The cost of purchasing x meter lengths of a certain rope is:
C=1/5x^2-8x+100 per rope.
What is the length of the cheapest rope that can be purchased?
minimum cost is at the vertex of the parabola, which occurs at x = 8/(2/5) = 20
To find the length of the cheapest rope that can be purchased, we need to minimize the cost function C.
The cost function is given by C = (1/5)x^2 - 8x + 100 per rope.
To minimize the cost function, we can use calculus. We need to find the critical points of the cost function by taking the derivative of C with respect to x and setting it equal to zero.
Let's find the derivative of C with respect to x:
dC/dx = (1/5)(2x) - 8
Setting the derivative equal to zero:
(1/5)(2x) - 8 = 0
Multiplying both sides of the equation by 5 to eliminate the fraction:
2x - 40 = 0
Adding 40 to both sides of the equation:
2x = 40
Dividing both sides of the equation by 2:
x = 20
So, the critical point is x = 20.
To determine if this critical point is a minimum or maximum, we can check the second derivative of C. If the second derivative is positive, then it is a minimum value.
Let's find the second derivative of C:
d²C/dx² = (1/5)(2)
Since the second derivative is positive (1/5)(2) > 0, x = 20 is indeed a minimum.
Therefore, the length of the cheapest rope that can be purchased is 20 meters.