Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

A builder wishes to construct a ramp 24 feet long that rises to a height of \(5.0\) feet above level ground. Approximate the angle that the ramp should make with the horizontal.

Step-by-Step Solution

Answer

The ramp should make an angle of approximately \(12.02^{\circ}\) with the horizontal.

Step 1: Identify the Right Triangle Components

The ramp forms a right triangle with the ground, where the ramp itself is the hypotenuse, which measures 24 feet, and the vertical rise is one of the legs, measuring 5 feet.

Step 2: Use the Sine Function to Find the Angle

The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. For this ramp, we know the opposite side is 5 feet and the hypotenuse is 24 feet. Therefore, we can write: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{24} \]

Step 3: Solve for the Angle Using the Inverse Sine Function

To find the angle \( \theta \), take the inverse sine (\( \sin^{-1} \)) of \( \frac{5}{24} \). This can be done using a calculator:\[ \theta = \sin^{-1}\left(\frac{5}{24}\right) \]

Step 4: Calculate the Angle

Using a calculator, compute \( \theta = \sin^{-1}(0.2083) \approx 12.02^{\circ} \).

Key Concepts

Right Triangle

Understanding right triangles is crucial for solving many trigonometry problems. A right triangle is a triangle that includes one 90-degree angle. This type of triangle has three sides: the hypotenuse, and two legs.

• The hypotenuse is the longest side, found opposite the right angle.
• The legs are the two shorter sides. One leg is adjacent to the angle you are interested in, while the other is opposite to it.

In the ramp problem, the hypotenuse is the 24-foot-long ramp and one leg is 5 feet tall. These components form a right triangle along with the horizontal ground, allowing us to use trigonometric functions to find unknown angles.

Sine Function

The sine function is one of the basic functions in trigonometry. It relates the ratio of the opposite side of a right triangle to its hypotenuse. More formally, for a given angle \( \theta \) in a right triangle, the sine function is defined as:
\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
When dealing with the ramp problem, we use this definition to set up an equation to solve for the angle. Here, the opposite side is 5 feet, and the hypotenuse is 24 feet, giving us the equation:
\[\sin(\theta) = \frac{5}{24}\]
This equation allows us to use the inverse sine function to determine the angle \( \theta \), completing the necessary trigonometric calculation.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles in right triangles when side lengths are known. These functions are essentially the reverse of the standard trigonometric functions like sine, cosine, and tangent.
The inverse sine function, denoted as \( \sin^{-1} \) or arcsin, helps find an angle \( \theta \) given the sine value. Specifically, it computes the angle that corresponds to a particular sine ratio.
In the example of calculating the ramp angle, we use the inverse sine to find \( \theta \):
\[\theta = \sin^{-1}\left(\frac{5}{24}\right)\]
By inputting the calculated ratio into a calculator, you find the angle to be approximately \( 12.02^{\circ} \). Thus, the ramp should form an angle of about 12 degrees with the ground to meet the specified height when 24 feet long.