Exercise
A machinist is required to manufacture a circular metal disk with area $1000$ cm$^2$.
What radius produces such a disk?
If the machinist is allowed an error tolerance of $\pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?
In terms of the $\epsilon$, $\delta$ definition of $\lim \limits_{x \to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $\epsilon$ is given? What is the corresponding value of $\delta$?
Solution
Note: I've decided to solve part (3) before part (2), as it helps set the stage of the solution of (2).
1. What radius produces such a disk?
$\pi r_0^2 = 1000 \implies r_0 = \sqrt{\frac{1000}{\pi}} \approx 17.8412$
3. In terms of the $\epsilon$, $\delta$ definition of $\lim \limits_{x \to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $\epsilon$ is given? What is the corresponding value of $\delta$?
$a = r_0 \approx 17.8412$
$L = \pi r_0^2 = 1000$
$x = r$
$f(x) = \pi r^2$
$|r - r_0| < \delta$
$|\pi r^2 - \pi r_0^2| < \epsilon$
2. If the machinist is allowed an error tolerance of $\pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?
$|\pi r^2 - \pi r_0^2| < \epsilon$ $\implies |\pi r^2 - 1000| < 5$ $\implies -5 < \pi r^2 - 1000 < 5$ $\implies 995 < \pi r^2 < 1005$ $\implies \frac{995}{\pi} < r^2 < \frac{1005}{\pi}$ $\implies \sqrt{\frac{995}{\pi}} < r < \sqrt{\frac{1005}{\pi}}$ $\implies 17.7966 < r < 17.8858$
$|r - r_0| < \delta \implies |r - 17.8412| < \delta$
$|17.7966 - 17.8412| < \delta \implies 0.0446 < \delta$
$|17.8858 - 17.8412| < \delta \implies 0.0446 < \delta$
$\delta = \min(0.0446, 0.0446) = 0.0446$
Answer
1. What radius produces such a disk?
$$r_0 = 17.8412 \text{ cm}$$
2. If the machinist is allowed an error tolerance of $\pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?
$$\delta = 0.0446 \text{ cm}$$
3. In terms of the $\epsilon$, $\delta$ definition of $\lim \limits_{x \to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $\epsilon$ is given? What is the corresponding value of $\delta$?
$$|r - r_0| < \delta$$
$$|\pi r^2 - \pi r_0^2| < \epsilon$$
Request
Is my answer correct? If not, in what part of my solution did I make a mistake?