Problem 15 - 5 Steps to A 5: AP Calculus AB 2021

Question

A plane lifts off from a runway at an angle of \(20^{\circ}\). If the speed of the plane is \(300 \mathrm{mph}\), how fast is the plane gaining altitude?

Step-by-Step Solution

Answer

Step 2: Calculate the speed of the plane in feet per minute Given the speed of the plane as 180 mph, we can now calculate its speed in feet per minute using the conversion formula: Speed (ft/min) = 180 mph * 5280 feet/mile * 1 hour/60 minutes Speed (ft/min) = 180 * 5280 / 60 Speed (ft/min) = 15840 ft/min Step 3: Use the sine function to determine the vertical component of the plane's speed To find the rate of altitude gain, we will use the sine function with the given angle of 15 degrees. The sine function helps us find the vertical component of the plane's speed (altitude gain): Altitude gain = Speed (ft/min) * sin(angle) Step 4: Calculate the altitude gain Since we have calculated the speed of the plane in feet per minute (15840 ft/min) and we know the climbing angle (15 degrees), we can now calculate the altitude gain: Altitude gain = 15840 ft/min * sin(15 degrees) Altitude gain ≈ 15840 * 0.2588 Altitude gain ≈ 4098 ft/min The plane is gaining altitude at a rate of approximately 4098 feet per minute.

Step 1: Convert the speed of the plane to feet per minute

Since we are measuring altitude in feet, we need to convert the plane's speed from miles per hour (mph) to feet per minute (ft/min). There are 5280 feet in a mile and 60 minutes in an hour, therefore we can use the following conversion formula: Speed (ft/min) = Speed (mph) * 5280 feet/mile * 1 hour/60 minutes

Step 2: Calculate the speed of the plane in feet per minute

Given the speed of the plane as 180 mph, we can now calculate its speed in feet per minute using the conversion formula: Speed (ft/min) = 180 mph * 5280 feet/mile * 1 hour/60 minutes Speed (ft/min) = 180 * 5280 / 60 Speed (ft/min) = 15840 ft/min

Step 3: Use the sine function to determine the vertical component of the plane's speed

To find the rate of altitude gain, we will use the sine function with the given angle of 15 degrees. The sine function helps us find the vertical component of the plane's speed (altitude gain): Altitude gain = Speed (ft/min) * sin(angle)

Step 4: Calculate the altitude gain

Since we have calculated the speed of the plane in feet per minute (15840 ft/min) and we know the climbing angle (15 degrees), we can now calculate the altitude gain: Altitude gain = 15840 ft/min * sin(15 degrees) Altitude gain ≈ 15840 * 0.2588 Altitude gain ≈ 4098 ft/min The plane is gaining altitude at a rate of approximately 4098 feet per minute.

Key Concepts

Trigonometry

Trigonometry is a critical branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. This study revolves around functions such as sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are particularly useful because they apply universally to any right triangle. For instance, in our exercise related to a plane taking off at an angle of \(20^{\circ}\), we're most interested in the sine function, which is often used in scenarios involving vertical height in right triangles.

• Sine (\(\sin \theta\)) = Opposite side / Hypotenuse
• Cosine (\(\cos \theta\)) = Adjacent side / Hypotenuse
• Tangent (\(\tan \theta\)) = Opposite side / Adjacent side

This understanding allows us to solve many practical problems involving angles and distances, such as determining how fast the plane gains altitude determined by the speed and angle of ascent.

Unit Conversion

Unit conversion is a fundamental skill in mathematics and the sciences, enabling us to translate measurements into units that are consistent with the problem we're solving. In this exercise, since altitude is often measured in feet, we need to convert the plane's speed from miles per hour (mph) to feet per minute (ft/min).Here's how you perform the conversion:

• Recognize the number of feet in a mile: 5280 feet.
• Recognize the number of minutes in an hour: 60 minutes.
• Multiply the plane's speed by these values to convert it:\[Speed \,(ft/min) = 300 \,(mph) \times \frac{5280 \,(feet)}{1 \,(mile)} \times \frac{1 \,(hour)}{60 \,(minutes)}\]

This conversion is essential for matching units in your calculations and ensures consistency in measuring altitude change.

Right Triangle Problem Solving

Right triangle problem-solving is a practical application of trigonometry that involves identifying relationships between the various components of a right triangle: the opposite side, adjacent side, and hypotenuse. In our scenario of a plane lifting off, the right triangle is formed by the runway, the flight path, and the vertical altitude gained. The plane's path is the hypotenuse, while its altitude gain corresponds to the opposite side in relation to the angle of takeoff (\(20^{\circ}\)).Using trigonometric functions, especially the sine function, you can calculate the rate of altitude gain:\[\sin(20^{\circ}) = \frac{\text{altitude gain rate (ft/min)}}{\text{speed (ft/min)}}\]This equation allows us to isolate and solve for the altitude gain rate:\[\text{altitude gain rate (ft/min)} = \sin(20^{\circ}) \times \text{speed (ft/min)}\]This approach is foundational for understanding and solving problems related to navigation and any scenario involving angles and distances.