# Thread: Optimization and L'Hospital's rule

1. ## Optimization and L'Hospital's rule

1) $y = \lim_{x\to\infty} x^{\frac{ln(7)}{1+ln(x)}}$
$ln(y) = \lim_{x\to\infty} \frac{ln(7)ln(x)}{1+lnx}$
Deriving the right side gives me:
$ln(y) = \lim_{x\to\infty} \frac{\frac{ln(x)}{7}+\frac{ln(7)}{x}}{\frac{1}{x} }$ $=> \frac{\infty + 0}{0}$

Since the top goes towards a positive number (infinity) and the bottom goes to 0, $ln(y) = \infty => y = e^{\infty}$

Is my logic wrong?

2) A cone-shaped paper drinking cup is to be made to hold 33 cm^3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Give your answers correct to two decimal places.)

$V = 33 = \frac{1}{3} \pi r^2h$
$SA = \pi rs + \pi r^2$ where $s = \sqrt{h^2+r^2}$
So
$SA = \pi r\sqrt{h^2+r^2}+ \pi r^2$
$h = \frac{99}{\pi r^2}$
$=> SA = \pi r \sqrt{(\frac{99}{\pi r^2})^2 + r^2}+ \pi r^2$

Am I starting this off the right way? Because if I am, this looks disgusting. I was trying to derive the Surface Area formula just in terms of r, and then finding the minimum I could plug it back into the volume formula to gind the minimum h?

Am I wrong in my approach?

2. Originally Posted by Open that Hampster!
1) $y = \lim_{x\to\infty} x^{\frac{ln(7)}{1+ln(x)}}$
$ln(y) = \lim_{x\to\infty} \frac{ln(7)ln(x)}{1+lnx}$
Deriving the right side gives me:
$ln(y) = \lim_{x\to\infty} \frac{\frac{ln(x)}{7}+\frac{ln(7)}{x}}{\frac{1}{x} }$ $=> \frac{\infty + 0}{0}$

Since the top goes towards a positive number (infinity) and the bottom goes to 0, $ln(y) = \infty => y = e^{\infty}$

Is my logic wrong?

2) A cone-shaped paper drinking cup is to be made to hold 33 cm^3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Give your answers correct to two decimal places.)

$V = 33 = \frac{1}{3} \pi r^2h$
$SA = \pi rs + \pi r^2$ where $s = \sqrt{h^2+r^2}$
So
$SA = \pi r\sqrt{h^2+r^2}+ \pi r^2$
$h = \frac{99}{\pi r^2}$
$=> SA = \pi r \sqrt{(\frac{99}{\pi r^2})^2 + r^2}+ \pi r^2$

Am I starting this off the right way? Because if I am, this looks disgusting. I was trying to derive the Surface Area formula just in terms of r, and then finding the minimum I could plug it back into the volume formula to gind the minimum h?

Am I wrong in my approach?
You don't need L.H

$\ln(y) = \lim_{x\to\infty} \frac{\ln(7) \ln(x)}{1+\ln(x)}= \lim_{x\to\infty} \frac{\ln(7)}{1+\frac{1}{\ln(x)}}= \ln(7)$

Your start seems fine on the cone except I don't think you need the

$\pi r^2$ term this would put a top on the cone cup.

### A cone-shaped paper drinking cup is to be made to hold 36 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. (Round your answers to two decimal places.)

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