This following question is from a calculus course.
A ball is supposed to be manufactured with radius 10cm. If it gets made with a radius of 9.97cm instead, it would take up approximately how much less volume (in cubic centimeters)? Do not use a calculator.
I got the wrong answer and am not sure how to do this problem. My wrong answer:
$$(4\pi*10^3)/3 - (4\pi*9.97^3)/3$$
Is not the difference of the two volumes how much less the volume with 9.97 radius is? How do I solve? Thank you.
Again I run into "cannot edit comments after five minutes!
Here is what I meant to write: A major error is that the volume of a sphere is given by $\frac{4}{3}\pi r^3$, not $\frac{4}{3}\pi^3 r$ as you have! I think you have misunderstood the question. Yes, what you have (if you cube r rather than $pi$) is the calculation you would do the exactly calculate the difference. But you haven't done the calculation. Also the problem said "approximation" and. by you title you were supposed to use differentiation!
For any function, f(x), the derivative is defined as $\lim_{h\to 0} \frac{f(x+h)- f(x)}{h}$. That difference in volumes can be approximated by the derivative, $4pi r^2= 4\pi(10^2)= 400\pi$, times 10- 9.97= 0.03.