Problem 26 - Precalculus: A Concise Course

Question

You observe a plane approaching overhead and assume that its speed is 550 miles per hour. The angle of elevation of the plane is \(16^{\circ}\) at one time and \(57^{\circ}\) one minute later. Approximate the altitude of the plane.

Step-by-Step Solution

Answer

The altitude of the plane is approximately 5.14 miles.

Step 1: Calculate the horizontal distance the plane travelled

We'll make use of the formula distance = speed * time to compute the horizontal distance the plane travelled. It is given that the plane was travelling at a speed of 550 miles per hour and moved for 1 minute (which is 1/60 of an hour). Therefore, by substituting these values into the formula, we get distance = 550 * (1/60) miles = \( \frac{550}{60} = 9.167 \) miles.

Step 2: Determine the lengths of the triangle's sides

We create a triangle where the horizontal distance the plane travelled represents the lower side of the two right-angled triangles formed by the angles of elevation at the two different times, and the altitude of the plane is the other side. Because the angles and the base of the triangle are known, we can make use of the properties of tan of an angle (which is the ratio of the side opposite to the angle to the side adjacent to the angle in a right triangle) to determine the length of the sides. In other words, tan(\(16^{\circ}\)) = altitude / horizontal distance and tan(\(57^{\circ}\)) = altitude / (horizontal distance + changed horizontal distance).The changed horizontal distance is the additional horizontal distance travelled by the plane in the one minute which is approximately 9.167 miles.

Step 3: Solve the equation for altitude

If we express altitude in both equations we mentioned in step 2, we can equate them to solve for altitude. Therefore, from equation tan(\(16^{\circ}\)) = altitude / 9.167 we rearrange to get altitude = 9.167 * tan(\(16^{\circ}\)). And from equation tan(\(57^{\circ}\)) = altitude / (9.167 + 9.167) we rearrange to get altitude = (9.167 * 2) * tan(\(57^{\circ}\)). Equating these and solving will give us the altitude of the plane.