Question

8) The cost, C, in dollars, of operating a concrete-cutting machine is modelled by
C = 2.2n? - 66n+ 655, where n is the number of minutes the machine is run. How long must the
machine run for the operating cost to be at a minimum? What is the minimum cost?

Answer 1

The machine must run for 15 minutes for the operating cost to be at a minimum

The minimum cost is $160

Step-by-step explanation:

The cost, C, in dollars, of operating a concrete-cutting machine is modeled by  C = 2.2n² - 66n + 655, where n is the number of minutes the machine is run.

We need to find how long the  machine must run for the operating cost to be at a minimum, and find the minimum cost

For the minimum cost:

• Differentiate the cost C ⇒ $\frac{dC}{dn}$ or C'
• Equate the differentiation by zero ⇒ C' = 0
• Solve for n ⇒ which obtain the minimum cost
• Substitute the value of n in C to find the minimum cost
• Remember the differentiation of $ax^{n}=a(n)x^{(n-1)}$

∵ C = 2.2n² - 66n + 655

- Find C'

∵ $C'=2.2(2)n^{(2-1)}-66(1)n^{(1-1)}$

∴ $C'=4.4n^{(1)}-66n^{(0)}$

- Remember $n^{(0)}=1$

∴ C' = 4.4 n - 66

- Equate C' by zero

∵ C' = 0 ⇒ for minimum cost

∴ 4.4 n - 66 = 0

- Add 66 to both sides

∴ 4.4 n = 66

- Divide both sides by 4.4

∴ n = 15 minutes

The machine must run for 15 minutes for the operating cost to be at a minimum

To find the minimum cost substitute n in C by 15

∵ C = 2.2n² - 66n + 655

∵ n = 15

∴ C = 2.2(15)² - 66(15) + 655

∴ C = $160

The minimum cost is $160

Learn more:

You can learn more about the differentiation in brainly.com/question/4279146

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