Question

Erin is testing how a metal pipe in a machine responds to rapid changes in temperature over a period of 30 seconds. During the test, the temperature of the metal pipe in degrees Fahrenheit at
t
seconds is modeled by the quadratic function
f
(
t
)
=
−
t
2
+
24
t
for
t
=
0
to
t
=
30
.

Using this model, which statement is true about the test?


The maximum temperature reached by the metal pipe during the experiment is 144°F.


The temperature is increasing for the duration of the experiment.


The maximum temperature reached by the metal pipe during the experiment is 12°F.


The initial temperature of the metal pipe is 24°F.

Answer 1

The first statement is true.

Explanation: f(t) = −t^2 + 24t is a downward-opening parabola with vertex at t = −b/(2a) = −24/(2·−1) = 12 seconds. The maximum temperature is f(12) = −12^2 + 24·12 = −144 + 288 = 144°F. Also f(0) = 0°F, so the initial temperature is not 24°F and the temperature is not increasing the whole time (it increases until 12 s, then decreases).