A 20 feet ladder is leaning against the outside of the house while a painter is painting. The ladder makes an angle of 72 with the ground. What is the distance between the foot of the ladder and the house? Round your answer to the nearest foot.
To find the distance between the foot of the ladder and the house, we can use trigonometry. In this case, we have the angle and the length of the ladder, and we need to find the adjacent side, which represents the distance between the foot of the ladder and the house.
We can use the cosine function, which relates the adjacent side to the hypotenuse and the angle. The cosine function is defined as:
cos(angle) = adjacent / hypotenuse
In this case, the angle is 72 degrees and the hypotenuse is the length of the ladder, which is 20 feet. Let's substitute those values into the formula:
cos(72) = adjacent / 20
To find the adjacent side, we can multiply both sides of the equation by 20:
20 * cos(72) = adjacent
Using a calculator, we can find the value of cos(72) is approximately 0.309. Multiply this by 20:
20 * 0.309 = 6.18
Therefore, the distance between the foot of the ladder and the house is approximately 6.18 feet (rounding to the nearest foot).
To find the distance between the foot of the ladder and the house, we need to use trigonometry. Specifically, we will use the sine function, since we have the opposite and hypotenuse sides of a right triangle. Here's how to do it step by step:
1. Draw a diagram: You have a right triangle with the ladder as the hypotenuse, the distance between the foot of the ladder and the house as the opposite side, and the ground as the adjacent side.
2. Identify the known values: The length of the ladder is 20 feet and the angle between the ladder and the ground is 72 degrees.
3. Use the sine function: The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, we have:
sin(72) = opposite / hypotenuse
sin(72) = distance / 20
4. Solve for the distance: Rearrange the equation to solve for the distance:
distance = 20 * sin(72)
5. Calculate the distance: Use a calculator to evaluate the sine of 72 degrees and multiply it by 20:
distance ≈ 19.33 feet
So, the distance between the foot of the ladder and the house is approximately 19.33 feet, rounded to the nearest foot.