Question

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

Answer 1

Answer:1.1405 foot

Explanation:

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

from diagram

`tan\theta =(21)/(57)`

and
`tan\left ( \theta +7.1\right )=(21+h)/(57)`

and we know
`tan\left ( A+B\right )=(tanA+tanB)/(1-tanAtanB)`

using above formula

`(tan\theta +tan7.1)/(1-tan\theta tan7.1)=(21+h)/(57)`

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

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

h=1.1405 foot

Answer 2

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)=(RQ)/(PQ)=(21)/(57)`

Which gives x to be:

`x=tan^(-1) ((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)=(SQ)/(PQ)`

`tan(7.1^0+20.22^0)=(SR+RQ)/(PQ)`

`tan(27.32^0)=(h+21)/(57)`

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

`h\approx8.45 feet`

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