You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway-to-the-West arch. This monument rises to a height of 192 m. You estimate your line of sight with the top of the arch to be 7.31 ° above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?
Assuming the arch is a perfect triangle, the distance from the base of the arch can be calculated using the law of sines. The angle of 7.31° is opposite the side of the triangle with length 192 m. The other two sides of the triangle are the distance from the base of the arch and the line of sight.
Distance = (192 m * sin(7.31°)) / sin(90°)
Distance = 0.9 km
To determine the distance from the base of the arch, you can use trigonometry and the angle of elevation. Here's how you can calculate it:
1. Start by drawing a right-angled triangle. The horizontal line represents the distance between you and the base of the arch, and the vertical line represents the height of the arch.
___________
/|
/ |
height / | distance
/ |
/ |
/_____|
angle θ
2. Identify the values you have: the height of the arch is 192 m, and the angle of elevation (angle θ) is 7.31°.
3. Now, you need to find the distance using the tangent function, which relates the opposite side (height) and the adjacent side (distance). The tangent of an angle is equal to the opposite side divided by the adjacent side.
tan(θ) = height / distance
4. Rearrange the equation to solve for distance:
distance = height / tan(θ)
5. Convert the height from meters to kilometers by dividing it by 1000.
height = 192 m / 1000 = 0.192 km
6. Now, substitute the values into the equation:
distance = 0.192 km / tan(7.31°)
7. Use a calculator to calculate the tangent of 7.31°:
tan(7.31°) ≈ 0.1272
8. Substitute the value of tan(7.31°) into the equation to find the distance:
distance ≈ 0.192 km / 0.1272
9. Calculate the result using a calculator:
distance ≈ 1.509 km
Therefore, you are approximately 1.509 kilometers away from the base of the arch.
To find the distance from the base of the arch, we need to use basic trigonometry. Given the height of the arch (192 m) and the angle of elevation (7.31°), we can use the tangent function.
We can use the formula:
Distance = height / tan(angle)
Plugging in the values:
Distance = 192 m / tan(7.31°)
Using a scientific calculator:
Distance ≈ 192 m / 0.1272
Distance ≈ 1509.43 m
Since the question asks for the distance in kilometers, we need to convert meters to kilometers.
Distance ≈ 1509.43 m / 1000
Distance ≈ 1.51 km
Therefore, you are approximately 1.51 kilometers away from the base of the arch.