Solving Proportionality: Cost of Running a Business w/ 18 Employees

[lektor]

Hi a few questions i want to check my answers for here,


3. Write down an expression, evaluate the constant of proportionality, and answer the question.
[Remember to work in SI units. Simplify answers to 1 d.p. only.]
(a) The annual cost (C) of running a business is partly constant and partly proportional to the number of employees.
If there are 20 employees, the annual cost is $400000.
If there are 25 employees, the annual cost is $470000.
Find the annual cost if there are 18 employees

What is meant by partly constant? how would this be written.

C = kN? or C = kN + c?

 

[assyrian_77]

The second one since I guess you mean a constant by c. The first one is as you can see "fully" proportional.

 

[lektor]

hrmk so finished that question and was hoping for some help on this next one.

In a bookstore, the selling price of a certain dictionary is $x. The total profit ($P) on the sale of the dictionary is partly proportional to the selling price of the dictionary and partly proportional to the square of the selling price of the dictionary. When x = 35, P = 78750; when x = 60, P = 60000.
(a) Express P in term of x.
(b) Find the total profit if the selling price of the dictionary is $65.

For a) I am not certain but this is my expression,

P = ( kx ) + (x^2) + c
P = x(k + x) + c

with my values of c = 107100 and k = -845 found from info given..

so when x = 65 P = 65(-845+65) + 107100
P = $56400

cheers, any corrections and pointers would be great.

 

[assyrian_77]

The total profit ($P) on the sale of the dictionary is partly proportional to the selling price of the dictionary and partly proportional to the square of the selling price of the dictionary.

It doesn't say partly constant, so why is 'c' in the equation? With a nonzero 'c' in your equation, you would have a profit even if the selling price is zero. No, think about it again. What should the equation be?

 

[lektor]

ahah woops, my bad..

So in that case it will be

P = x . x^2
P = Kx^3

However when i solve for k the answers are different..

 

[HallsofIvy]

I find the phrasing "partly proportional to the selling price and partly proportional to the square of the selling price" confusing but I'm sure "P= k x (x²)= x³" is not intended since that would simply be "proportional to the cube of the selling price". You were closer with your first idea: P= kx+ (x²)+ c. As Assyrian_77 said, drop the c since nothing is said about a constant- and remember that "proportional to the square of the selling price is "cx²". You don't have a constant of proportionality on the x².