Help please, we didn't learn this in class

Question

The profit (in dollars) for a certain commodity, n, where n is in units, is given by the function:

P(n)= -25n^2+350n+1275

A) What is the input variable and its units ?

B) What is the output variable and its units ?

Answer 1

P(n) = -25n^2 +350n +1275

output is P. It's units are in dollars

Profits is a downward opening parabola with vertex (7, 2550) for maximum profit at n=7, P =$2,550

input is "n" which in business or economics usually would be an integer number of products

but that would mean P(0) = $1,275 profit for zero production which is strange

-25n^2 + 350n + 1275 = 0

n = -350/-50 + or - (1/50)sqr(350^2 [phone])

n = 7 + or - (1/50)sqr(122500 +510000)

n = 7 + or - 722.7/50

n = 7 + or - 14.45

n = 21.45 or - 7.45 for zero profits or about 21 units

usually you find maximum profit by taking the derivative & setting it = 0

P'(n) = -50n +350=0

50n = 350

n =350/50 = 7 units to minimize profit

max profit = -25(7)^2 +350(7)+ 1275 =1275 + 1275= 2550

side note, also virtually irrelevant to the problem is:

P = profits = Revenue -Costs. Usually there are variable costs and fixed costs. VC and FC

P(n) = R(n) -VC(n) - 1275

P(n) = R(n) -C(n) = R(n) - VC(n) - FC where FC would = 1275. Fixed Costs are a constant, not a function of n, the input n.

but

R(0) - VC(0) -1275 = 1275 where V(0) = 0

if P(0) = 1275 then R(0) = 1275 + 0 +1275 = $2,550

at zero production, the Revenue is somehow = twice the fixed costs

P(0) = 1275 = R(0) -VC(0) - 1275

R(0) - VC(0) = 2550

R(0) - 0 = 2550

R(0) = 2550

either Revenue at zero production = $2,550

or

VC(0) is negative, Variable "Costs" <0

IF R(0) = 0, then VC(0) = -$2,500, meaning variable "costs are negative at zero production, somehow creating revenue at zero production.

or some combination of Revenue positive with no sales and

variable "cost's negative generating revenue with no production.

which is doubly strange