Question

The height of a drone above ground is monitored for 10 seconds. The height of the drone when the monitoring begins is 40 feet. For the first 6 seconds, the height of the drone follows a quadratic pattern. The drone reaches is minimum height of 10 feet 4 seconds into monitoring. The drone's height then increases at a steady rate of 20 feet per second for the remainder of the time it is monitored.

Which piecewise function models the height of the drone x seconds after monitoring begins?

(1 point)

f(x)={158(x−4)2+1020xifif0≤x≤66<x≤10


f(x)={158(x−4)2+1020x+10ifif0≤x≤66<x≤10


f(x)={158(x−4)2+1020x−2052ifif0≤x≤66<x≤10


f(x)={154(x−4)2+1020x+2052ifif0≤x≤66<x≤10

Answer 1

Let f(x)=a(x−4)^2+10 on [0,6] (vertex (4,10)). Use f(0)=40:
40 = a(−4)^2 + 10 → 16a = 30 → a = 15/8.

So for 0≤x≤6: f(x) = (15/8)(x−4)^2 + 10.

Find f(6) = (15/8)(2)^2 + 10 = 15/2 + 10 = 35/2. For x>6 the height increases 20 ft/s, so
f(x) = 35/2 + 20(x−6) = 20x − 205/2 for 6<x≤10.

Thus the piecewise function is
f(x) = { (15/8)(x−4)^2 + 10, 0≤x≤6; 20x − 205/2, 6<x≤10 }.

This matches the third option.