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nasty-shy [4]
3 years ago
14

A statue is mounted on top of a 21 foot hill. From the base of the hill to where you are standing is 57feet and the statue subte

nds an angle of 7.1° to where you are standing. Find the height of the statue.

Mathematics
2 answers:
Mariulka [41]3 years ago
8 0

Answer:1.1405 foot

Step-by-step explanation:

Let height of statue be h and angle subtended by base of statue is \theta

from diagram

tan\theta =\frac{21}{57}

and tan\left ( \theta +7.1\right )=\frac{21+h}{57}

and we know tan\left ( A+B\right )=\frac{tanA+tanB}{1-tanAtanB}

using above formula

\frac{tan\theta +tan7.1}{1-tan\theta tan7.1}=\frac{21+h}{57}

57\left ( 0.3684+tan7.1\right )=\left ( 21+h\right )\left ( 1-0.3684\times tan7.1\right )

21+tan7.1=\left ( 21+h\right )\left ( 0.95411\right )

h=1.1405 foot

puteri [66]3 years ago
5 0

Please find the attached diagram for a better understanding of the question.

As we can see from the diagram,

RQ = 21 feet = height of the hill

PQ = 57 feet = Distance between you and the base of the hill

SR= h=height of the statue

\angle SPR=7.1^0=Angle subtended by the statue to where you are standing.

\angle x=\angleRPQ which is unknown.

Let us begin solving now. The first step is to find the angle \angle x which can be found by using the following trigonometric ratio in \Delta PQR:

tan(x)=\frac{RQ}{PQ}=\frac{21}{57}

Which gives x to be:

x=tan^{-1} (\frac{21}{57})\approx 20.22^{0}

Now, we know that \angle x and \angle SPR will get added to give us the complete angle \angle SPQ in the right triangle \Delta PQS.

We can again use the tan trigonometric ratio in \Delta PQS to solve for the height of the statue, h.

This can be done as:

tan(\angle SPQ)=\frac{SQ}{PQ}

tan(7.1^0+20.22^0)=\frac{SR+RQ}{PQ}

tan(27.32^0)=\frac{h+21}{57}

\therefore h+21=57\times tan(27.32^0)

h\approx8.45 feet

Thus, the height of the statue is approximately, 8.45 feet.

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