Derivative and marginal profit

[stanton]

Homework Statement

A company's Marginal profit is P'(x) = -0.0105x^2 + 0.14x + 25 where x represents hundreds of items and profit is thousands of dollars
a) Estimate the change in profit it the production is raised from 80 to 81 hundred units. Should the manager increase production?
b)How many hundreds of items should the company be making in order to maximize profit?

Homework Equations

∆y≈dy=f '(c)dx

The Attempt at a Solution

∆P = dP = P'(c) dx
C=8000, dx=100
dP= [-0.01059(8000)^2 + 0.14(8000) + 25](100)
So solving this equation I got dP = -67085500
And since the profit is expressed in thousands of dollars, I divided my answer by 1000. So I got $ -67085.5 for the answer to question (a). So I wrote profit [increase] by $ 67085.5
Am I doing right? Because my prof explained a sample probelm similar to this, and the answer was positive, so she wrote :decreases by xxx. And I thought if the answer is negative, I should write opposite from her. So I concluded it is increasing. But I am not sure.

And about problem (b), will I be able to get maximum if I set P'(x)=0 and have a test?

 

[Mark44]

One thing I see right away is that you are using different units than are given in the problem. x is in units of hundreds, and the marginal profit is in terms of thousands of dollars. Do your calculations in terms of those units, and then, later, do your conversions. For example, C = 80 and dx = 1. See what you get with those numbers.

 

[stanton]

Oops! Thank you so much. the answer was -31, using 80 as c and dx as 1. So profit has $ 31000 increased?

 

[Mark44]

The profit will decrease by about $31,000.

 

[stanton]

Thank you so much. :)