Question

a hot - air balloon is flying at an altitude of 150 feet (ft). when the pilot heats the air in the balloon it rises at a constant rate. the altitude of the balloon is a function of the number of minutes it rises.
describe the input and output of the function.
input (x - value): number of minutes
output (y - value): altitude of the balloon
when the y - values increase as the x - values increase, the result is a positive rate of change.
what is the rate of change of the function?
the rate of change is
the altitude increases by 7 feet every minute.

Explanation:

Step1: Identify two points

From the graph, at \( t = 0 \) min, altitude \( y = 150 \) ft. At \( t = 1 \) min, altitude \( y = 550 \) ft.

Step2: Calculate rate of change

Rate of change (slope) is \( \frac{\Delta y}{\Delta t}=\frac{550 - 150}{1 - 0}=\frac{400}{1}=400 \). Wait, but let's check another pair. At \( t = 1 \), \( y = 550 \); \( t = 2 \), \( y = 950 \). \( \frac{950 - 550}{2 - 1}=\frac{400}{1}=400 \). So the rate of change is 400 feet per minute. Wait, maybe the initial altitude was misread? Wait, the first label: 0 min, what's the altitude? Wait, the user's image: 0 min, maybe 150? Wait, the text says "flying at an altitude of 150 feet". Then at 1 min, 550? Wait 550 - 150 = 400. At 2 min, 950 - 550 = 400. 1350 - 950 = 400. 1750 - 1350 = 400. So the rate of change is 400. Wait, the original input had a wrong number? Wait, the user's problem: "The altitude increases by 7 feet" – no, that's incorrect. Let's recalculate.

Wait, let's list the times and altitudes:

• \( t = 0 \): 150 ft (given: "flying at an altitude of 150 feet")
• \( t = 1 \): 550 ft (from image: 1 min --- 550 ft)
• \( t = 2 \): 950 ft
• \( t = 3 \): 1350 ft
• \( t = 4 \): 1750 ft

So the change in altitude per minute: \( 550 - 150 = 400 \) (from t=0 to t=1), \( 950 - 550 = 400 \) (t=1 to t=2), etc. So rate of change is 400 feet per minute.

Answer:

The rate of change is 400 feet per minute. So the correct rate of change is 400, meaning the altitude increases by 400 feet every minute.