Question

The function \(d(t)=50 \cos \left(\frac{\pi t}{39}\right)+60\) represents the distance \(d\), in miles, from the airport after \(t\) minutes of an airplane asked to fly in a circular holding pattern. (a) What is the plane's average distance from the airport over one cycle? (b) How long does it take the plane to complete one cycle in the holding pattern? (c) What is the plane's speed, in miles per hour, while in the holding pattern? (d) If the plane travels 1.8 miles per gallon of fuel, how much fuel is used in one cycle of the holding pattern?

Short answer

Average distance: 60 miles, Time for one cycle: 78 minutes, Speed: 290 mph, Fuel used: 209.44 gallons.

Step-by-step solution

Understand the Problem

We are given a function that models the distance of a plane from the airport as it flies in a circular holding pattern. We need to calculate the average distance over one cycle, the time to complete one cycle, the plane's speed, and the fuel used.

Average Distance from the Airport

The average value of a cosine function over one period is the same as the midpoint between its maximum and minimum values. The function given is \(d(t) = 50 \cos \left( \frac{\pi t}{39} \right) + 60\). The amplitude is 50 and the vertical shift is 60. Therefore, the average distance is 60.

Time to Complete One Cycle

The period of the cosine function in the form \cos(kt) is given by \(T = \frac{2\pi}{k}\). Here, \frac{\pi}{39} is the coefficient of \cos, so \(T = \frac{2\pi}{\frac{\pi}{39}} = 78 \text{ minutes}\).

Plane's Speed

To find the speed, we need to calculate the distance traveled in one cycle. As the plane travels in a circular path with a radius equal to the average distance (60 miles), the circumference (total distance traveled in one cycle) is calculated as \(C = 2 \pi \times 60 = 120 \pi \text{ miles}\). The speed is then \(\frac{120 \pi}{78} \approx 4.83 \text{ miles per minute}\). To convert this to mph: \(4.83 \times 60 \approx 290 \text{ mph}\).

Fuel Used in One Cycle

The plane travels 120 \pi miles in one cycle. Given fuel efficiency is 1.8 miles per gallon, the fuel used is \(\frac{120 \pi}{1.8} \approx 209.44 \text{ gallons}\).

Key concepts

cosine function

The cosine function appears frequently in trigonometry and is represented as \(\text{cos}(x)\). In our exercise, the distance function given is \(d(t) = 50 \cos \left( \frac{\pi t}{39} \right) + 60\). Here, the cosine function helps to describe the periodic motion of the airplane around a holding pattern. The coefficient 50 represents the amplitude, which indicates how far the plane can deviate from its average distance due to the circular motion. The number 60 represents the vertical shift, meaning the average distance is shifted upwards by 60 miles from the base cosine graph. Cosine functions are crucial in modeling cyclical behaviors due to their repetitive wave-like pattern, and they can efficiently represent scenarios like circular movements or oscillations.

period of trigonometric functions

The period of a trigonometric function defines how long it takes for the function to complete one full cycle. For the function \(d(t) = 50 \cos \left( \frac{\pi t}{39} \right) + 60\), the key term is the coefficient of \(t\) within the cosine, which is \(\frac{\pi}{39}\). The formula for the period \(T\) of the cosine function \(\text{cos}(kt)\) is given by \(\frac{2\pi}{k}\). Here \(k = \frac{\pi}{39}\). Calculating the period gives \(T = \frac{2\pi}{\frac{\pi}{39}} = 78 \text{ minutes}\). Thus, it takes the plane 78 minutes to complete one full circular journey around the holding pattern, reflecting the cyclical nature of trigonometric functions.

average value of a function

The average value of a function over a specific interval gives us a meaningful measure of its central tendency during that interval. For the cosine function provided, the average value over one period is crucial. The function \(d(t) = 50 \cos \left(\frac{\pi t}{39}\right) + 60\) has an amplitude of 50 and a vertical shift of 60. The average value of a cosine function over one complete period is the midpoint between its maximum and minimum values. Thus, regardless of the amplitude, this average value centers around the vertical shift, which is 60 miles in our problem. This indicates that over one cycle, the plane's average distance from the airport remains consistently 60 miles.

speed calculation

To determine an object's speed from its motion over a set path, we use the basic definition of speed, which is distance divided by time. For the plane flying in a circular holding pattern, we first calculate the total distance traveled in one complete cycle. Given the radius is 60 miles, the circumference of the circle (distance traveled in one cycle) is \(C = 2 \pi \times 60 = 120 \pi \text{ miles}\). Knowing that one cycle takes 78 minutes, the speed in miles per minute is \(\frac{120 \pi}{78} \approx 4.83 \text{ miles per minute}\). To convert this into miles per hour (mph), multiply by 60, giving \(\text{Speed} \approx 290 \text{mph}\). Speed calculation is essential in analyzing how fast an object travels over a period and aids in determining other related factors such as fuel consumption.

fuel consumption calculation

Fuel consumption calculation helps in determining the amount of fuel an airplane uses over a certain distance or period. In our exercise, the plane travels a circular path with a circumference of \(120 \pi \text{ miles}\) in one cycle. Given the plane's fuel efficiency is 1.8 miles per gallon, we can find the total fuel used by dividing the total distance by the fuel efficiency. Thus, \(\text{Fuel Used} = \frac{120 \pi}{1.8} \approx 209.44 \text{ gallons}\). Understanding fuel consumption is vital for planning and operational efficiency, ensuring that the airplane maintains sufficient fuel reserves for its journey.