- #1
Chocolaty
- 48
- 0
Ok we're doing optimization, i don't understand why you have to derive, it just doesn't come naturally to me, like it's not something i instinctively think of doing...
My example
A company wants to fence a rectangular piece of land that is bordered on one side by a road and by a river on the opposite side. There will be no fence along the river. The fence along the road costs $3 per foot and the fence along the other two sites costs $2 per foot. If the rectangular piece of land must have an area of 10800 square feet, find the dimensions that will give the minimum cost. What is the minimum cost?
I understand the need to use 2 equations like in related rates to help reduce, in this case, the number of variables in the cost function.
Eq#1: Min C = 3x + 4y
Eq#2: xy = 10800
y = 10800/x
Min C = 3x + 4(10800/x)
Min C = 3x + 43200/x [x > 0]
At this point... he derives, but why?
C' = 3 - 43200/(x^2)
Now he finds the critical numbers, why? And also, how can you just take the right side of the equation of C' to do that?
CN: 3 = 43200/(x^2)
3x^2 = 43200
x^2 = 14400
x = 120
Now he finds the second derivitive, again, why?
C'' = 86400/(x^3)
C(120) > 0 => x = 120 will give Min C
Ans: x = 120, y = 90, Min C = $720
My example
A company wants to fence a rectangular piece of land that is bordered on one side by a road and by a river on the opposite side. There will be no fence along the river. The fence along the road costs $3 per foot and the fence along the other two sites costs $2 per foot. If the rectangular piece of land must have an area of 10800 square feet, find the dimensions that will give the minimum cost. What is the minimum cost?
I understand the need to use 2 equations like in related rates to help reduce, in this case, the number of variables in the cost function.
Eq#1: Min C = 3x + 4y
Eq#2: xy = 10800
y = 10800/x
Min C = 3x + 4(10800/x)
Min C = 3x + 43200/x [x > 0]
At this point... he derives, but why?
C' = 3 - 43200/(x^2)
Now he finds the critical numbers, why? And also, how can you just take the right side of the equation of C' to do that?
CN: 3 = 43200/(x^2)
3x^2 = 43200
x^2 = 14400
x = 120
Now he finds the second derivitive, again, why?
C'' = 86400/(x^3)
C(120) > 0 => x = 120 will give Min C
Ans: x = 120, y = 90, Min C = $720