A couple calc questions [optimization, max/min/inflection]

**jelloish** · December 7th 2008, 07:19 AM

The depth of water at my favorite surfing spot varies from 4 to 1 feet, depending on time. Last sunday, high tide occurred at 5:00 a.m. and the next high tide occurred at 6:30 p.m. Obtain a cosine model describing the depth of the water as a function of time t in hours since 5 a.m. Sunday morning. Find the fastest rate at which the tide was rising on Sunday. Waht what time (after 5 a.m Sunday) did that first occur?
-I managed to get the cosine model, but I'm not sure where to go from there. I have y=1.5cos(13.5t)+5.5 as the model.

The function f(x) = x cos( ln x ) has infinitely many critical points. Find the smallest x-value in the interval [1,∞) for which f(x) a relative minimum point, a relative maximum point, and an inflection point.
-I'm not sure where to go if the function has infinitely many critical points. I know you're supposed to derivate, set the function equal to zero (finding critical points), and then check to see where it has max/mins, but if it has infinitely many I'm not sure where to go with that.

Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a sphere of radius a.
-I drew the diagram but I'm having a hard time trying to figure out where to go from there.

**Skalkaz** · December 7th 2008, 07:53 AM

Dear jelloish,

I can see a contradiction:
1) varies from 4 to 1 feet
2) y=1.5cos(13.5t)+5.5 --> from 4 to 7

The function f(x) = x cos( ln x ) has indeed infinitely many critical points what is easy to find with the derivation. But it hasn't global min or max because of x multiplying.

"I drew the diagram..." --> That is very well!
Let be r the radius of cylinder. Than try to find out the high (h) of cylinder from the fact: "can be inscribed in a sphere of radius a". Oke, now express the volume of cylinder on depending r:
V(r) = r*r*pi*h(r)
This is the function. Where is its maximum?

**skeeter** · December 7th 2008, 07:57 AM

correction for your model ...

$y = 1.5\cos\left(\frac{4\pi}{3} \cdot t\right) + 2.5$

$\frac{dy}{dt}$ is the rate of change of the tide level

$\frac{dy}{dt} = -2\pi \sin\left(\frac{4\pi}{3} \cdot t\right)$

this is the function you want to maximize ... it will have a maximum positive value when $\sin\left(\frac{4\pi}{3} \cdot t\right) = -1$ .

for the second problem, find the smallest critical value that is $\geq 1$

$f(x) = x \cdot \cos(\ln{x})$

$f'(x) = -x \sin(\ln{x}) \frac{1}{x} + \cos(\ln{x})$

$f'(x) = \cos(\ln{x}) - \sin(\ln{x})$

$f''(x) = -\frac{1}{x} [\sin(\ln{x}) + \cos(\ln{x})]$

for critical values where f'(x) = 0 , think where $\cos(\ln{x}) = \sin(\ln{x})$ ...

$\ln{x} = \frac{\pi}{4} \, , \, \frac{5\pi}{4} \, , \, \frac{9\pi}{4} \, ...$

the first max is at $x = e^{\frac{\pi}{4}}$ and the first min is at $x = e^{\frac{5\pi}{4}}$ .
you can use the 2nd derivative test to confirm.

for the first inflection point ...

$\sin(\ln{x}) = -\cos(\ln{x})$

$\ln{x} = \frac{3\pi}{4}$

$x = e^{\frac{3\pi}{4}}$

**jelloish** · December 7th 2008, 09:25 AM

Correction! I must have copied the question down wrong, for the tide question, the spot varies from 4 to 17 feet. I forgot the 7 when copying down the problem and the 1 when I was solving the problem.

I have the correct model now- y=6.5cos[13.5t]=10.5. Skeeter, how did you get 4pi/3 for the center part of the model?

**jelloish** · December 7th 2008, 09:29 AM

Skeeter, I got to the part cos(lnx)=sin(lnx) by myself when solving the third problem but I wasn't sure how to solve that. How exactly did you get the e^pi/4, etc.?

**jelloish** · December 7th 2008, 09:34 AM

Originally Posted by Skalkaz

"I drew the diagram..." --> That is very well!
Let be r the radius of cylinder. Than try to find out the high (h) of cylinder from the fact: "can be inscribed in a sphere of radius a". Oke, now express the volume of cylinder on depending r:
V(r) = r*r*pi*h(r)
This is the function. Where is its maximum?

I'm not sure how to find the max because there aren't any numerical values. I know you find the derivative of the function, find critical points, and test them to see if they're max/mins, but if I can't find the actual "points" because there are no values, I'm not sure how to go about doing that.

If V(r)=r^2*pi* h(r), how do I derivate h(r)?

**Skalkaz** · December 7th 2008, 10:24 AM

sin(ln(x)) = cos(ln(x)) --> tan(ln(x)) = 1

**Skalkaz** · December 7th 2008, 10:31 AM

If you make the profil as sketch, you will give a circle with radius a and rectangle with side 2*r and h(r). Isn't true?

Try to find a connection between this 3 values and express h(r). Viva geometry!!!

**skeeter** · December 7th 2008, 12:45 PM

my mistake ... I saw 5:00 AM and thought the next high tide was at 6:30 AM, not PM as I see now.

period = 13.5

$B = \frac{2\pi}{13.5} = \frac{4\pi}{27}$

so ... readjusting your amplitude and period

$y = 6.5\cos\left(\frac{4\pi}{27} \cdot t\right) + 10.5$

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