# PreCalc

The temperature T(t) varies sinusoidally on a certain day in December. The minimum temperature is 35 degrees Fahrenheit at midnight. The maximum temperature is 50 degrees Fahrenheit at noon. Let t be the number of hours since midnight (t=o at midnight).
a.) Sketch and label a graph showing exactly two periods of T(t)beginning at t=0.
b.)Determine a function for T(t) using the cosine function.
c.)Determine a function for T(t) using the sine function.
d.) Use your equation(s) to find the temperature at 1 am.
e.) I want to go on a bike ride, but I prefer to ride when the temperature is at least 45 degrees Fahrenheit. What is the earliest time of day that I can leave for my ride? How long can I stay out before I get cold?

1. 👍 0
2. 👎 0
3. 👁 344
asked by Katy
1. the max-min value is 50-35=15, so the amplitude is 7.5. The center line is (35+50)/2 = 37.5

y = 7.5 sin(x) + 37.5

y has a max at t=12, and the period is 24 hours, so the minimum k is a t t=0.

cos(x) has a max at x=0, so we have -cos(x) and thus

y = -7.5 cos(π/12 x) + 37.5

Now use that to answer the other questions. recall that cos(x) = sin(π/2 - x).

1. 👍 0
2. 👎 0
posted by Steve
2. when t=0 , temp = 35
when t = 12, temp = 50
Period = 24 hours or k = π/12

how about
Temp = 7.5cos (π/12)(t + 12) + 42.5

check:
when t = 0, Temp = 7.5cos(π) + 42.5 = 35
when t = 12, Temp = 7.5cos(2π) + 42.5 = 50

You try it with a sine curve, be aware that your phase shift will have to be different.

a 1:00 am , t = 1
temp = 7.5cos (π/12)(13) + 42.5 = 35.26° F

for T ≥ 45
7.5cos (π/12)(t+12) + 42.5 = 45
7.5cos (π/12)(t+12) = 2.5
cos (π/12)(t+12) = .3333...
(π/12)(t+12) = 1.231 or (π/12)(t+12) = 2π - 1.231 = 5.052
t+12 = 4.702 or t+12 = 19.297
t = -7.298 or t = 7.297 because of the symmetry of the curve
and
t = 19.297
t = 7.297 hrs = appr 7:18 am
t = 19.297 = appr 7:18 pm

so the temp is above 45° F from 7:18 am to 7:18 pm

check my arithmetic

1. 👍 0
2. 👎 0
posted by Reiny
3. Reiny is right: 42.5, not 37.5

but I'm sure you caught my error.

Note how he offset his function by using a phase shift, which has the same effect as changing the sign as I did, since shifting by 1/2 period does that flip:

cos(x+pi) = -cos(x)

1. 👍 0
2. 👎 0
posted by Steve

## Similar Questions

1. ### Algebra

Stella is recording temperatures every day for 5 days. On the first day, Stella recorded a temperature of 0 degrees Farenheit. A. On the second day, the temperature was 3 degrees Fahrenheit above the temperature on the first day.

asked by Ms. Stephie on September 18, 2014
2. ### math

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 55 and 85 degrees during the day and the average daily temperature first occurs at 10 AM. How many hours after

asked by pika on July 22, 2019
3. ### math-precalculus

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 73 and 87 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after

asked by j on April 27, 2017
4. ### Algebra 1

the temperature at noon in los angeles on a summer day was 88 degrees fahrenheit. during the day, the temperature varies from this by as much as 7.5 degrees fahrenheit. write and solve an absolute-value inequality to find the

asked by Katie on September 11, 2014
5. ### Calculus

The temperature of a cup of coffee varies according to Newton's Law of Cooling: dT/dt = -k(T - A), where T is the temperature of the tea, A is the room temperature, and k is a positive constant. If the water cools from 100°C to

asked by Erika on June 4, 2019
6. ### Trigonometry

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 95 degrees occurs at 3 PM and the average temperature for the day is 80 degrees. Find the temperature, to the nearest

asked by Julie on November 1, 2016
7. ### ap chem

Like all equilibrium constants, Kw varies somewhat with temperature. Given that Kw is 5.02×10−13 at some temperature, compute the pH of a neutral aqueous solution at that temperature. 006

asked by Anonymous on February 14, 2011
8. ### AP Chemistry

Like all equilibrium constants, Kw varies somewhat with temperature. Given that Kw is 7.51×10−13 at some temperature, compute the pH of a neutral aqueous solution at that temperature.

asked by Gabriella on January 24, 2014
9. ### calculus

The temperature of a hard boiled egg varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A, where T is the temperature of the egg, A is the room temperature, and k is a positive

asked by Anonymous on July 17, 2015
10. ### Science

Ed wants to determine if there is a trend in air temperature changes during April. Which of the following procedures should he follow? a) Measure the temperature every hour for 1 day; b) Measure the temperature at noon every day

asked by Mia on February 21, 2012

More Similar Questions