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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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Most popular questions from this chapter

Use the Pythagorean Theorem to solve Exercises 39-46. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. The base of a 20-foot ladder is 15 feet from the house. How far up the house does the ladder reach?

Describe everyday objects that approximate points, lines, and planes.

Use the Pythagorean Theorem to solve Exercises 39-46. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. Picky, Picky, Picky This problem appeared on a high school exit exam: Alex is building a ramp for a bike competition. He has two rectangular boards. One board is six meters and the other is five meters long. If the ramp has to form a right triangle, what should its height be? Students were asked to select the correct answer from the following options: 3 meters; 4 meters; \(3.3\) meters; 78 meters. a. Among the available choices, which option best expresses the ramp's height? How many feet, to the nearest tenth of a foot, is this? Does a bike competition that requires riders to jump off these heights seem realistic? (ouch!) b. Express the ramp's height to the nearest hundredth of a meter. By how many centimeters does this differ from the "correct" answer on the test? How many inches, to the nearest half inch, is this? Is it likely that a carpenter with a tape measure would make this error? c. According to the problem, Alex has boards that measure 5 meters and 6 meters. A 6 -meter board? How many feet, to the nearest tenth of a foot, is this? When was the last time you found a board of this length at Home Depot? (Source: The New York Times, April 24, 2005)

A 200 -foot cliff drops vertically into the ocean. If the angle of elevation of a ship to the top of the cliff is \(22.3^{\circ}\), how far off shore, to the nearest foot, is the ship?

What happens to the volume of a sphere if its radius is doubled?
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