{6}{5} \][/tex]
Simplifying the left side:
[tex]\[ 6 = \sqrt{h} \][/tex]
3. Solve for [tex]\( h \)[/tex]:
To find [tex]\( h \)[/tex], square both sides of the equation:
[tex]\[ (\sqrt{h})^2 = 6^2 \][/tex]
[tex]\[ h = 36 \][/tex]
4. Round to the nearest whole number if necessary:
In this case, [tex]\( h = 36 \)[/tex] is already a whole number.
Thus, the height of the mast is approximately 36 feet.
