Question

1. (4 marks) the cost $c(x)$, in dollars, of producing $x$ items is given by $c(x)=2x^{2}+5x+100$

(a) find the average rate of change of the cost from $x = 10$ to $x = 20$.
(b) what does this rate represent?

Explanation:

Step1: Calculate $C(10)$

Substitute $x=10$ into $C(x)$:
$C(10)=2(10)^2 + 5(10) + 100 = 200 + 50 + 100 = 350$

Step2: Calculate $C(20)$

Substitute $x=20$ into $C(x)$:
$C(20)=2(20)^2 + 5(20) + 100 = 800 + 100 + 100 = 1000$

Step3: Compute average rate of change

Use formula $\frac{C(20)-C(10)}{20-10}$:
$\frac{1000 - 350}{20 - 10} = \frac{650}{10} = 65$

Answer:

(a) 65
(b) This rate represents the average increase in cost (in dollars) per additional item produced when increasing production from 10 items to 20 items.