Write Linear Equations Review

Question

A hot air balloon descends at a rate of 2 meters per second. After 15 seconds, it is 620 meters off the ground. Write an equation that represents the relationship between the time and the height of the hot air balloon. What is the height of the balloon after 35 seconds?

Answer 1

Hi, Cassie.

In this problem, you are given two pieces of information:

1. The rate of descent is 2 meters per second. This information tells you the hot air balloon's rate of change: -2 m/s. (The rate of change is negative because the balloon is descending.) In terms of a graph, the rate of change is the slope of the line representing the hot air balloon's height h (in meters) at time t (in seconds).
2. You know that at time t = 15 seconds, the hot air balloon is at height h = 620 meters. On a graph, this fact is represented by the point (t, h) = (15, 620).

What you need to determine is the hot air balloon's height at time t = 35 seconds. There are several ways to obtain the answer, but since this problem is titled "Write Linear Equations Review," I'm going to assume that you're expected to write a linear equation. In this case, you know the slope of the line and a point on the line, so it makes sense to use point-slope form. The general form is y - y₁ = m(x - x₁). In this problem, I'm using t in place of x and h in place of y. So, I can write the equation h - 620 = -2(t - 15). Simplify this equation to get h - 620 = -2t + 30, or h = -2t + 650. With this equation, you can substitute 35 for t to get h = -2(35) + 650 = 580. So, the hot air balloon's height is 580 meters after 35 seconds.

As a check on this answer, think about the fact that the hot air balloon is losing 2 meters of height every second. At t = 15 seconds, it was at a height of 620 meters. At t = 35 seconds, you know that 20 seconds have passed and the hot air balloon has lost 20(2) = 40 meters of height. So, the balloon's height is 620 - 40 = 580 meters after 35 seconds. This confirms the answer obtained using the point-slope form of a linear equation.

Doug