The cosat C, in dollars, of operating a concrete-cutting machine is modelled by C=2.2n^2-66n+655, where n is the number of minutes the machine is run. How long must trhe machine run for the operating cost to be at a minimum? What is the4 minimum cost?
Why did the concrete-cutting machine go to therapy? Because it had a lot of unresolved cost issues!
To find the minimum operating cost, we need to determine the value of 'n' that minimizes the function C = 2.2n^2 - 66n + 655. This can be done by finding the vertex of the quadratic equation.
The formula for the x-coordinate of the vertex of a quadratic function in the form of f(x) = ax^2 + bx + c is given by x = -b/(2a).
In this case, a = 2.2 and b = -66. Plugging these values into the formula, we find:
n = -(-66) / (2 * 2.2)
= 66 / 4.4
≈ 15
So, the machine must run for approximately 15 minutes to minimize the operating cost.
To find the minimum cost, we can substitute this value of 'n' back into the cost equation:
C = 2.2(15)^2 - 66(15) + 655
= 2.2(225) - 990 + 655
= 495 - 990 + 655
= 160
The minimum cost of operating the machine is $160. I hope this quadratic journey brought a bit of humor to your day!
To find the minimum operating cost, we need to find the value of n that minimizes the cost function C = 2.2n^2 - 66n + 655.
Step 1: Identify the coefficient of the n^2 term.
In this case, the coefficient is 2.2.
Step 2: Use the formula for the x-coordinate of the vertex of a quadratic function.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a is the coefficient of the n^2 term and b is the coefficient of the n term.
In our case, a = 2.2 and b = -66.
x = -(-66) / (2 * 2.2)
x = 66 / 4.4
x = 15
Step 3: Determine the minimum cost.
To find the minimum cost, substitute the value of n from step 2 back into the cost function C.
C = 2.2n^2 - 66n + 655
C = 2.2(15)^2 - 66(15) + 655
C = 2.2(225) - 990 + 655
C = 495 - 990 + 655
C = 160
Step 4: Find the units of measurement.
In this case, the units of measurement are dollars (as given in the problem statement).
Therefore, the machine must run for 15 minutes for the operating cost to be at a minimum. The minimum operating cost is $160.
To find the minimum cost and the duration the machine must run for the operating cost to be at a minimum, we need to use the concept of derivatives.
The cost function is given as: C = 2.2n^2 - 66n + 655
Step 1: Take the derivative of the cost function with respect to 'n'
To find the minimum, we need to differentiate the cost function with respect to 'n'.
dC/dn = 4.4n - 66
Step 2: Set the derivative equal to zero and solve for 'n'
To find the minimum or maximum, we need to set the derivative equal to zero.
4.4n - 66 = 0
4.4n = 66
n = 66 / 4.4
n = 15
Step 3: Determine the minimum cost
To find the minimum cost, substitute the value of 'n' obtained from the previous step back into the cost function.
C = 2.2(15)^2 - 66(15) + 655
C = 495 - 990 + 655
C = 160
Therefore, the machine must run for 15 minutes to achieve the minimum cost of $160.