Question

The desert temperature, H, oscillates daily between 30ºF at 4am and 70ºF at 4pm. Write a possible formula for H in terms of t, measured in hours from 4am. Use the form . H = A cos(Bt) + C.. A=. B=. C=

Answer 1

The formula for the desert temperature H in terms of time t (hours from 4am) is H = 20 cos(πt/6) + 50, with A = 20, B = π/6, and C = 50.

Step-by-step explanation:

The student asks for a formula for the temperature of the desert, H, as a function of time, t, using the form H = A cos(Bt) + C. To write this formula, we need to determine the values of A, B, and C.

To find A, which is the amplitude, we calculate the difference between the maximum and minimum temperatures and then divide by 2: A = (70°F - 30°F) / 2 = 20°F.

Next, to find B, which represents the frequency, we need to consider that the oscillation period is 12 hours (from 4am to 4pm). Therefore, B = 2π / period = 2π / 12 = π/6.

The value of C, which is the vertical shift, is the average of the maximum and minimum temperatures: C = (70°F + 30°F) / 2 = 50°F.

So, our final formula for H in terms of t measured in hours from 4am is H = 20 cos(πt/6) + 50.

Answer 2

The find step is to find C which is the vertical shift. This is the average between the maximum and minimum. So its 30+70=100/2= 50. the amplitude (A) take the average of the difference between max. temp. and min. value So 70-30= 40/2= 20, but it becomes -20 because the graph is shifted up 50 degrees and decreases to 30 at 4a.m. Then the period is 2pi/B which is 2pi/24, which equals pi/12. A=-20 B= pi/12 C=50