Maximizing Volume, Area, and Rent: Optimization Problems Homework

[pinkerpikachu]

Homework Statement

If you can help me answer ANY of these, it will be very appreciated. thanks in advance.

1. The following problem was stated and solved in the work Nova stereometria vinariorum, published in 1615 by the astronomer Johannes Kepler. What are the dimensions of the cylinder of largest volume that can be inscribed in a sphere of radius R? (Hint: Show that the volume of an inscribed cylinder is 2pix(R^2 - x^2), where x is one-half the height of the cylinder.

2. Find the angle theta that maximizes the area of the trapezoid with a base of length 4 and sides of length 2, as in Figure 16: http://tinypic.com/r/2iw4zh3/7

3. Optimal Price: Let r be the monthly rent per unit in an apartment building with 100 units. A survey reveal that all units can be rented when r= $900 and that one unit becomes vacant with each $10 increase in rent. Suppose that the average monthly maintenance per occupied unit is $100 per month.
a) Show that the number of units rented is n = 190 - (r/10) for 900 is less than r which is less than 1,900
b) Find a formula for the net cash intake (revenue minus maintenance) and determine the rent r that maximizes intake

4. Use calculus to show that among all right triangles with hypotenuse of length 1, the isosceles triangle has maximum area. Can you see more directly why this must be true by reasoning from Figure 22? http://tinypic.com/r/30i9bmo/7

5. Find the area of the largest isosceles triangle that can be inscribed in a circle of radius r.

6. The problem is to put a "roof" of side s on an attic room of height h and width b. Find the smallest length s for which this is possible. http://tinypic.com/r/2hi2lxy/7

Homework Equations

For 1: Volume of a sphere is 4/3[tex]\pi[/tex]r^2 and Volume of a cylinder is
=[tex]\pi[/tex]hr^2

For 2: Area of a trapezoid = 1/2(b1 +b2)h

For 4: a^2 + b^2 = c^2

For 5:= 1/2a^2sqrt((b^2)/(a^2)-1/4).

For 6: I don't really know, wouldn't it be another isosceles triangle?

The Attempt at a Solution

Okay for 1, I've tried solving for R in the equation for volume and plugging it into the equation for the volume of a cylinder, but this doesn't give me the hint's answer.
Am I using the right equations? and the right method?
R is not only the sphere's radius but also the radius of the cylinder, correct? and if x is half the height of the cylinder isn't it also another radius?

2) For 2, I'm not really sure how to fit the two angles into the problem. I know for optimization problems you essentially need to find two equations and then plug one into the other, but I can't seem to find/understand the relationship between the area of the trapezoid to the angles given...

3) I don't really get it, where does the 190 come in...or rather how does the equation come together?

4) Okay, this one makes sense to me, but I'm not sure how I'm supposed to go about proving it? Should I prove that it is right by disproving the figure in the image (which is obviously not an isoceles triangle...). For the figure am I supposed to set up two similar right triangles?

5) I need help deciding what my two equations should be...The equation for the area of a triangle and for a circle?

6) I was thinking similar triangles again, but they're not right triangle...or rather they look equilateral, but I'm not sure.


So for these problems, once I get started I can usually figure the problem out, its just that I have a hard time figuring out the relationships between all the variables.

thanks for any help :)

 

[mighty2000]

Hello there! I am happy to help you with these problems. Let's go through them one by one:

1. For this problem, you are correct in using the equations for the volume of a sphere and a cylinder. However, the key to solving this problem is to use the hint given to you. Remember that x is half the height of the cylinder, so the full height of the cylinder is 2x. Can you find the maximum volume of the cylinder by using this information and the equations for the volume of a sphere and a cylinder?

2. To solve this problem, let's first draw a picture of the trapezoid. We know that the base is 4 and the sides are 2, so we can label the angles as shown in the picture. Now, we want to find the angle that maximizes the area. Can you write an equation for the area of the trapezoid in terms of theta? Then, can you use calculus to find the value of theta that maximizes the area?

3. For part a), you can use the information given to find the number of units rented at a particular rent r. For example, when r = $900, all units are rented, so n = 100. When r increases to $910, one unit becomes vacant, so n = 99. Can you use this information to come up with an equation for n in terms of r? For part b), you can use the formula for net cash intake (revenue - maintenance) and take the derivative with respect to r. Set this derivative equal to 0 and solve for r to find the rent that maximizes the net cash intake.

4. You are on the right track! To prove that the isosceles triangle has the maximum area, you can use the Pythagorean theorem (a^2 + b^2 = c^2) and the fact that the sum of the angles in a triangle is 180 degrees. Can you come up with an equation for the area of the triangle in terms of one of the angles? Then, you can take the derivative with respect to that angle and set it equal to 0 to find the angle that maximizes the area.

5. For this problem, you can use the equation for the area of a triangle and the equation for the area of a circle. Remember that the isosceles triangle has two equal sides, so you can set the length