Precalculus question?

Question

Suppose T(t)=23sin(2π(t-7)/24)+66 is the temperature (in degrees Fahrenheit) at time t, where t is measured in hours after midnight on Sunday. You paint the exterior door to your house at 5 p.m. on Monday. The paint information states that 24 hours of 75 F drying time is required; i.e., you can only count time periods when the temperature is at least 75 F. When will the door be dry?

Answer 1

Painting at t=17

T(17) >75 but only for like 1/2 hour

after that...

the temperature is at least 75 degrees

hours 32 to 41

56 to 65

80 to 89 <--- almost 90

will be dry at approximately hour 85.5

that's 3 days later... and then....

85.5 - 24*3 = 85.5 - 72 = 13.5

so around 1:30 on the 4th day

graphing the given time function at DESMOS dot com is very helpful as you can

see at what hours for time t the temperature is at or above 75 degrees. You can ZOOM in

Answer 2

Note that the sin curve starts at 7AM (66 degrees). Picturing a sin curve, the above 75 temperature will occur at the intersection of the upward hump and a horizontal line at 75. We solve for time in a 24 hour period that the function equals 75:

23sin(2π(t-7)/24) +66 = 75

2π(t - 7) / 24 = sin^-1((75-66)/23)

t = 7 + 24/(2π) sin^-1((75-66)/23) (make sure you are in radians, one answer will be the lower bound and pi minus that answer will be the upper bound.)

t = 7 + (1.536 to 10.464)

For 24 hours of drying we need 2 24 hour periods past 7AM +( 24- 2(10.464-1.536) + 1.536 hours

time left to dry drying occurs

1.536 hours after 7

Total time will be 7 +2*24 + 7.68 = 62.68 hours

14 hours and (.68*60) minutes = 2:41 PM on 3rd day.

Nice problem!