Newton's Law of Cooling states that the rate of change of the temperature of an object, T, is proportional to the difference of
1 answer:
Answer:
305 °F
Step-by-step explanation:
The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.
<h3>Cooling Model</h3>The solution to the differential equation described by Newton's law of cooling is the exponential equation ...
y = ab^t +c
where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.
For this problem, the ambient temperature is c=80, and the differences of interest are ...
a = 1200 -80 = 1120
b = (830 -80)/1120 = 75/112
Using these values in the model gives ...
y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t
Note that units of time are hours.
<h3>Application</h3>We want y when t=4.
y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212
The core temperature after 4 hours is about 305 °F.
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<em>Additional comment</em>
The differential equation will have a solution of the form ...
where k = ln(75/112) ≈ -0.40101
In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.
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f(1)=1-11+36-36, not 0
f(2)=8-44+72-36=-36+72-36=0. Here we have our first root (2).
Now we cad divide f(x) by (x-2) which will give us a quadratic expression, which we can factorize easily (if the discriminant is non negative).
We can also try some other factors of 36. Indeed we can check that
f(3)=27-99+108-36=135-135=0,
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Thus, f(x)=(x-2)(x-3)(x-6). Note that if we expanded the right hand side expression, the constant term would be the product of the constants 2, 3, 6.
This is the reason why in the first place we looked at the factors of 36 for the possible zeros of f(x).
Thus, the x-intercepts are (2, 0), (3, 0), (6, 0), or 2, 3, 6.
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Note that f(0)<f(2) because f(0)=-36 and f(2)=0. This means that at the left side, the graph is coming from - infinity. Similarly,
we can check that f(10)=1000-1100+360-36=224> f(6). That is, to the right of our rightmost root, the graph is getting larger.
Thus, the end behaviors are: the graph goes to + infinity as x goes to + infinity,
and it goes to minus infinity as x goes to - infinity.
To find the x-intercepts, we need to factorize the function. A very good idea is to first try the factors of 36:
f(1)=1-11+36-36, not 0
f(2)=8-44+72-36=-36+72-36=0. Here we have our first root (2).
Now we cad divide f(x) by (x-2) which will give us a quadratic expression, which we can factorize easily (if the discriminant is non negative).
We can also try some other factors of 36. Indeed we can check that
f(3)=27-99+108-36=135-135=0,
and
f(6)=216-396+216-36=0.
Thus, f(x)=(x-2)(x-3)(x-6). Note that if we expanded the right hand side expression, the constant term would be the product of the constants 2, 3, 6.
This is the reason why in the first place we looked at the factors of 36 for the possible zeros of f(x).
Thus, the x-intercepts are (2, 0), (3, 0), (6, 0), or 2, 3, 6.
The y-intercept is f(0), which is -36.
Note that f(0)<f(2) because f(0)=-36 and f(2)=0. This means that at the left side, the graph is coming from - infinity. Similarly,
we can check that f(10)=1000-1100+360-36=224> f(6). That is, to the right of our rightmost root, the graph is getting larger.
Thus, the end behaviors are: the graph goes to + infinity as x goes to + infinity,
and it goes to minus infinity as x goes to - infinity.