The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t), where t is in min

Question

1 month ago

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The temperature, H, in degrees Celsius, of a cup of coffee placed on the kitchen counter is given by H = f(t), where t is in min

utes since the coffee was put on the counter.
a) Is f'(t) positive or negative? Why?
b) What are the units of f'(35)?
c) Suppose that |f'(35)| = 1.5 and f(35) = 68. Fill in the blanks (including units where needed) and select the appropriate terms to complete the following statement about the temperature of the coffee in this case.
At ____ minutes after the coffee was put on the counter, its temperature is ____ and will ____ by about ____ in the next 30 seconds.

Mathematics

1 answer:

Leona [12.6K] · Answer 1 · 2026-02-25T07:38:07+03:00

Response:

a

f'(t) is negative

b

The unit is degC/min

c

At 35 minutes since the coffee was placed on the counter, the temperature is 68, and it will drop by approximately 0.75 in the next 30 seconds.

Detailed explanation:

According to the question, we are informed that

The temperature of a cup of coffee on a counter is represented by the equation

$H= f(t)$

where t indicates the time in minutes since the coffee was set on the counter.

In general, f(t)' denotes the derivative of f(t), indicating the change in temperature over time for the coffee

As the coffee rests on the counter, its temperature tends to decrease after a few minutes; thus, f(t)' is negative.

The temperature is measured in degrees Celsius while time is expressed in minutes.

Therefore, the change in temperature with respect to time, f(35)', will be given in degrees Celsius per minute (i.e., degC/min).

The question states that

$|f(35)'| = 1.5$

which means the rate of temperature change after 35 minutes is 1.5.

And

$|f(35)| = 68$

indicating that the coffee's temperature after 35 minutes is 67 degrees Celsius.

Subsequently, after another t= 30 seconds = $\frac{30}{60}= 0.5 \ minutes$ the rate at which the coffee's temperature changes is

$R = |f(35)'| * t$

=> $R = 1.5 * 0.5$

=> $R = 0.75$

Thus

At 35 minutes since the coffee was placed on the counter, its temperature is 68 and will drop by roughly 0.75 in the next 30 seconds.