A) Find the marginal cost at a production level of x = 20,000 items, given that cost is given by, C(x)=20,000+7x−

A)

The marginal cost at a production level of x = 20,000 items:

C(x)=20,000+7x−.00005x2CCostMarginal(x)=dCdx=ddx[20,000+7x−.00005x2]=[0+7−.00005(2x)][ Differentiate with repsect to x]⇒CCostMarginal(x)=7−.0001x⇒CCostMarginal(20,000)=7−.0001(20,000)=7−2=$5[ This is the marginal cost at sells level x=20,000]C(x)=20,000+7x−.00005x2CMarginalCost(x)=dCdx=ddx[20,000+7x−.00005x2]=[0+7−.00005(2x)][ Differentiate with repsect to x]⇒CMarginalCost(x)=7−.0001x⇒CMarginalCost(20,000)=7−.0001(20,000)=7−2=$5[ This is the marginal cost at sells level x=20,000]

Therefore, the marginal cost at a production level of x = 20,000 items

CCostMarginal(20,000)=$5CMarginalCost(20,000)=$5

B)

The unit of marginal cost is $/item

C) The average cost of producing 20,000 items:

C(x)=20,000+7x−.00005x2CCostAverage(x)=??∫200000C(x)dx20000??=120000∫200000[20,000+7x−.00005x2]dx=120000[20,000x+7x22−.00005x33]200000[ integrate with repsect to x]=120000[20,000(20000)+7(20000)22−.00005(20000)33]−0=120000[1267066666.66666]=$63353.33[ This is the average cost of 20,000units]C(x)=20,000+7x−.00005x2CAverageCost(x)=[∫020000C(x)dx20000]=120000∫020000[20,000+7x−.00005x2]dx=120000[20,000x+7x22−.00005x33]020000[ integrate with repsect to x]=120000[20,000(20000)+7(20000)22−.00005(20000)33]−0=120000[1267066666.66666]=$63353.33[ This is the average cost of 20,000units]

Therefore, the average cost of x = 20,000 items is

CCostAverage(20,000)=$63353.33