Tamika leans a 30-foot ladder against a wall so that it forms an angle of 65degrees ∘ with the ground. How high up the wall does the ladder reach? Round your answer to the nearest hundredth of a foot if necessary.
Question
Tamika leans a 30-foot ladder against a wall so that it forms an angle of 65degrees ∘ with the ground. How high up the wall does the ladder reach? Round your answer to the nearest hundredth of a foot if necessary.
Solution
To solve this problem, we can use the sine function from 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 is the hypotenuse, the wall is the opposite side, and the ground is the adjacent side. We are trying to find the height up the wall, which is the length of the opposite side.
The formula for sine is:
sin(angle) = opposite/hypotenuse
We can rearrange this formula to solve for the opposite side:
opposite = sin(angle) * hypotenuse
Substituting the given values:
opposite = sin(65 degrees) * 30 feet
Using a calculator, sin(65 degrees) is approximately 0.9063.
So,
opposite = 0.9063 * 30 feet = 27.19 feet
Therefore, the ladder reaches approximately 27.19 feet up the wall.
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