The table displays the number of hours of daylight, H, in Seattle, Washington, d
ID: 3052081 • Letter: T
Question
The table displays the number of hours of daylight, H, in Seattle, Washington, d days after March 21, 2010. Use the table of values to answer the following questions. (Hint: Be sure to use Radian mode on your calculator.) Days after March 21,2010 Number of Daylight Hours Power Usage 30 60 90 120 150 180 210 240 270 300 330 360 12.20 14.07 15.45 15.98 15.53 14.21 12.36 10.48 9.04 8.43 8.80 10.06 11.88 (a) Find an equation for the sinusoidal function, mc, to model the number of daylight hours in Seattle (b) Use H(d) to estimate when the number of daylight hours will be 13.5 hours. (Enter your answers as a comma-separated list. Round your answers to one decimal place.) (c) Find the average rate of change of M(d) from -30 to d. 120. (Enter your answers as a comma-separated list. Round your answer to three decimal places.) hr/day What does this rate mean?Explanation / Answer
(a) The sinusoidal function can be of the form Hd = A Sin d + B, where A and B are constants to be determined.
The maximum daylight is 15.98, the minimum is 8.43, so range = 7.55.
Therefore, the sinusoid goes from -3.775 to 3.775 with zero in the middle.
A = 3.775
However, the function has been shifted vertically. 15.98 - 3.775 = 12.205. So B = 12.205
The equation can now be written as
Hd = 3.775 Sin d + 12.205
(b) Hd = 13.5
3.775 Sin d + 12.205 = 13.5
Sin d = (13.5-12.205)/3.775 = 0.343
d = Sin-1 (0.343) = 20.05975 ~ 20.1
d can have the values 20.1 and 180-20.1 = 159.9.
Daylight hours will be 13.5 on 20.1 and 159.9th days.
(c) d = 30, Hd = 14.07
d = 120, Hd = 15.53
Change in Hd = 15.53 - 14.07 = 1.46
Change in days = 120-30 = 90
Average rate of change of daylight hours per day = 1.46/90 = 0.016
The rate means that on an average, in the range 30th to 120th day, the daylight hours changes by 0.016 hours/day.
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