Mathematics Problem

I am having trouble with the whole of this question for some reason and I can't understand where I am going wrong.

Two students, Alan and Betty, work part time washing dishes in a local restaurant. After a particularly busy night each is faced with a mountain of identical dinner plates to wash. They both have their own sink to work at. Water is supplied to each sink at 60°C.

This is too hot for Alan who adds cold water until the temperature of the water in the sink is 50°C. Betty, however, has a pair of rubber gloves and can stand the hotter water. The temperature in the kitchen is 20°C.

Both students are studying an engineering degree course at their university and know that a possible mathematical model for the washing up process is

T(n) = p + qe^(-0.02n)

where T °C is the temperature of the water and n is the number of plates washed, while p and q are constants.

(a) Work out the values of p and q

(i) for Alan (ii) for Betty

The students begin washing up and keep going until the water temperature in each sink drops to 25°C.

(b) How many whole plates have been washed.

(i) by Alan (ii) by Betty?

(Give your answers to the nearest plate).

Alan, who has washed fewer plates, resumes the job and continues until Betty’s total has been matched.

(c) What is now the temperature of the water in Alan’s sink?
(Give your answer to 3SF)

I understand part (b) which is just using Natural Logs and part (c) which just uses the numbers from the previous 2 parts but part (a) is throwing me off.

As the number of plates washed at the start would be 0, my equation to solve for Alan would be 50 = p + q and for Betty would be 60 = p + q which are incorrect due to 2 unknowns and I cannot figure out how to solve this question.

Reply 1

ghostwalker

17

Original post by D556mm

...

Hint: Consider

\displaystyle \lim_{n\to\infty}T(n)

Reply 2

D556mm

OP

Original post by ghostwalker

Hint: Consider

\displaystyle \lim_{n\to\infty}T(n)

So .. 50 = p + qe^(-0.02 * a large value) ?
I tried 1 million and got that 50 = p but then that would mean in the next part that you would have to take a Natural Log of a negative which is undefined

:s-smilie:

Reply 3

ghostwalker

17

Original post by D556mm

So .. 50 = p + qe^(-0.02 * a large value) ?
I tried 1 million and got that 50 = p but then that would mean in the next part that you would have to take a Natural Log of a negative which is undefined

:s-smilie:

So you're telling me that after Alan has done an infinite number of plates the water temperature is still 50!

And you don't need the log of a negative number when doing part b). Post your working, once you've sorted a).

Reply 4

KSP

9

I think you need to consider both the limit as n tends to infinity and also the case when n=0. Remember that the water temperature can't fall below room temperature either. That should give you enough information to determine p and q.

(edited 11 years ago)

Reply 5

D556mm

OP

What I'm getting here is:

n tends to infinity:
50 = p + qe^(-0.02*1000000)
e^(-0.02*1000000) = 0
50 = p + (q*0)
50 = p which would mean q = 0

n tends to 0:
50 = p + qe^(-0.02*0)
e^(-0.02*0) = e^0 = 1
50 = p + (1*q)
50 = p + q (many values for both p and q)

Apologies for all these problems, I've just been stuck on it for the past 2 days and I can't get my head around it at all :s-smilie:

Reply 6

F1Addict

12

Original post by D556mm

What I'm getting here is:

n tends to infinity:
50 = p + qe^(-0.02*1000000)
e^(-0.02*1000000) = 0
50 = p + (q*0)
50 = p which would mean q = 0

n tends to 0:
50 = p + qe^(-0.02*0)
e^(-0.02*0) = e^0 = 1
50 = p + (1*q)
50 = p + q (many values for both p and q)

Apologies for all these problems, I've just been stuck on it for the past 2 days and I can't get my head around it at all :s-smilie:

The temperature of the water after an infinite number of plates have been washed is not 50. Think about it, with each plate washed, the temperature drops a little. What temperature will the water eventually settle to after washing enough plates?

\displaystyle \lim_{n\to\infty} p+qe^{-0.02n} = p \not= 50

For Alan:

T(0)=50 \not= T(\infty)

Same for Betty, but note that she has a different T(0) to Alan.

(edited 11 years ago)

Reply 7

ghostwalker

17

Original post by D556mm

...

OK, for Alan:

For n=0: Since the water starts at 50 degrees, we have.

50 = p + q.

As n tends to infinity: The water will have cooled down to ambient(room) temperature, so:

20 = p + 0

And solve simultaneously.

Reply 8

D556mm

OP

Ok, I think I got it.

Alan:
p = 20, q = 30
Washes 89.59 dishes = 90 dishes (or would that be rounded down as it is for wholes plates?)

Betty:
p = 20 and q = 40
Washes 103.97 = 104 (again, rounded down or up?)

and finally Alan's temp when matched with the number of dishes that Betty washed (I used 104 as n) would be 23.7°C (3sf)

Reply 9

ghostwalker

17

Original post by D556mm

...

They seem reasonable. I'm not checking arithmetic at this level. You can plug values back into the appropriate equation to check.

As to whether you round up or down, I don't know. The instructions in the question seem contradictory to me if your decimal part is >= 0.5.