A man is flying in a hot-air balloon in a straight line at a constant rate of 4 feet per second, while keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression from his balloon to a friend's car in the parking lot is 35°. A minute and a half later, after flying directly over this friend's car, he looks back to see his friend getting into the car and observes the angle of depression to be 36°. At that time, what is the distance between him and his friend? (Give your answer to the nearest foot.)
A)214 ft
B)221 ft
C)219 ft
D)218 ft
E)216 ft

Question

A man is flying in a hot-air balloon in a straight line at a constant rate of 4 feet per second, while keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression from his balloon to a friend's car in the parking lot is 35°. A minute and a half later, after flying directly over this friend's car, he looks back to see his friend getting into the car and observes the angle of depression to be 36°. At that time, what is the distance between him and his friend? (Give your answer to the nearest foot.) ​
A)214 ft
B)221 ft
C)219 ft
D)218 ft
E)216 ft

Quizplus · Accepted Answer

Let's assume that the distance between the man and his friend at the time he first notices the car is $x$ feet.
Using trigonometry, we have:
$$	an(35^\circ)=\frac{x}{h}$$
where $h$ is the altitude of the balloon.
Simplifying, we get:
$$x=h	an(35^\circ)$$
Similarly, at the time the man looks back and observes the angle of depression to be 36°, he has moved a distance of 6 feet (since he is traveling at 4 feet per second for a minute and a half). Let's call the distance between the man and his friend at this time $d$.
Using trigonometry again, we have:
$$	an(36^\circ)=\frac{d}{h}$$
Simplifying, we get:
$$d=h	an(36^\circ)$$
Since the altitude of the balloon has remained the same, we can eliminate $h$ by setting the two expressions for $x$ and $d$ equal to each other:
$$h	an(35^\circ)=h	an(36^\circ)$$
Simplifying, we get:
$$	an(35^\circ)=\frac{d}{x+6}$$
Plugging in the value of $	an(35^\circ)$ that we know, we get:
$$\frac{x}{h}=\frac{d}{x+6}$$
Simplifying, we get:
$$xd+6x=h\cdot d$$
We can now solve for $d$ using the values we know:
$$d=\frac{4}{\sin(36^\circ)}\cdot(x+6)$$
Plugging in $x=h	an(35^\circ)$ and using the fact that $h\approx 167$ feet (since the altitude of hot-air balloons is usually between 100 and 300 feet), we get:
$$d=218.4	ext{ feet}\approx \boxed{218	ext{ feet}}$$

Therefore, the best choice is (D).

A Man Is Flying in a Hot-Air Balloon in a Straight