Final answer:
To find the linear function that models the plane’s descent, calculate the slope using the change in elevation over time, then use the slope and one point to solve for the y-intercept. The resulting linear function is h(t) = -500t + 5000, with t in minutes and h in feet.
Step-by-step explanation:
The student is asking to write a linear function that models the plane’s elevation h as a function of the time t it has been descending. We are given two points in time: two minutes after descent with an elevation of 4000 ft, and five minutes after descent with an elevation of 2500 ft.
First, we calculate the slope of the line which represents the rate of change of elevation with respect to time. The slope m is given by the change in altitude divided by the change in time (Δh/Δt). So the slope m = (2500 ft - 4000 ft) / (5 min - 2 min) = (-1500 ft) / (3 min) = -500 ft/min.
Now, we need a point to complete the linear equation in the form h(t) = mt + b. Using one of the points, let's take t=2 and h=4000 ft, and substitute these values into the equation with our slope to find b. So 4000 ft = (-500 ft/min)(2 min) + b, which leads us to b = 4000 ft + 1000 ft = 5000 ft.
Finally, the linear function that models the plane's elevation h as a function of the descent time t is:
h(t) = -500t + 5000