# Math Help - Optimization

1. ## Optimization

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.

2. Originally Posted by 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.
I presume that the formula is:
P = 0.1g + 1000/(1 + g)

To find the minimum we take the first derivative:
dP/dg = 0.1 + 1000*(-1)/(1 + g)^2

then set it equal to 0:
0.1 + 1000*(-1)/(1 + g)^2 = 0

0.1 = 1000/(1 + g)^2

0.1(1 + g)^2 = 1000

(1 + g)^2 = 1000/0.1 = 10000

1 + g = 100 <-- Technically (+/-)100, but since g can't be negative...

g = 100 - 1 = 99

I'll let you prove that this value of g provides a minimum.

-Dan