A manufacturer is in the business of producing small figurines. He finds that the daily cost in pesos,P, of producing n figurines is given by P= n2-120 +4200. How many figurines produced per day so that the cost will be minimum?
I will assume a typo and that you meant:
P= n^2-120n +4200
This is a parabola when graphed.
Both of your questions are answered once you find the vertex of that parabola.
What method have you learned to find that vertex ?
To find the number of figurines produced per day that will result in the minimum cost, we need to identify the value of n that minimizes the cost function P.
The cost function is given by the equation: P = n^2 - 120n + 4200.
To minimize this function, we can use calculus. We need to find the critical points by taking the derivative of P with respect to n and setting it equal to zero.
1. Differentiate the cost function, P, with respect to n:
dP/dn = 2n - 120
2. Set the derivative equal to zero to find the critical points:
2n - 120 = 0
3. Solve for n:
2n = 120
n = 60
After finding the critical point, we need to confirm that it indeed corresponds to the minimum point of the cost function. To do this, we can check the second derivative of the cost function.
4. Differentiate the derivative, dP/dn, with respect to n:
d^2P/dn^2 = 2
Since the second derivative is positive (2 > 0), we can conclude that the critical point n = 60 corresponds to a minimum point on the cost function.
Therefore, the minimum cost will be achieved when the manufacturer produces 60 figurines per day.