Question

A statue is mounted on top of a 16 foot hill. From the base of the hill to where you are standing is 77 feet and the statue subtends an angle of 13.2° to where you are standing. Find the height of the statue.

Answer 1

Consider right triangle with vertices B - base of the hill, S - top of the statue and Y - you. In this triangle angle B is right and angle Y is 13.2°. If h is a height of the statue, then the legs YB and BS have lengths 77 ft and 16+h ft.

You have lengths of two legs and measure of one acute angle, then you can use tangent to find h:

$\tan 13.2^{\circ}=\dfrac{\text{opposite leg}}{\text{adjacent leg}} =\dfrac{16+h}{77}, \\ \\ 0.2345=\dfrac{16+h}{77},\\ \\ 16+h=0.2345\cdot 77=18.0565,\\ h=18.0565-16=2.0565$ ft.

Answer: the height of the statue is 2.0565 ft.

Answer 2

The height of the statue is 19.8 feet

Further explanation

Firstly , let us learn about trigonometry in mathematics.

Suppose the ΔABC is a right triangle and ∠A is 90°.

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

There are several trigonometric identities that need to be recalled, i.e.

$cosec ~ A = \frac{1}{sin ~ A}$

$sec ~ A = \frac{1}{cos ~ A}$

$cot ~ A = \frac{1}{tan ~ A}$

$tan ~ A = \frac{sin ~ A}{cos ~ A}$

Let us now tackle the problem!

Look at ΔABC in the attachment.

We will use the following formula to find ∠BAC:

tan ∠BAC = opposite / adjacent

$\tan \angle BAC = \frac{BC}{AB}$

$\tan \angle BAC = \frac{16}{77}$

$\angle BAC = tan^{-1} \frac{16}{77}$

$\large {\boxed{ \angle BAC \approx 11.7^o } }$

Look at ΔABD in the attachment.

We will use the following formula to find BD :

tan ∠BAD = opposite / adjacent

$\tan \angle BAD = \frac{BD}{AB}$

$\tan (11.7^o + 13.2^o) = \frac{BD}{77}$

$\tan 24.9^o = \frac{BD}{77}$

$BD = 77 \times \tan 24.9^o$

$\large {\boxed{ BD \approx 35.8 ~ feet } }$

Finally , the height of the statue is equal to CD :

The height of the statue = CD = x

$CD = BD - BC$

$CD = 35.8 - 16$

$\large {\boxed {CD = 19.8 ~ feet } }$

Learn more

• Calculate Angle in Triangle : https://brainly.com/question/12438587
• Periodic Functions and Trigonometry : https://brainly.com/question/9718382
• Trigonometry Formula : https://brainly.com/question/12668178

Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse , Triangle , Fraction , Lowest , Function , Angle