Question

A man is flying in a hot-air balloon in a straight line at a constant rate of 6 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 37°. 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 38°. At that time, what is the distance between him and his friend?

Answer 1

Answer: 336.4447 feet

Step-by-step explanation: from the picture I attached to this answer, you will see how I represented the question in a diagram

Point A being the point of his friends car, point B being the point of the hot air balloon when he noticed the angle of depression to his friends car to be 37 degrees, point C being the point of the hot air balloon after passing his friends car and noticing the angle of depression to be 38 degrees

From my diagram, I labeled y as the distance between B and C and we are told he traveled when a speed of 6 feet per second with a constant altitude, and after 1 and a half minutes he reached point C, which is 90 seconds

To get value of y we multiply the speed and the time, 90 multiplied by 6 which will give 540 feet

From my diagram, I calculated the angle inside the triangle to be 105, we all know the sum of angles in a straight line to be 180, and also knowing alternate angles, we have 37 and 38 degree as the angle outside the triangle at that point, so adding both angle 37+38=75 and subtracting that from 180 we get 105

So the get the distance between him and his friends when the angle of depression is 38 degree, which is the distance between point A and C which I labeled x, we use the sin rule

Sin rule states that the ratio between the length of a side of a triangle and the sin of the angle opposite it is constant for all sides of the triangle,

The steps are also solved in the picture I sent

So we have the ratio of x and sin37 is also equal to the ratio of 540 and sin105

So x divided by sin37 equals 540 divided by sin105

Sin37 equals 0.6018, sin105 equals 0.9659

So making x the subject of formula

We get x will be equal to (540*0.6018)/0.9659 which will give you 336.4447 feet