exponents? Degrees? Word Problem from GMAT software

[guest612]

GMATPREP CAT 2:

The temperature of a certain cup of coffee 10 minutes after it was poured was 120 degrees Fahrenheit. If the temperature F of the coffee t minutes after it was poured can be determined by the formula: F = 120 (2^(-at)) + 160, where F is in degrees Fahrenheit and a is a constant, then the temperature of the coffee 30 minutes after it was pourced was how many degrees Fahrenheit?

A. 65
B. 75
C. 80
D. 85
E. 90

Correct Answer: B. 75.

Please help! I've tried to plug in 120 for F, and 10 for t, and couldn't get to any answer close to the provided answer choices. Can you please tell me how to best solve this? Secondly, if this question comes up and I'm kind of stuck is there a way to eliminate the answer choices? I understand it may not work for this question since the values are not that far apart.

Thank you so much in advance!

[StaceyKoprince]

Can you double-check that you have inputted the problem completely correctly?

Specifically, what you've typed would give us the function:
120 = 120 (2^(-10a)) + 160
This can't work - we could simplify one step to get
-40 = 120 (2^(-10a))
and now the left side is negative and the right side is positive - they can't equal each other.

In fact, the whole term 120 (2^(-10a)) is always non-negative (regardless of the value or sign of a), so when you add the 160 at the end, the minimum temperature we could have here is 160 and that doesn't make sense. Is it supposed to say -160 instead of +160?

[guest612]

I'm sorry. It's + 60 (not +160).

[StaceyKoprince]

Okay, that makes a lot more sense!

F = 120 (2^(-at)) + 60

First data point is (10,120); use that to find a.

120 = 120(2^(-10a)) + 60
60 = 120(2^(-10a))
1/2 = 2^(-10a)
1/2 = 1/(2^(10a))
(1/2)^1 = (1/2)^(10a)
drop the bases and set the exponents equal to each other
1 = 10a
1/10 = a

Then plug that in to find F when t=30:
F = 120(2^(-30*1/10)) + 60
F = 120(2^-3) + 60
F = 120(1/8) + 60
F = 15 + 60
F = 75

[guest612]

thank you. you make it look so easy! :)

[StaceyKoprince]

It's only easy once you already know how to do it. :)

[nidhishnair]

Okay, that makes a lot more sense!

F = 120 (2^(-at)) + 60

First data point is (10,120); use that to find a.

120 = 120(2^(-10a)) + 60
60 = 120(2^(-10a))
1/2 = 2^(-10a)
1/2 = 1/(2^(10a))
(1/2)^1 = (1/2)^(10a)
drop the bases and set the exponents equal to each other
1 = 10a
1/10 = a

Then plug that in to find F when t=30:
F = 120(2^(-30*1/10)) + 60
F = 120(2^-3) + 60
F = 120(1/8) + 60
F = 15 + 60
F = 75

Hi Stacey,
An improvisation=>I think we can eliminate a couple of steps here by just finding till 2^-a10.
Since 30 mins is 3 *10 mins.
120 = 120(2^(-10a)) + 60
60 = 120(2^(-10a))
1/2 = 2^(-10a)

cubing both sides => 1/8=2^(-a30)

Plug this back in the eqn.
F = 120(1/8) + 60
F = 15 + 60
F = 75

We dont have to deal with exponents and it can maybe save some valuable time :D

[RonPurewal]

nidhishnair -- yes, that approach will save time.

there are 2 reasons why we (instructors) don't generally show such solutions on the forum, though:
1/ they are usually highly particular -- i.e., they are "clever" ways to streamline the single problem at hand, but it's unlikely that they will be of use on any future problem;
2/ they have a huge intuitive component ("the lightbulb has to go on", so to speak) and aren't as teachable as more standard methods.

but, yes, your reasoning is valid.

[BernardK777]

Why do you cube 1/2 = 2^(-10a) as oppose to multiplying by 3 if your only multiplying the time by 3 (not cubing it)? Thanks.

[tim]

Please see Ron's post. The solution you're referring to is not a standard solution and requires some insight to make it work. If it doesn't make sense to you, there's not much we'll be able to do to show you how to make that connection. That's why we recommend you stick to the other solution unless this new one just comes to you independently.