Problem 35 - 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 long is the wire \(a ?\)

Step-by-Step Solution

Answer

The wire is approximately 1122.47 feet long.

Step 1: Understanding the Problem

We have a right triangle formed by the wire, the height of 195 feet, and the ground. The wire connects the top of the 195 feet height to the ground. We need to find the length of this wire, which we'll call \( a \).

Step 2: Identify Relevant Angles

We are given two angles: the angle of inclination from the point where the basket would touch the ground to the base of the launch pad (1°), and from the same point to the point where the wire is attached (10°). The difference in these angles is important for identifying the angle that relates to the wire.

Step 3: Calculate the Angle Related to the Wire

The angle associated with the wire is the difference between the total angle to the top of the wire (\(10^{\circ}\)) and the angle to the base (\(1^{\circ}\)). This gives us \(\theta = 10^{\circ} - 1^{\circ} = 9^{\circ}\).

Step 4: Apply the Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, \(\tan(9^{\circ}) = \frac{195}{b}\), where \(b\) is the adjacent side along the ground.

Step 5: Solve the Tangent Equation

To find \(b\), rearrange the equation: \(b = \frac{195}{\tan(9^{\circ})}\). Calculate \(b\) to use later in finding \(a\).

Step 6: Use the Sine Function to Find the Wire Length

Use the sine function with the total wire angle \(10^{\circ}\): \(\sin(10^{\circ}) = \frac{195}{a}\). Rearrange the equation to solve for \(a\): \(a = \frac{195}{\sin(10^{\circ})}\).

Step 7: Calculate the Length of the Wire

Calculate \(a\) using the previous step: \(a = \frac{195}{\sin(10^{\circ})} \approx 1122.47\) feet.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This is known as the right angle. In any right triangle, the sides have specific names based on their position relative to the right angle: the side opposite the right angle is the hypotenuse, while the other two sides are known as the adjacent and opposite sides.

Right triangles are the basis for many trigonometric relationships and are essential in solving problems involving distances or heights, like the problem involving the escape basket here. By identifying the right triangle in a problem, you can apply various trigonometric functions to find missing measurements easily.

Angle of Inclination

The angle of inclination is the angle formed by a horizontal line and the line in question, typically a slope or an incline. It describes the steepness of a slope and is commonly used in engineering and physics to assess structures like ramps, trails, and, in our example, the wire for the escape basket.

In the problem we're examining, two angles of inclination are given from a common point on the ground to two distinct points on the launch pad. These angles help differentiate the base of the launch pad from the higher point where the wire is attached. Calculating the difference between these angles helps us find the specific angle for analyzing the wire's path.

Tangent Function

The tangent function in trigonometry relates the angle in a right triangle to the lengths of the opposite and adjacent sides. It is defined as \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \). This function is particularly useful when you know the height and need to find the base of a triangle, like in our escape basket scenario.

• When given a vertical height and an angle, you can use the tangent function to calculate the horizontal distance, or adjacent side.
• It is effective for problems that involve slope and distance calculations.

By rearranging the equation and solving for the required side, you can fill in the missing dimension of the triangle, aiding in the design and safety requirements of the setup, such as determining the escape wire's length across a distance.

Sine Function

The sine function is another fundamental tool in trigonometry. It relates an angle in a right triangle to the lengths of the hypotenuse and the opposite side. Mathematically, it is expressed as \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).

In scenarios where the problem involves finding the length of the hypotenuse, such as the escape wire, the sine function becomes extremely handy. You use the known vertical height (opposite side) and the angle from the horizontal to determine the hypotenuse's length—or in this case, the entire wire.

• By rearranging the sine formula, you can solve for the hypotenuse: \( a = \frac{195}{\sin(10^{\circ})} \).
• This approach allows you to find the wire’s actual stretch over the ground efficiently.

Using the sine function helps bridge the gap between theoretical geometric angles and real-world applications, ensuring safe and accurate implementation.