A car on a straight and level road is approaching a mountain. At some point on the trip, the angle of elevation from the car to the top of the mountain is measured to be 23°29'. Find the height of the mountain to the nearest foot if the car is 16,424.7 feet from the center of the mountain at the base when measuring the angle of elevation. Show your equation and work to solve.
To find the height of the mountain, we can use trigonometry. Let's assume the height of the mountain is h feet.
We have a right triangle formed by the car, the top of the mountain, and the center of the mountain at the base. The angle of elevation (θ) is given as 23°29', and the distance from the car to the center of the mountain (adjacent side) is given as 16,424.7 feet.
In a right triangle, the tangent function relates the angle of elevation to the opposite side (height) and the adjacent side (distance to the base). The formula for tangent is:
tan(θ) = opposite side / adjacent side
Substituting the known values into the equation:
tan(23°29') = h / 16,424.7
Now, we can solve the equation for h by isolating it. Multiply both sides of the equation by 16,424.7:
16,424.7 * tan(23°29') = h
Using a scientific calculator, evaluate tan(23°29') to find its decimal value. Multiply this value by 16,424.7, and you'll get the height of the mountain to the nearest foot.