Question

The angle of elevation from me to the top of a hill is 51 degrees. The angle of elevation from me to the top of a tree is 57 degrees. The tree stands 20 ft off the hill. How high is the top of the tree from the base of the hill?

Answer 1

To find the height of the hill, we can use the tangent function in trigonometry and solve two equations simultaneously.

Step-by-step explanation:

To solve this problem, we can use trigonometry and some basic geometry. Let's assume that the distance from the base of the hill to the top of the hill is represented by the variable 'x'. We can create a right triangle with the hill, tree, and the line of sight from you to the top of the tree. The height of the hill can be found by using the tangent function:

tan(51) = x / h

Similarly, the height of the tree can also be found using the tangent function:

tan(57) = (x + 20) / h

Now, we can solve these two equations simultaneously to find the values of 'x' and 'h', which represent the distance from the base of the hill to the top of the hill and the height of the hill, respectively.

Answer 2

Approximately
`101\; \rm ft` (assuming that the height of the base of the hill is the same as that of the observer.)

Step-by-step explanation:

Refer to the diagram attached.

• Let
  `\rm O` denote the observer.
• Let
  `\rm A` denote the top of the tree.
• Let
  `\rm R` denote the base of the tree.
• Let
  `\rm B` denote the point where line
  `\rm AR` (a vertical line) and the horizontal line going through
  `\rm O` meets.
  `\angle \rm B\hat{A}R = 90^\circ`.

Angles:

• Angle of elevation of the base of the tree as it appears to the observer:
  `\angle \rm B\hat{O}R = 51^\circ`.
• Angle of elevation of the top of the tree as it appears to the observer:
  `\angle \rm B\hat{O}A = 57^\circ`.

Let the length of segment
`\rm BR` (vertical distance between the base of the tree and the base of the hill) be
`x\; \rm ft`.

The question is asking for the length of segment
`\rm AB`. Notice that the length of this segment is
`\mathrm{AB} = (x + 20)\; \rm ft`.

The length of segment
`\rm OB` could be represented in two ways:

• In right triangle
  `\rm \triangle OBR` as the side adjacent to
  `\angle \rm B\hat{O}R = 51^\circ`.
• In right triangle
  `\rm \triangle OBA` as the side adjacent to
  `\angle \rm B\hat{O}A = 57^\circ`.

For example, in right triangle
`\rm \triangle OBR`, the length of the side opposite to
`\angle \rm B\hat{O}R = 51^\circ` is segment
`\rm BR`. The length of that segment is
`x\; \rm ft`.

`\begin{aligned}\tan{\left(\angle\mathrm{B\hat{O}R}\right)} = \frac{\,\rm {BR}\,}{\,\rm {OB}\,} \; \genfrac{}{}{0em}{}{\leftarrow \text{opposite}}{\leftarrow \text{adjacent}}\end{aligned}`.

Rearrange to find an expression for the length of
`\rm OB` (in
`\rm ft`) in terms of
`x`:

`\begin{aligned}\mathrm{OB} &= \frac{\mathrm{BR}}{\tan{\left(\angle\mathrm{B\hat{O}R}\right)}} \\ &= (x)/(\tan\left(51^\circ\right))\approx 0.810\, x\end{aligned}`.

Similarly, in right triangle
`\rm \triangle OBA`:

`\begin{aligned}\mathrm{OB} &= \frac{\mathrm{AB}}{\tan{\left(\angle\mathrm{B\hat{O}A}\right)}} \\ &= (x + 20)/(\tan\left(57^\circ\right))\approx 0.649\, (x + 20)\end{aligned}`.

Equate the right-hand side of these two equations:

`0.810\, x \approx 0.649\, (x + 20)`.

Solve for
`x`:

`x \approx 81\; \rm ft`.

Hence, the height of the top of this tree relative to the base of the hill would be
`(x + 20)\; {\rm ft}\approx 101\; \rm ft`.