Question

A weather balloon is carrying instruments into the atmosphere to measure various parameters. The height of the weather balloon is recorded for the first five seconds of its rise into the air. The data is shown in the table below.

t(sec) | 0 | 1 | 2 | 3 | 4 | 5 |
h(ft) | 27 | 42 | 57 | 72 | 87 | 102 |

(a) How does the data in the table indicate that the height of the balloon is a linear function of time?
(c) Give a formula for the height, h, as a linear function of the time, 1, it has been in the air.
(b) What is the average rate of change of the height? Include proper units.
(d) How high above the ground does your model from (c) predict the balloon will be after 5
minutes of rising?

Answer 1

The data in the table indicates that the height of the weather balloon is a linear function of time because there is a constant increase in the height for each second. The average rate of change of the height is 15 ft/sec, and the height of the balloon after 5 minutes of rising is predicted to be 4527 ft.

Step-by-step explanation:

(a) The data in the table indicates that the height of the weather balloon is a linear function of time because there is a constant increase in the height for each second. The difference between the heights at consecutive time intervals is always the same.

(c) We can represent the height, h, as a linear function of time, t, using the equation h = mt + b, where m is the slope of the line (the rate of change of height with respect to time) and b is the y-intercept (the initial height of the balloon).

(b) To find the average rate of change of the height, we divide the change in height by the change in time. In this case, the change in height is 102 ft - 27 ft = 75 ft, and the change in time is 5 sec - 0 sec = 5 sec. So, the average rate of change is 75 ft / 5 sec = 15 ft/sec.

(d) To predict the height of the balloon after 5 minutes (300 seconds) of rising, we can substitute t = 300 into the formula from (c). Using the equation h = mt + b, we find h = 15 ft/sec * 300 sec + 27 ft = 4527 ft.

Answer 2

The increase in balloon height is consistent with each time interval, indicating a linear relationship. The linear function for height as a function of time is h(t) = 15t + 27, with the average rate of change being 15 feet per second. After 5 minutes, the model predicts the balloon would be at 4527 feet.

Step-by-step explanation:

(a) The data in the table indicates that the height of the weather balloon is a linear function of time because the increase in height is consistent for each one-second interval. This means we can observe a constant difference (15 feet per second) between the height measurements at each time interval, which suggests a straight line if plotted on a graph.

(c) To give a formula for the height, h, as a linear function of the time, t, we can use the two points (0, 27) and (1, 42) to find the slope (m) of the line: m = (42 - 27) / (1 - 0) = 15. Since the balloon starts at 27 feet, the y-intercept (b) is 27. The linear function is h(t) = 15t + 27.

(b) The average rate of change of height is the slope of the linear function, which is 15 feet per second, as calculated from the height differences between successive time intervals.

(d) After 5 minutes (300 seconds) of rising, the height of the balloon can be predicted using the linear model: h(300) = 15(300) + 27 = 4500 + 27 = 4527 feet.