Question

Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be approximated by the formula P= -2x^2 + 8x + 2, where x is the number of units produced per week, in thousands.

a.How many units should the company produce per week to earn the maximum profit?

b.Find the maximum weekly profit

Answer 1

PART A:

To find the maximum point of any function we need to know where it's derivative equals zero: \[P(\max) \rightarrow \frac{ dP }{ dx }=0 \]
So to start we need to find the derivative: \[\frac{ dP }{ dx } = \frac{d}{dx}(-2x^{2}+8x+2)=-4x+8\] We can then find the maximum by setting this equal to zero: \[-4x+8=0\] and solving for x \[4x=8\]\[x=2\]

So, for maximum profit they should produce 2 units/week.


PART B

We know how man units they need to produce and we know the profit function. To find the profit amount we need to evaluate the profit function at x=2.
\[-2x^{2}+8x+2\]\[-2(2)^{2}+8(2)+2\]\[10\]

The maximum profit will be 10 * $100 = $1,000/week when 2 units are produced.

Answer 2

Is there a simpler way to explain this to me? @twvogels

Answer 3

Maybe. What part don't you understand?

Answer 4

For starters: To find the maximum point of any function we need to know where it's derivative equals zero:
P(max)→dP/dx=0

Answer 5

Have you taken Calculus 1 yet? (limits, derivatives etc.)

That's the fastest way to solve it but there are other ways to do it.

Answer 6

Nope.

Answer 7

Hmm...okay. You can also solve it by graphing the function. If you do that you can read the maximum right off the chart.

Answer 8

It will need to be fairly accurate though. Use a calculator or computer, it will be pretty hard to do it neatly by hand.

Answer 9

I have graphed it, but I don't get an answer that makes since. I keep getting a negative.

Answer 10

Okay hold on. I'll graph it and see what I get. It'll just take a second.

Answer 11

If you look at that graph you can read off the two numbers you need.

Answer 12

2, 10?

Answer 13

Yup, exactly. Just the coordinates of the vertex.

Answer 14

2 units, for a max profit of 10.

Answer 15

But that doesn't make since
These are the options for my answers:
a.4,000 units; $300
b.2,000 units; $1000
c.4,000 units; $200
d.2,000 units; $200

Answer 16

Oh the scale is just different. They say in the question: "weekly profit, P, in hundreds of dollars" and "number of units produced per week, in thousands"

So it not 2 and 10, it's 2*1000= 2000 and 10 * $100 = $1000.

Answer 17

OOOHHH! OK, I got it. Thanks. :)

Answer 18

No problem.