Expert: Paul Klarreich - 10/16/2007

Question
Hi Paul, I am in college algebra and I found a situation at work which I applied algebra to.  I created a math problem out of it, but I couldn't figure out how to create a function that relates time elapsed to average turnaround time.  Here it is.

A small business is trying to find out how long it will take for their average turnaround time to reach 14 days.  Turnaround time is calculated every 3 months.  Their current average turnaround time is 6.17 days.  Their current efficiency ratio is 0.0285.

The following formula can be used to calculate average turnaround time.

Y = RX + 3.32

Y = Average turnaround time

R = efficiency ratio

X = number of orders over 3 months


Every three months the company receives 20 more orders than the last three months.  The efficiency ratio decreases by 0.0015 every three months.  

a. Create a function of x, in which x represents time (x values measured by 3 months) and f(x) represents average turnaround time (in days).

b. Use the function to find how long it will take for the turnaround time to reach 14 days.

Answer
Questioner:   Adam
Category:  Advanced Math
Private:  No
 
Subject:  Complex algebra
Question:  Hi Paul, I am in college algebra and I found a situation at work which I applied algebra to.  I created a math problem out of it, but I couldn't figure out how to create a function that relates time elapsed to average turnaround time.  Here it is.

A small business is trying to find out how long it will take for their average turnaround time to reach 14 days.  Turnaround time is calculated every 3 months.  Their current average turnaround time is 6.17 days.  Their current efficiency ratio is 0.0285.

The following formula can be used to calculate average turnaround time.

Y = RX + 3.32

Y = Average turnaround time

R = efficiency ratio

X = number of orders over 3 months


Every three months the company receives 20 more orders than the last three months.  The efficiency ratio decreases by 0.0015 every three months.  

a. Create a function of t, in which t represents time (t-values measured by 3 months) and f(t) represents average turnaround time (in days).

b. Use the function to find how long it will take for the turnaround time to reach 14 days.
............................................
Hi, Adam,

Let  t = elapsed time in 3-month intervals. (Don't use x.)
(to go with these items as you correctly noted.)
    R = eff.ratio
    X = number of orders per interval.
    Y = turnaround time in intervals.

Now take each sentence and write a conclusion using your variables:

******* Sentence 1 ******
Their current average turnaround time is 6.17 days.  

Conclusion: At  t = 0,  Y = 6.17

******* Sentence 2 ******
Their current efficiency ratio is 0.0285.

Conclusion: At t = 0,  R = 0.0285

******* Sentence 3 ******
Every three months the company receives 20 more orders than the last three months.

Conclusion:  

X is a linear function of time with slope = 20.

X = 20t + b

Since we don't know the current number of orders, that will have to do.  We might be able to compute 'b' from other facts.

******* Sentence 4 ******
The efficiency ratio decreases by 0.0015 every three months.  

Conclusion:

R is a linear function of time with slope = -0.0015
(MINUS, because it is decreasing)

R = -0.0015t + c

But if we also use Sentence 2, we find that  c = 0.0285, so

R = -0.0015t + 0.0825

**************************
Now put things together:


Y = RX + 3.32

Y = (-0.0015t + 0.0825)(20t + b) + 3.32  [Equation 5]

We don't have b, yet.  But we have not used Sentence 1 yet, either.  Now we use it.  Put  Y = 6.17  at  t = 0:

6.17 = (0.0825)(b) + 3.32

and solve this equation for b.

6.17 = 0.0825 b + 3.32

0.0825 b = 6.17 - 3.32

0.0825 b = 2.85

b = 2.85/0.825

I'll leave the rest to you:  You will compute that b and put it back into equation 5, and THAT WILL BE YOUR FUNCTION.

Finally, you will take your function, set Y = 14, then solve for t.  You will need the quadratic formula and you will have to push some buttons on your calculator.

[I think Excel is good for that.  Program two cells with the two (plus and minus) forms of the Q.F.]

You will get two solutions for t.  It will be up to you to decide which one applies to the practical situation.  If you are not sure, check with your broker.