A hill is inclined at 18 degrees to the horizontal. It runs down to the beach so its base is at sea level. (a) If I walk 1.2 km up the hill, what is my height above sea level? (b... A hill is inclined at 18 degrees to the horizontal. It runs down to the beach so its base is at sea level. (a) If I walk 1.2 km up the hill, what is my height above sea level? (b) If I am 500 meters above sea level, how far have I walked up the hill?
Understand the Problem
This problem involves trigonometry, where we are given the angle of inclination of a hill and some distances related to walking up the hill. We need to use trigonometric functions (sine) to find the height above sea level in part (a) and the distance walked up the hill in part (b). Part (a) asks us to find the height I am above sea level if I walk 1.2km up the hill. Part (b) asks us to find the distance walked up the hill given that the height is 500m above sea level.
Answer
a) $370.8$ m b) $1618.12$ m
Answer for screen readers
a) The height above sea level is approximately $370.8$ meters. b) The distance walked up the hill is approximately $1618.12$ meters.
Steps to Solve
- Part (a): Define variables and recall trigonometric relationships
Define $h$ as the height above sea level and $d$ as the distance walked up the hill. The angle of inclination is given as $18^\circ$. The relationship between the height, distance, and angle is given by the sine function: $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $$ In this case, $\sin(18^\circ) = \frac{h}{d}$.
- Part (a): Calculate height above sea level
We are given $d = 1.2 \text{ km} = 1200 \text{ m}$. We need to find $h$. Using the sine relationship: $$ h = d \cdot \sin(18^\circ) $$ $$ h = 1200 \cdot \sin(18^\circ) $$ $$ h \approx 1200 \cdot 0.3090 $$ $$ h \approx 370.8 \text{ m} $$
- Part (b): Find the distance walked up the hill
We are given $h = 500 \text{ m}$ and need to find $d$. Using the sine relationship: $$ \sin(18^\circ) = \frac{h}{d} $$ $$ d = \frac{h}{\sin(18^\circ)} $$ $$ d = \frac{500}{\sin(18^\circ)} $$ $$ d \approx \frac{500}{0.3090} $$ $$ d \approx 1618.12 \text{ m} $$
a) The height above sea level is approximately $370.8$ meters. b) The distance walked up the hill is approximately $1618.12$ meters.
More Information
The sine function relates the angle of inclination to the ratio of the height and the distance walked along the hill. This relationship allows us to calculate either the height or the distance if we know the other.
Tips
- Forgetting to convert kilometers to meters in part (a) could lead to an incorrect answer.
- Using the wrong trigonometric function (e.g., cosine or tangent) instead of sine.
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