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Chapter 10: Problem 47

A wheelchair ramp is to be built beside the steps to the campus library. Find the angle of elevation of the 23 -foot ramp, to the nearest tenth of a degree, if its final height is 6 feet.

Short Answer

Expert verified
The angle of elevation to the nearest tenth of a degree is approximately 15 degrees.

Step by step solution

01

Labeling the Triangle

Firstly, visualizing and labeling the right triangle is important. The 23-foot ramp is the hypotenuse (\(c\)), and the raise or the final height of the ramp is the opposite side (\(a\)), as it's directly opposite to the angle we are searching for.
02

Setup the Trigonometric Equation

The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, we can set up the equation based on the sine formula, \(\sin(\Theta) = \frac{a}{c}\), where \( \Theta \) is the angle we need to find.
03

Substitute the Values into the Equation

Let's substitute the known values into the equation to get \(\sin(\Theta) = \frac{6}{23}\).
04

Find the Angle

To find \(\Theta\), the angle of elevation, use the arcsine function, which is the inverse of the sine function. So, \(\Theta = \arcsin(\frac{6}{23})\).
05

Convert the Result to Degree Format

Most scientific calculators will give the result in radians. If that's the case, convert the radian measurement to degrees by multiplying by \(\frac{180}{\pi}\).

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