The Correct Answer and Explanation is:
The correct answer is 19.1 ft.
This problem can be solved using trigonometry by recognizing that the ladder, the building, and the ground form a right-angled triangle. The goal is to find the height h, which is the side opposite the given angle.
First, we must identify the components of the right triangle in relation to the 73-degree angle. The 20-foot ladder is the longest side of the triangle, positioned opposite the right angle, which makes it the hypotenuse. The height h is the side directly across from the 73-degree angle, making it the opposite side. The distance b along the ground is the side next to the 73-degree angle, making it the adjacent side.
To find the height h, we need a trigonometric function that relates the side we want to find (opposite) with the side we know (hypotenuse). The mnemonic SOH CAH TOA helps us choose the correct function. SOH stands for Sine equals Opposite over Hypotenuse. This is the perfect choice for our problem.
The formula is:
sin(angle) = Opposite / Hypotenuse
Plugging in the known values from the problem:
sin(73°) = h / 20
From the provided table of values, we know that sin(73°) is approximately 0.9563. We can substitute this value into our equation:
0.9563 = h / 20
To solve for h, we need to isolate it. We can do this by multiplying both sides of the equation by 20:
h = 20 * 0.9563
Performing the multiplication gives us:
h ≈ 19.126 feet
When we compare this result to the available options, the value 19.126 ft is closest to 19.1 ft. Therefore, the approximate height of the ladder on the building is 19.1 feet.
