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The function gives the cost to manufacture x items. C(x) = 10,000 + 20x + 30,000 x ; x = 100 Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h = 10 and 1. (Use smaller values of h to check your estimates.) HINT [See Example 1.] (Round your answer to two decimal places.)

Respuesta :

Answer:

Therefore, it can be estimated that the average rate of change of the total cost to manufacture  100  items is  $ 30020 per item (interpreted units).

Step-by-step explanation:

Average rate of change through the formula[tex]=\frac{f\left ( b \right )-f\left ( a \right )}{b-a}[/tex]

given that,

[tex]c\left ( x \right )=10,000+20x+30,000x[/tex]

[tex]x=100[/tex]

[tex]h=10,1[/tex]

To find the average cost per unit of manufacturing  [tex]h[/tex] more items, the strategy is to apply the average rate of change to the function for  [tex]x[/tex] and the given  [tex]h[/tex]  present.

when, [tex]h[/tex][tex]=10[/tex]

Average cost per unit manufacturing[tex]=\frac{C\left (100+10 \right )-C\left ( 100 \right )}{\left ( 100+10 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{C\left (110 \right )-C\left ( 100 \right )}{\left ( 110 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{\left ( \left ( 10,000+20\left ( 110 \right )+30,000\left ( 110 \right ) \right ) -\left ( 10,000+20\left ( 100 \right ) +30,000\left ( 100 \right )\right )\right )}{10}[/tex]

[tex]=\frac{3312200-3012000}{10}[/tex]

[tex]=\frac{300200}{10}[/tex]

[tex]=30020[/tex]

when [tex]h=1[/tex]

Average cost per unit manufacturing[tex]=\frac{C\left (100+1 \right )-C\left ( 100 \right )}{\left ( 100+1 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{C\left (101 \right )-C\left ( 100 \right )}{\left ( 101 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{\left ( \left ( 10,000+20\left ( 101 \right )+30,000\left ( 101 \right ) \right ) -\left ( 10,000+20\left ( 100 \right ) +30,000\left ( 100 \right )\right )\right )}{1}[/tex]

[tex]=\frac{3042020-3012000}{1}[/tex]

[tex]=\frac{30020}{1}[/tex]

if [tex]h=0.1[/tex]

Average cost per unit manufacturing[tex]=\frac{C\left (100+0.1 \right )-C\left ( 100 \right )}{\left ( 100+0.1 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{C\left (100.1 \right )-C\left ( 100 \right )}{\left ( 100.1 \right )-\left ( 100 \right )}[/tex]

[tex]=\frac{\left ( \left ( 10,000+20\left ( 100.1 \right )+30,000\left ( 100.1 \right ) \right ) -\left ( 10,000+20\left ( 100 \right ) +30,000\left ( 100 \right )\right )\right )}{0.1}[/tex]

[tex]=\frac{3015002-3012000}{0.1}[/tex]

[tex]=\frac{3002}{0.1}[/tex]

[tex]=30020[/tex]

Therefore, it can be estimated that the average rate of change of the total cost to manufacture  100  items is  $ 30020 per item (interpreted units).