# Exponential function help

1. Aug 8, 2014

### kevinshen18

So this question is from Khan Academy. I understood the first part and chose the correct function, but the second question(from 40 degrees to 30 degrees change) explanation confused me.
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QUESTION:
Ajay made a steaming pot of stew. When he turned the stove off, the stew’s temperature was 70° Celsius, and it started cooling down to the room temperature, which was 20° Celsius.
The following expressions give the temperature of the stew, m minutes after Ajay turned the stove off. They are all equivalent, save for some rounding of the coefficients.
Which expression best highlights the amount of time it takes the difference between the stew’s temperature and the room’s temperature to drop by 50%?

20+50·0.50^12m

20+0.5·100^0.92m

20+50·0.20^052m

How long will it take the stew's temperature to go from 40° to 30°? Round your answer, if necessary, to two decimal places.

SOLUTION/EXPLANATION:
All the expressions are either given in the form 20+A·B^am where a is a constant number, or they can be quickly transformed to this form. In this form, A·B^am is the difference between the stew's temperature and the room's temperature. This implies that the difference between the temperatures is multiplied by B when a·m=1.
For instance, let's consider the expression 20+50·0.20^052m. The expression 0.052m is equal to 1 when m is approx = 19.23 minutes. This tells us that the difference between the temperatures is multiplied by 0.2 every 19.23 minutes. This is interesting information, but what we really want to know is how often the difference drops by 50%, not how often it drops by 80%.
The difference between the temperatures dropping by 50% is the same as the difference multiplied by 0.5. So if the expression contains 0.5 at the base of the exponentiation, we know this expression highlights the amount of time it takes the difference between the temperatures to drop by 50%.
The expression that has 0.5 at the base is 20+50·0.50^12m, so this is the expression we are looking for.
20+50·0.50^12m implies that the difference between the temperatures is multiplied by 0.5 when 0.12m=1. In other words, the difference drops by 50% when m is approx = 8.33 minutes.
Since the room's temperature is 20°, the corresponding temperature differences for the stew's temperature being 40° and 30° are differences of 20° and 10°.
We know that the percent rate of change of exponential expressions is constant. This means that the time it takes the difference between the temperatures to drop by 50% is the same whether the absolute change is from 50° to 25° or from 20° to 10°. Therefore, the amount of time it will take the stew's temperature to go from 40° to 30° is 8.33 minutes.
The expression that best highlights the amount of time it takes the difference between the stew’s temperature and the room’s temperature to drop by 50% is:
20+50·0.50^12m
The amount of time it will take the stew's temperature to go from 40° to 30° is 8.33 minutes.
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I got confused when they said the temperature would take the same amount of time to go from 20 to 10 and 40 to 30. But 40 to 30 isn't a -50% change. How does that work?

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2. Aug 8, 2014

### Simon Bridge

... that is because you have misread what they said.
You have to realise that the target temperature is 20deg. It is the difference, between the actual temperature and the target, that has to halve.

So the stew going from 40deg to 30deg means that the temperature difference goes from (40-20)deg to (30-20)deg. which is 20-10deg ... which is a halving of the difference.

3. Aug 9, 2014

### kevinshen18

Ah Okay, I see! Thank you!

4. Aug 9, 2014

### Simon Bridge

No worries: it gets confusing when they talk like that :)