If a cup of coffee has temperature
96 degrees Celsius in a room
where the ambient air temperature is 21 degrees Celsius,
according to Newton's Law of Cooling,
the temperature of the coffee after t minutes
is given by T of t.
What is the average temperature of the coffee
during the first 32 minutes?
To find the average temperature over the first 32 minutes,
we want to find the average function value
over the closed interval from zero to 32.
Therefore, we can use the average value formula given below
where the average function value, y sub av,
is equal to one, divided by the quantity b minus a,
times the integral of f of x from a to b.
Or in our case, a is equal to zero, b is equal to 32,
and the integral of f of x is equal to t of t.
So the average temperature capital t sub av
is equal to one,
divided by b minus a, that would be 32,
times the integral of t of t, which is 21
plus 75 times e
raised to the power of negative t divided by 50,
from zero to 32.
Let's begin evaluating this.
I'll begin by determining the anti-derivative
of 21 plus 75 times e,
raised to the power of negative t divided by 50.
Notice to integrate the second term,
we will have to perform u substitution.
The integral of 21 with respect to t would be
21t, but then integrate the second term,
let u be equal to negative t divided by 50.
So if u is equal to negative t divided by 50,
notice that differential u is equal to negative 1/50 dt.
So if we solve this equation for dt,
we'd multiply both sides by negative 50.
Notice at negative 50du is equal to dt.
When we integrate this,
we'll have an extra factor of negative 50.
So we'll have plus 75 and then times negative 50
times e to the u or e to the negative t divided by 50.
Let's go ahead and simplify this here,
75 times negative 50 is negative 3,750.
So we can write this as 1 over 32
and then we have 21t minus 3,750
times e raised to the power of -t divided by 50.
And now we'll evaluate this at 32 and then zero
and then find the difference.
Let's go ahead and do this on the next slide.
So when t is 32 we would have 21
times 32 minus 3,750
times e raised to the power of negative 32 divided by 50.
And then when t is zero,
we'd have 21 times zero
and then minus 3,750
times e raised to the power of, well is t is zero,
the exponent would be zero.
Let's go ahead and evaluate this on the calculator.
So we have 21 times 32
and then minus 3,750
times e
raised to the power of negative 32 divided by 50.
Running to two decimal places,
this would be approximately negative 1,305 point 35.
And then over here this would be zero.
e to the zero is equal to one, so we'd have negative 3,750.
So we'd have negative 1,305 point 35.
This becomes plus 3,750.
So we have one divided by 32
and then times negative 1,305
point 35
plus 3,750.
Running to one decimal place,
the average temperature would be
approximately 76 point 4 degrees Celsius.
I hope you found this helpful.