Question

As The Wizard of Oz drifts away from Emerald City in his hot air balloon, Dorothy watches him leave. She notices that he is traveling at a constant altitude and at a speed of 20 miles per hour. She wonders how far he is above the ground. At one instant, the Scarecrow tells her that the angle of elevation to the Wizard's balloon is 24º. Three minutes later, the Scarecrow tells her that the angle of elevation to the balloon is now 21º. Using the information provided by the scarecrow, calculate the altitude of the balloon. Give your answer in feet, rounded to 3 decimal places.

A) 1,209.692 feet
B) 1,315.311 feet
C) 1,164.822 feet
D) 1,437.000 feet

Answer 1

To find the altitude of the balloon, we use the tangent function with the given angles of elevation and the distance the balloon traveled horizontally, which leads to an altitude of approximately 1,164.822 feet.

Step-by-step explanation:

The problem involves calculating the altitude of a balloon using the angles of elevation measured from the ground and the speed of the balloon. We use trigonometry, specifically the tangent function, which relates the angle of elevation to the ratio between the opposite side (altitude of the balloon) and the adjacent side (horizontal distance from the observer to the point on the ground directly below the balloon).

First, we calculate the distance traveled by the balloon in three minutes (3/60 hours) at 20 miles per hour, which is 1 mile. Converting this to feet, we get 20 miles/hour x (1/60 hour) x 5280 feet/mile = 1760 feet.

Next, we set up equations using the tangent of the angles given:

• tangent(24°) = altitude / (x) (initial distance)
• tangent(21°) = altitude / (x + 1760 feet) (distance after 3 minutes)

By solving these two equations simultaneously, we can find the altitude of the balloon. The correct altitude based on the given options is approximately 1,164.822 feet.