A 16 ft. ladder is leaning against a house. If the ladder reaches 12 ft. up the wall of the house, find the angle (to the nearest degree) that the ladder makes with the ground.
Question
A 16 ft. ladder is leaning against a house. If the ladder reaches 12 ft. up the wall of the house, find the angle (to the nearest degree) that the ladder makes with the ground.
Solution
To solve this problem, we can use trigonometry. Specifically, we'll use the cosine function, which is defined as the adjacent side divided by the hypotenuse in a right triangle.
Step 1: Identify the sides of the triangle. The ladder forms the hypotenuse of the triangle, the height up the wall is the adjacent side, and the distance from the base of the house to the base of the ladder is the opposite side.
Step 2: Use the Pythagorean theorem to find the length of the opposite side. The Pythagorean theorem is a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the other two sides. We know that the ladder (c) is 16 ft and the height up the wall (a) is 12 ft. So, we can solve for b:
12^2 + b^2 = 16^2 144 + b^2 = 256 b^2 = 256 - 144 b^2 = 112 b = sqrt(112) = approximately 10.58 ft
Step 3: Use the cosine function to find the angle. The cosine of an angle is equal to the adjacent side divided by the hypotenuse. So, we have:
cos(theta) = adjacent/hypotenuse cos(theta) = 12/16 cos(theta) = 0.75
Step 4: Use the inverse cosine function to find the angle. The inverse cosine of 0.75 is approximately 41.41 degrees.
So, the ladder makes an angle of approximately 41 degrees with the ground.
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