Questions: If a car is climbing a hill at a 15° angle, how many horizontal feet will the car travel every 1,000 feet of road if the car gains 259 feet of vertical feet? Round your answer to the nearest foot.

Solution Steps

To solve the problem, we first need to convert the angle of the hill from degrees to radians. The angle given is \(15^\circ\). The conversion from degrees to radians is done using the formula:

\[ \text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180} \]

Substituting the given angle:

\[ \text{angle in radians} = 15 \times \frac{\pi}{180} \approx 0.2618 \]

We use the tangent function to relate the vertical gain and the horizontal distance. The tangent of an angle in a right triangle is the ratio of the opposite side (vertical gain) to the adjacent side (horizontal distance). Therefore, we have:

\[ \tan(\theta) = \frac{\text{vertical gain}}{\text{horizontal distance}} \]

Rearranging for horizontal distance:

\[ \text{horizontal distance} = \frac{\text{vertical gain}}{\tan(\theta)} \]

Substituting the known values:

\[ \text{horizontal distance} = \frac{259}{\tan(0.2618)} \approx 966.6012 \]

Finally, we round the calculated horizontal distance to the nearest foot:

\[ \text{horizontal distance rounded} = 967 \]

Final Answer