Could the "rate" refer to the rate at which the balls were balls being sold which would be the instantaneous rate of change at x=3, and therefore the slope of the tangent at x=3? This would make more sense going by what we have recently covered.....This is what I wrote...
Assuming the question is asking for the rate of balls being sold, as in the tangent at x=3, the following is true:
I agree that the intention of the question was most likely that you should find the slope of the tangent - but just what is that slope? P(x) is weekly profit in dollars. The question you will be answering is, "
What is the rate of change of profit with respect to sales volume, when sales equal 300 cans per week?" Keep track of units.
\(\displaystyle P(x) = 180x - 2x^2 \; \; \; \; \; \)gives Profit ($ per week)
\(\displaystyle P(3) = 180(3) - 2(3)x^2 \)
\(\displaystyle P(3) = $522\; per\ week\)
The slope of the tangent of P(x) at (3, 522) is
\(\displaystyle \displaystyle \lim_{h\rightarrow 0} \dfrac{P(3+h) - P(3)}{h}\; \; \; \; \)P is $/week and h is 100 cans/week, so the ratio is $/(100 cans)
\(\displaystyle \displaystyle \lim_{h\rightarrow 0} \dfrac{180(3+h)-180(3)-2(3+h)^2+2(3)^2}{h}\)
\(\displaystyle \displaystyle \lim_{h\rightarrow 0} \dfrac{180\ h -12\ h - 2\ h^2}{h}\)
\(\displaystyle \displaystyle \lim_{h\rightarrow 0} \dfrac{178\ h - 2\ h^2}{h}\)
\(\displaystyle \displaystyle \lim_{h\rightarrow 0} \left( 178 - 2\ h\right) = $178 \; per\ 100\ cans\)
BTW, you had an arithmetic error, 180-12=178. Other than that, your method to find the slope was correct - too bad that isn't what the question actually asked for - but probably intended.
