word problems

Question

A bowl of pudding  at 76 degrees F is placed in a refrigerator that maintains a constant temperature of 35 degrees F. The temperature T of  the pudding t minutes after it is placed in the refrigerator is given by 

 

T(t)= 35+41 e^-0.058t     Find the temperature of the pudding 15 minutes after it is placed in the refrigerator. (round to the nearest tent of a degree) Please show all work. 

Answer 1

This word problem directly gives us an equation for temperature as a function of time (meaning that if we plug in any value of time, t, in minutes, we can calculate the expression to determine the temperature, T, at that time.

 

T(15) = 35 + 41 e^(-.058(15))

 

T(15) = 35 + 41 e^(-.87)

 

T(15) = 35 + 41 (.41895)

 

T(15) = 35 + 17.177

 

T(15) = 52.2, rounded to the nearest tenth of a degree

Answer 2

To find temperature after 15 minutes (i.e. T(15)), plug in t=15 into the equation:

 

T(15) = 35 + 41e^(-0.058*15)

         = 35 + 41e^(-.87)

         = 35 + 41(.412)

         = 52 F

Answer 3

T(t)= 35+41 e^-0.058t  which, for t=15 is

T(15)= 35+41×e^(-0.058×15)=35+41×e^(-.87)=35+41×.41895155=52.1770135 or to the nearest tenth of a degree 52.2 

Answer 4

T(t) = 35 + (41)e^(-0.058)t

 

T(t) = 35 + (41)e^{(-0.058)(15 minutes)} = 35 + (41)e^-0.87 = 35 + (41)(0.419) = 52.2°

 

 

Answer 5

So you need to apply your equation:

 

T(t) = 35 + 41e^{-0.058 t}

 

to find T.

 

They want you to figure out the temperature of the pudding, and they have told you they decided to call it T.

 

The notation T(t) means T is a function of t.  In this case, they have said the equation describes the temperature of pudding placed in the refrigerator t minutes ago.  A function is a relationship between an input variable and an output variable, such that for every input value there is exactly one possible output value.  Thus, writing the equation as a function "T of (t)" reveals that at every moment t (the input), there is one pudding temperature T (the output), and the equation defines how T changes as time passes.

 

e is just a particular number, just as pi's value to four digits is 3.141... and infinitely many digits after that which matter very little to most people.  It doesn't have a really simple name, but it is known as the base of the natural logarithm, or just "e".  e's actual value to four digits is 2.718.... and there are infinitely many digits of precision past that, which matter little to most of us.

 

You should have an "e^{^}" or button available if you have a suitable type of calculator for this question.  For that reason, many people don't even actually remember what number e is to four digits, because they can always compute e^1 if they really need to know.  

 

To make sure you are using an appropriate calculator correctly, try to compute e^1 yourself. You should be able to see that e = 2.718... as I've stated.

 

If you correctly raise e to the second power, you'll be computing e * e (the product of two copies of e), and you should get 7.389...  and a bunch more digits.

 

 

Then, what you need to do to apply the equation you've been given is to raise the number e to the power of :

the product of 0.058, a constant, and t, the number of minutes since the pudding was placed in the refrigerator at the moment in the story of the problem that you are being asked to find the temperature of the pudding.

 

In the story, the pudding was placed in the refrigerator 15 minutes ago, so in your problem, t = 15. In other words, use "(15)" in the equation instead of "t", in other words, replace "t" with "15".

 

You'll want to raise "e" to the power of that product (-0.058 and t).

 

If you don't have an e^ button, realize that computing the negative power of something is like computing the reciprocal of the positive power of that same thing.  

 

In other words to compute e^-x, you can just compute 1/(e^x).  Also, you can use the y^x button on a calculator that has one, if you know that for your question y = e = 2.718.

 

In case you need more digits of precision, here is e to twelve digits

e = 2.718281828459...

 

I hope you can see how to do the rest.   If not, please post again.

 

I'd give more details if you give more details on why you think this problem is hard or what it is about the problem that you don't understand.