Problem 36 - Precalculus

Question

On the launch pad at Kennedy Space Center, there is an escape basket that can hold four astronauts. The basket slides down a wire that is attached 195 feet high, above the base of the launch pad. The angle of inclination measured from where the basket would touch the ground to the base of the launch pad is \(1^{\circ},\) and the angle of inclination from that same point to where the wire is attached is \(10^{\circ}\). NASA. How far from the launch pad does the basket touch the ground? That is, find \(b\).

Step-by-Step Solution

Answer

The basket touches the ground approximately 1229.7 feet from the launch pad.

Step 1: Understanding the Problem

We have a right triangle where the height of the attachment point on the launch pad is 195 feet, the angle of inclination to the top is \(10^{\circ}\), and the angle to the base is \(1^{\circ}\). We need to find the distance \(b\), where the basket touches the ground.

Step 2: Identify the Relevant Angles

The right triangle includes a small angle at the base of \(1^{\circ}\) and a larger angle at the top of \(10^{\circ}\). Therefore, the remaining angle on the ground side of the right triangle is the angle \(10^{\circ} - 1^{\circ} = 9^{\circ}\).

Step 3: Use Trigonometric Functions

We can use the tangent function to find the distance \(b\). The tangent of an angle in a right triangle is the opposite side divided by the adjacent side. Here, the opposite side is the height of 195 feet and the adjacent side is \(b\).

Step 4: Set Up the Tangent Equation

Using the tangent of the \(9^{\circ}\) angle, we set up the equation: \(\tan(9^{\circ}) = \frac{195}{b}\).

Step 5: Solve for \(b\)

Rearrange the equation to solve for \(b\): \(b = \frac{195}{\tan(9^{\circ})}\). Calculate \(\tan(9^{\circ})\) using a calculator and solve: \(b \approx \frac{195}{0.1584} \approx 1229.7\) feet.

Key Concepts

Understanding Right Triangles

Right triangles are a fundamental concept in trigonometry. They have one angle that is exactly 90 degrees. The two sides forming this right angle are called the legs, and the side opposite the right angle is known as the hypotenuse. Each right triangle follows the Pythagorean theorem:\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse.

In the exercise, we have a right triangle involving the launch pad, wire, and the ground. The height at which the wire is attached acts as one side of this right triangle. The key point to remember about right triangles in trigonometry is the way angles and sides are interrelated. Knowing any two dimensions (sides or angles) allows you to calculate the third.

The Tangent Function in Right Triangles

The tangent function is one of the three primary trigonometric functions, often abbreviated as \( \tan \). It relates to the angles and sides of a right triangle. Specifically, \( \tan(\theta) \) in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side:

\[\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\]
In our problem, \( \tan(\theta) \) is used to find the distance \( b \), where the basket touches the ground. By rearranging the formula to solve for \( b \):

• Identify the angle \( \theta = 9^{\circ} \)
• Know that the opposite side is the height, 195 feet
• Adjacent side is what we want to find

This relationship is crucial because it builds a link between angles and physical measurements, like heights or distances.

Angles of Inclination

Angles of inclination help us understand how steep a slope is. In trigonometry, they are the angles formed by the horizontal ground line and some line that goes upwards, like the wire attached to the basket in our exercise. The angle of inclination was critical in setting up the right triangle and calculating distances.

In the given problem, the basket's escape path is a line that forms a \( 10^{\circ} \) inclination to the wire attachment point and a \( 1^{\circ} \) angle at the base with the ground. The difference between these angles, \( 9^{\circ} \), is the critical angle necessary for using the tangent function. Understanding angles of inclination is essential, as they determine how far one needs to move horizontally, given a specific elevation change. Angles like these are common in everyday engineering and physics problems related to slopes, ramps, and inclines.