Applied Maximum and Minimum Problems help

[hank]

Ok, I have two problems I am completely lost on, and four I'm pretty sure I got but could use someone to double check:

//Problems completely lost on:
1) Show that the right circular cylinder of greatest volume that can be inscribed in a right circular cone has volume that is 4/9 the volume of the cone.

Comments:
I think the equations I need to use for this problem are the equation for the volume of a cylinder, and I need to utilize the formula for similar triangles.

Formula for similar triangles:
I decide to solve for h, and I get h = H(R-r) / R

Then, I need to plug in h into the formula for the volume of a Cylinder:

V=pi * r^2 * (H(R-R) / R) = (H * pi / R) * (r^2R - r^3).

Now, I need to take the derivative, but it gets into a huge jumbled mess, and I can't get through that.

2) A cone-shaped paper drinking cup is to hold 10 cm^3 of water. Find the height and radius of the cup that will require the least amount of paper.

Comments:
I believe I need to utilize the formula for the Volume of a cone and for the surface area of a cone.

I will first need to restate l in terms of r and h for the surface area of a cone using the pythagorean theorem
S = pi * r * l

becomes
S = pi * r * sqrt(r^2 + h^2)

10 = 1/3 * pi * r^2 * h.

I solve for h.
h = 30 / (pi * r^2)

I utilize the surface area formula, substituting in for h:
S = pi * r * sqrt(r^2 + (30/(pi * r^2)^2)

Now it's time to take the derivative and I get into such a huge mess, I don't believe I've set the problem up right. I even attempt to use the properties of logs to simplify it and it doesn't help.

//Problems i think I have solved, but could use verification:
1) A field has boundary a right triangle with hypotenuse along a straight stream. A fence bounds the other two sides of the field. Find the dimensions of the field with maximum area that can be closed using 1000 ft of fence.

My answer: 500 ft by 500ft by 50 sqrt(2) ft

2) A firm determines that x units of its product can be sold daily at p dollars per unit where x = 1000 - p. The cost of producing x units per day is C(x) = 3000 + 20x.
a) find the revenue function R(x).

My answer: R(x) = 1000x - x^2

b) Find the profit function P(x).

My answer: P(x) = 1080x - x^2 - 3000.

c) Assuming that the production capacity is at most 500 units per day, determine how many units the company must produce and sell each day to maximize the profit.

My answer: 500 units.

d) Find the maximum profit.

My answer: $287,000.

e) What price per unit must be charged to obtain the maximum profit?

My answer: $500.

3) Find all points on the curve y = sqrt(x) for 0 <= x <= 3 that are closest to, and at the greatest distance from, the point (2,0).

My answer: closest point (3/2, 7/4), furthest point (3, 4).

Anyway, I know it's a lot, but if anyone can help with any of it, I would appreciate it greatly.

 

[galactus]

2) A cone-shaped paper drinking cup is to hold 10 cm^3 of water. Find the height and radius of the cup that will require the least amount of paper.

Comments:
I believe I need to utilize the formula for the Volume of a cone and for the surface area of a cone.

I will first need to restate l in terms of r and h for the surface area of a cone using the pythagorean theorem
S = pi * r * l

becomes
S = pi * r * sqrt(r^2 + h^2)

10 = 1/3 * pi * r^2 * h.

I solve for h.
h = 30 / (pi * r^2)

I utilize the surface area formula, substituting in for h:
S = pi * r * sqrt(r^2 + (30/(pi * r^2)^2)

Now it's time to take the derivative and I get into such a huge mess, I don't believe I've set the problem up right. I even attempt to use the properties of logs to simplify it and it doesn't help.

//Problems i think I have solved, but could use verification:

Very good. You have the correct function. I'll start from where you left off.

\(\displaystyle \L\\A={\pi}r\sqrt{r^{2}+\frac{900}{{\pi}^{2}r^{4}}}\)

Now, you were having trouble differentiating?.

Let's simplify first to make it easier:

Let \(\displaystyle \L\\S=A^{2}\)

\(\displaystyle \L\\S={\pi}^{2}r^{2}\left(r^{2}+\frac{900}{{\pi}^{2}r^{4}}\right)={\pi}^{2}r^{4}+\frac{900}{r^{2}}\)

That's easier than that monster radical we had, ain't it?.

Now, can you find \(\displaystyle \L\\\frac{dS}{dr}\). I believe so. You did good finding your maximizing function.