Question

While visiting Yosemite National Forrest, Joe approximated the angle of elevation to the top of a hill to be 25 degrees. After walking 350 ft closer, he guessed that the angle of elevation had increased by 14 degrees. Approximately how tall is the hill?A. 475 feetB. 385 feetC. 608 feetD. 202 feet

Answer 1

The line sketch of the movement and positioning of Joe and the hill are shown below.

Brief description of the sketch made.

From the sketch, point A is where Joe started from heading towards the hill.

The hill is represented by BC and the height is h

25 degrees was the initial angle of elevation.

After moving 350 ft closer to the hill, the angle of elevation increased by 14 degrees to 25 + 14 = 39 degrees.

From triangle BCD,

Using the trigonometric function of tan,

`\begin{gathered} \tan =\frac{opposite}{\text{adjacent}} \\ \tan 39=(h)/(x) \\ 0.8098=(h)/(x) \\ \text{Making h subject of the formula,} \\ h=0.8098x \end{gathered}`

From triangle ABC,

Using the trigonometric function of tan,

`\begin{gathered} \tan =\frac{opposite}{\text{adjacent}} \\ \tan 25=(h)/(350+x) \\ \text{Substituting the value of h gotten from the previous triangle,} \\ \tan 25=(0.8098x)/(350+x) \\ 0.4663=(0.8098x)/(350+x) \\ C\text{ ross multiplying,} \\ 0.4663(350+x)=0.8098x \\ 163.205+0.4663x=0.8098x \\ C\text{ ollecting the like terms,} \\ 163.205=0.8098x-0.4663x \\ 163.205=0.3435x \\ \text{Dividing both sides by 0.3435 to get x,} \\ x=(163.205)/(0.3435) \\ x=475.124ft \\ \\ \text{The height, h of the hill is;} \\ h=0.8098x \\ h=0.8098*475.124 \\ h=384.755ft \\ h\approx385ft \end{gathered}`

Therefore, the height of the hill is 385 feet

The correct answer is option B.