4 otimization questions

[aaasssaaa]

1. The "gait" of an animal is a measure of how jerky or smooth the animal's motion appers when it is running. Gait, g can be shown to be related to the power, P, necessary for the animal to run at a given speed. For an animal 1m long, running at a velocity of 10m/s, the power is given by
P= 0.1g+ (1000/1+g)

Determine the gait that minimizes the required power for the animal to run.

2. Jeff is making a toy box for his nephew. He will use pine for the square base and sides. but oak for the top. Pine costs $16/m squared and oak costs $20/m squared. If the volume of the toy box is to be 1.3 m cubed, fined the amount of oak that is needed if the cost of all the wood for the toy box is to be as low as possible.

3.Corn Silos are usually in the shape of a cylindar surmounted by a hemisphere. If the average yield on a given farm requires that the silo contain 1000m cubed of corn, what dimension of the silo would use the minimum amount of materials?

4. A cylindrical kite frame is to be constructed from a 4m length of light bendable rod. The frame will be made up of two circles joined by four straight rods of equal length. In order to maximize lift, the kite frame must be constructed to maximize the volume of the cylinder. Into what lengths should the pieces be cut in order to optimize the kits's flight?

 

[pka]

Can you do any of these?
Can you even start any of these.
If so, please show us some effort on your part!
If not, why were you given these problems?

 

[soroban]

Hello, aaasssaaa!

Here's #4 . . .

4) A cylindrical kite frame is to be constructed from a 4m length of light bendable rod.
The frame will be made up of two circles joined by four straight rods of equal length.
In order to maximize lift, the kite frame must be constructed
to maximize the volume of the cylinder.
Into what lengths should the pieces be cut in order to optimize the kite's flight?

Let \(\displaystyle r\) = radius of the circles.
Let \(\displaystyle h\) = length of the straight rods.

The circumference of one circle is: \(\displaystyle \,2\pi r\)
. . The two circles will use \(\displaystyle 4\pi r\) meters.
The four straight rods have a total length of \(\displaystyle 4h\) meters.

The total length of the rods is: \(\displaystyle \:4\pi r\,+\,4h\:=\:4\;\;\Rightarrow\;\;h\:=\:1\,-\,\pi r\;\) [1]

The volume of the cylinder is: \(\displaystyle \:V \:=\:\pi r^2h\;\) [2]

Substitute [1] into [2]: \(\displaystyle \:V \:=\:\pi r^2(1\,-\,\pi r)\;\;\Rightarrow\;\;V \:=\:\pi(r^2\,-\,\pi r^3)\)

Differentiate and equate to zero: \(\displaystyle \:V' \:=\:\pi(2r\,-\,3\pi r^2)\:=\:0\)

Then: \(\displaystyle \:\pi r(2\,-\,3\pi r)\:=\:\;\;\Rightarrow\;\;r\:=\:0,\:\frac{2}{3\pi}\)

Obviously, \(\displaystyle r\,=\,0\) gives minimum volume.
. . Hence, \(\displaystyle \L r\,=\,\frac{2}{3\pi}\) gives maximum volume.

The circles will have circumference: \(\displaystyle 2\pi\left((\frac{2}{3\pi}\right) \:=\:\frac{4}{3}\) meters.

The straight rods are: \(\displaystyle \:h\:=\:1 - \pi\left(\frac{2}{3\pi}\right)\:=\:\frac{1}{3}\) meters.


Therefore, cut the four-meter rod into:
. . two lengths of \(\displaystyle \frac{4}{3}\) m for the circles
. . and four lengths of \(\displaystyle \frac{1}{3}\) m for the straight rods.

 

[aaasssaaa]

1. not sure but is g equal to 99
3. is it 10/√π

these are the questions i have lesft that i still cant solve