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 7.37 ° above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?
Ht. = Y = 192 m.
Dist. = X m.
A = 7.37o.
tan7.37 = Y/X = 193/X
X = 193/tan7.37 = 1492 m. = 1.492 km.
To find the distance to the base of the arch, we can use trigonometry. We can use the tangent function since we have the opposite side (height of the arch) and the adjacent side (distance to the base of the arch).
The tangent of an angle is equal to the opposite side divided by the adjacent side:
tan(angle) = opposite / adjacent
In this case, the angle is 7.37° and the opposite side is the height of the arch (192 m). We want to find the adjacent side, which represents the distance to the base of the arch.
So we can rearrange the formula to solve for the adjacent side:
adjacent = opposite / tan(angle)
Plugging in the values, we get:
adjacent = 192 m / tan(7.37°)
Now we need to convert the distance to kilometers. 1 kilometer is equal to 1000 meters, so we divide the distance in meters by 1000 to get the distance in kilometers.
adjacent (in kilometers) = (192 m / tan(7.37°)) / 1000
Using a calculator, we find that the distance to the base of the arch is approximately 6.676 kilometers.
To find the distance from the base of the arch, we can use trigonometry. We will use the tangent function as follows:
tan(angle) = opposite / adjacent
In this case, the opposite side is the height of the arch (192 m), and the angle is 7.37°. We need to find the adjacent side, which will be the distance from the base of the arch.
Therefore, we can rearrange the equation to solve for the adjacent side:
adjacent = opposite / tan(angle)
Plugging in the values:
adjacent = 192 m / tan(7.37°)
Before calculating the tangent, we need to convert the angle to radians because most calculators work in radians. To convert degrees to radians, we use the following formula:
radians = degrees * pi / 180
With that in mind, we get:
radians = 7.37° * π / 180 ≈ 0.1281 radians
Now, we can calculate the tangent:
adjacent = 192 m / tan(0.1281 radians) ≈ 8682.6 m
To convert this distance to kilometers, we divide by 1000:
distance in kilometers = 8682.6 m / 1000 ≈ 8.68 km
Therefore, you are approximately 8.68 kilometers away from the base of the arch.