A hot air balloon is at an altitude of h meters. The balloon begins to descend at a consta… — SAT

A hot air balloon is at an altitude of $h$ meters. The balloon begins to descend at a constant rate. After $3$ minutes, the altitude of the balloon is $540$ meters. After $8$ minutes, the altitude of the balloon is $390$ meters. If the altitude of the balloon is modeled by a linear function of time, what was the initial altitude of the balloon, in meters?

Answer: 630 meters

Answer

The initial altitude of the balloon was 630 meters.

To find the initial altitude, we model the balloon's descent as a linear equation of the form

h(t) = mt + b

, where

h(t)

is the altitude at time

t

m

is the constant rate of change, and

b

is the initial altitude. Using the given points

(3, 540)

and

(8, 390)

, we find the slope

m = \frac{390 - 540}{8 - 3} = -30

meters per minute. Substituting

m = -30

and the point

(3, 540)

into

h(t) = mt + b

gives

540 = -30(3) + b

. Solving for

b

yields

b = 540 + 90 = 630

meters.

Step-by-Step Solution

Identify two data points from the problem context

The coordinate points are

(3, 540)

and

(8, 390)

To define the linear relationship, we need at least two coordinate points representing (time, altitude)

Calculate the slope (

m

) of the linear function

m = \frac{390 - 540}{8 - 3} = \frac{-150}{5} = -30

The rate of change represents the speed at which the balloon descends each minute

Solve for the vertical intercept (

b

) using the slope-intercept form

y = mx + b

540 = -30(3) + b \implies b = 630

The initial altitude corresponds to the altitude at time

t = 0

, which is the vertical intercept of the linear function

Key Concept

Linear Functions and Graphs

Question

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