A 26-foot ladder leaning against the side of a house makes a 60∘60∘ angle with the ground. How far up the side of the house does the ladder reach?
Question
A 26-foot ladder leaning against the side of a house makes a 60° angle with the ground. How far up the side of the house does the ladder reach?
Solution
To solve this problem, we can use the sine function in trigonometry. The sine of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse.
In this case, the ladder acts as the hypotenuse of a right triangle, the ground acts as one leg, and the side of the house acts as the other leg. We are trying to find the length of the leg along the side of the house, which is the opposite side of the 60-degree angle.
The sine of a 60-degree angle is √3/2. So, we can set up the equation:
sin(60) = opposite/hypotenuse √3/2 = opposite/26
To solve for the opposite side, we multiply both sides by 26:
opposite = 26 * √3/2
This simplifies to:
opposite = 13√3 feet
So, the ladder reaches 13√3 feet up the side of the house.
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