The temperature, t, in degrees celsius, in a warehouse changes according to the function t(h) = 18 + 4sin(pi/12 (x = 8)).
where h is the number of hours since midnight. at what rate is the temperature changing at 9 a.m.?

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MrRoyal MrRoyal · Answer 1 · 2022-06-14T19:27:33+02:00

The rate at which the temperature is changing at 9 a.m. is 19.03 degrees Celsius per hour

The function is given as:

[tex]t(h) = 18 + 4\sin(\pi/12(x - 8))[/tex]

9a.m is 9 hours since midnight.

This means that

x = 9

So, we have:

[tex]t(h) = 18 + 4\sin(\pi/12(9 - 8))[/tex]

Evaluate

[tex]t(h) = 18 + 4\sin(0.262)[/tex]

Evaluate the expression

t(h) = 19.03

Hence, the rate at which the temperature is changing at 9 a.m. is 19.03 degrees Celsius per hour

The rate of change of temperature with respect to time is the derivative of t(h) with respect to h.

t'(h) = 4(π/12)cos(π/12(h -8)) = π/3·cos(π/12(h -8))

For h = 9 hours after midnight, the rate of change is ...

t'(9) = π/3·cos(π/12(9 -8)) = π/3·cos(π/12) ≈ (3.14159)(0.965926)/3

t'(9) ≈ 1.01152

The rate of change of temperature at 9 a.m. is about 1.01 °C/hour.