To determine the height \( h \) of the observation deck of the Empire State Building given that you can see about 27 miles on a clear day, we will use the provided formula:
[tex]\[
d = \frac{5}{6} \sqrt{h}
\][/tex]
We are given that \( d = 27 \) miles. To find \( h \), we need to rearrange the formula and solve for \( h \).
First, start with the given formula:
[tex]\[
d = \frac{5}{6} \sqrt{h}
\][/tex]
Plug in the given distance \( d = 27 \):
[tex]\[
27 = \frac{5}{6} \sqrt{h}
\][/tex]
To isolate \( \sqrt{h} \), multiply both sides of the equation by \( \frac{6}{5} \):
[tex]\[
27 \times \frac{6}{5} = \sqrt{h}
\][/tex]
Calculate the left side of the equation:
[tex]\[
27 \times \frac{6}{5} = 32.4
\][/tex]
So we have:
[tex]\[
\sqrt{h} = 32.4
\][/tex]
Now, to find \( h \), square both sides of the equation:
[tex]\[
(\sqrt{h})^2 = 32.4^2
\][/tex]
[tex]\[
h = 32.4^2
\][/tex]
Calculate the result:
[tex]\[
h = 1049.76
\][/tex]
Thus, the estimated height \( h \) of the observation deck is approximately 1050 feet.
Comparing this to the given options:
- A. 4.3 miles
- B. 875 feet
- C. 1050 feet
- D. 506 feet
The correct answer is:
C. 1050 feet
