Martin is painting his house and needs to reach a point 9 feet above the ground. He places a 10.5-foot-long ladder against the house so that it forms an angle of with the ground.
a. Describe the triangle that is formed by the ground, the house, and the ladder. Identify the hypotenuse, the adjacent side, and the opposite side.
b. To the nearest tenth of a degree, find the angle the ladder forms with the ground. Show your work.
a. The triangle formed by the ground, the house, and the ladder is a right triangle. In a right triangle, the side opposite the right angle is called the hypotenuse. In this case, the ladder forms the hypotenuse. The side adjacent to the angle formed between the ladder and the ground is called the adjacent side. In this case, the side representing the house is the adjacent side. Lastly, the side opposite to the angle formed between the ladder and the ground is called the opposite side. In this case, the side representing the height Martin needs to reach is the opposite side.
b. We know that the ladder forms an angle with the ground. Let's call this angle θ. To find the angle θ, we can use the inverse trigonometric function cosine, which relates the adjacent side and the hypotenuse in a right triangle:
cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is the length of the house (9 feet) and the hypotenuse is the length of the ladder (10.5 feet). Plugging these values into the equation, we get:
cos(θ) = 9/10.5
Now, we can solve for θ by taking the inverse cosine (or arccosine) of both sides:
θ = arccos(9/10.5)
Calculating this on a calculator, we find that θ is approximately 37.1 degrees. Therefore, to the nearest tenth of a degree, the angle the ladder forms with the ground is 37.1 degrees.