Analyzing a Cost Function

The weekly cost \(C\), in dollars, of manufacturing \(x\) lightbulbs is \[ C ( x ) =7500+ \sqrt{125x} \]

1. Find the average rate of change of the weekly cost \(C\) of manufacturing from \( 100\) to \(101\) lightbulbs.
2. Find the average rate of change of the weekly cost \(C\) of manufacturing from \( 1000\) to \(1001\) lightbulbs.
3. Interpret the results from parts (a) and (b).

Solution (a) The weekly cost of manufacturing \(100\) lightbulbs is \[ C ( 100 ) =7500+ \sqrt{125\cdot 100}=7500+ \sqrt{12{,}500}\approx \$7611.80 \]

The weekly cost of manufacturing \(101\) lightbulbs is \[ C ( 101 ) =7500+ \sqrt{125\cdot 101}=7500+ \sqrt{12{,}625}\approx \$7612.36 \]

The average rate of change of the weekly cost \(C\) from \(100\) to \(101\) is \[ \dfrac{\Delta C}{\Delta x}=\dfrac{C ( 101 ) -C ( 100 ) }{ 101-100}\approx \dfrac{7612.36-7611.80}{1}=\$0.56 \]

(b) The weekly cost of manufacturing \(1000\) lightbulbs is \[ C ( 1000) =7500+ \sqrt{125\cdot 1000}=7500+ \sqrt{125{,}000}\approx \$7853.55 \]

The weekly cost of manufacturing \(1001\) lightbulbs is \[ C ( 1001) =7500+ \sqrt{125\cdot 1001}=7500+ \sqrt{125{,}125}\approx \$7853.73 \]

The average rate of change of the weekly cost \(C\) from \(1000\) to \(1001\) is \[ \dfrac{\Delta C}{\Delta x}=\dfrac{C ( 1001) -C ( 1000) }{ 1001-1000}\approx \dfrac{7853.73-7853.55}{1}=\$0.18 \]

(c) Part (a) tells us that the cost of manufacturing the 101st lightbulb is $0.56. From (b) we learn that the cost of manufacturing the 1001st lightbulb is only $0.18. The unit cost per lightbulb decreases as the number of lightbulbs manufactured per week increases.