Question

Can someone please answer. There is one problem. There's s picture. Thank you!

Answer 1

The correct answer is the first option, 23.1 miles.

Knowing that the sum of the triangle's interior angles should equal a measure of 180°, you can start off by solving for the missing interior angle, near the library.

Since this is a right triangle (meaning one of the angles is 90°), and the other angle is given (57°), simply subtract these two values from 180° to get $180-(57+90)=33\textdegree.$

Next, use the Law of Sines (stated below) which will allow you to solve for the missing distance from the radio tower to the library in miles, using the angle measures and height from the tower to the balloon that you've found.

$\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$

For this problem, we'll let the height of the hot air balloon be $a=15$ and its respective opposite angle be $A=33\textdegree$, and let the missing x-distance be $b$, with its respective opposite angle $B=57\textdegree$. Note, we're solving for $b$.

$\frac{\sin(33)}{15} = \frac{\sin(57)}{b} \\ \\ b\times sin(33)=15\times \sin(57) \\ \\ b=\frac{15\times \sin(57)}{sin(33)} \\ \\ b\approx \frac{15\times 0.84}{.54} \\ \\ b\approx \frac{12.6}{.54} \\ \\ b\approx \frac{12.6}{.54} \\ \\ b\approx 23.3$

Because there was a lot of approximation in the above operations, the answer doesn't exactly match any of the answer options in the question. However, it's very close to the first option, 23.1 miles, so that must be the correct answer.