Answer:
(a) 25 degrees
(b) -11 degrees
(c) 38 degrees
Step-by-step explanation:
The temperature function is:
[tex]T(t) = -t^2+4t+34[/tex]
(a) The average value for a temperature is:
[tex]M=\frac{1}{b-a}* \int\limits^b_a {f(x)} \, dx[/tex]
In this particular case, the average temperature is:
[tex]M=\frac{1}{9-0}* \int\limits^9_0 {T(t)} \, dt \\M=\frac{1}{9}* \int\limits^9_0 {(-t^2+4t+34)} \, dt \\M=\frac{1}{9}* {(-\frac{t^3}{3}+2t^2+34t)}|_0^9\\M=\frac{1}{9}*( {(-\frac{9^3}{3}+2*(9^2)+34*9)-0)[/tex]
[tex]M=25[/tex]
The average temperature is 25 degrees.
(b) The expression is a parabola that is concave down, therefore there are no local minimums, which means that the minimum temperature will be at one of the extremities of the interval:
[tex]T(0) = -0^2+4*0+34=34\\T(9) = -9^2+9*4+34=-11[/tex]
The minimum temperature is -11 degrees.
(c) The maximum temperature will occur at the point for which the derivate of the temperature function is zero:
[tex]T(t) = -t^2+4t+34\\T'(t)=-2t+4=0\\2t=4\\t=2[/tex]
At t = 2, the temperature is:
[tex]T(2) = -2^2+4*2+34=38[/tex]
The maximum temperature is 38 degrees.
