URGENT HELP PLEASE!! 4. Consider the temperature in a classroom throughout any given day. Suppose that the temperature constantly cycles between 68°F and 80°F and is given by the sinusoidal function

Question

f(t)= Acos(Bt)+C, where t=the number of hours after midnight. The temperature starts at 68°F at midnight and first returns to 68°F at 8am.

A.) Sketch the graph of the function for two full periods.

B.) Find values for A,B, and C and write f(t).

C.) For what values of t, 0 ≤ t ≤ 4, is the temperature in the classroom exactly 72°F?

D.) Since it has two components, electromagnetism is often described in terms of complex numbers. Suppose the energy company providing the power to maintain the classroom’s temperature uses the complex number z=2-4i for some of its calculations. Write s in polar form.

E.) In general, why do we bother writing complex numbers in polar form?

Answer 1

A) Sketch (description):

The graph is a cosine curve with period 8 hours. It starts at 68°F at t = 0, reaches a maximum of 76°F at t = 4, and returns to 68°F at t = 8. This pattern repeats, so two full periods run from t = 0 to t = 16.

B) Find A, B, and C:

The period is 8 hours, so

2π / B = 8

B = π / 4

Use f(0) = 68:

A + C = 68

The midline is 72, so C = 72.

Then A = −4.

The function is:

f(t) = −4 cos((π/4)t) + 72

C) When is the temperature exactly 72°F for 0 ≤ t ≤ 4?

Set f(t) = 72:

−4 cos((π/4)t) + 72 = 72

cos((π/4)t) = 0

(π/4)t = π/2

t = 2

D) Write z = 2 − 4i in polar form:

Magnitude:

r = sqrt(2^2 + (−4)^2) = sqrt(20) = 2 sqrt(5)

Angle:

θ = arctan(−4 / 2) = arctan(−2)

Since z is in Quadrant IV, θ = −arctan(2)

Polar form:

z = 2 sqrt(5)(cos(−arctan(2)) + i sin(−arctan(2)))

or

z = 2 sqrt(5) e^(−i arctan(2))

E) Why use polar form?

Polar form makes multiplication, division, powers, and roots of complex numbers easier. Magnitudes multiply or divide directly, and angles add or subtract, which is especially useful in electromagnetism and AC circuit analysis.