Can you answer this math question please

Question

Suppose the temperature in degrees Celsius over a 10-hour period is given by T(t) = -t^2 + 2t + 33

a) Find the average temperature. b) Find the minimum temperature. c) Find the maximum temperature.

Answer 1

Hello, Ege,

I may not have this correct, but here are my thoughts.

If we graph the equation we find that an inverted parabola that has a maximum at t = 1 hour, for a temperature of 34C. We can also find the maximum by obtaining the first derivative of the equation and setting it to zero (where the slope is zero when the parabola reaches the peak and starts down).

T'(t) = -2t + 2

0 = -2t + 2

t = 1

The temperature at t = 0 hours is 33 C and at 10 hours is -47C, which will be the minimum over that time frame, due to the parabolic nature of the curve at that point.. (Where is this place? Earth or Mars?) There may be a more elegant way to calculate average temperature, but I can only offer:

(33 + (-47))/2 = -14/2 = -7C

At 50 hours; the temperature, according to this equation, would be - 2367C. It is only valid for the time range of 10 hours, at best.

Bob

Answer 2

a) Average temperature T_av = 1/10∫₀¹⁰( - t² + 2t + 33)dt = 1/10(- t³/3 + t² + 33t)₀¹⁰ = 1/10(- 1000/3 + 100 + 330) = 9.67°C

b) T'(t) = -2t + 2 = 0; t = 1 hour

T''(t) = -2 < 0. So we have max at t = 1 hour, maxT = T(1) = 34°C

c) T(0) = 33°C; T(10) = - 100 + 20 + 33 = - 47°C. So, minT = T(10) = - 47°C