Question

2. Assume that each situation can be expressed as a linear cost function and find the appropriate cost function. (a) Fixed cost, $100; 50 items cost $1600 to produce. (b) Fixed cost, $400; 10 items cost $650 to produce. (c) Fixed cost, $1000; 40 items cost $2000 to produce. (d) Fixed cost, $8500; 75 items cost $11,875 to produce. (e) Marginal cost, $50; 80 items cost $4500 to produce. (f)Marginal cost, $120; 100 items cost $15,800 to produce. (g) Marginal cost, $90; 150 items cost $16,000 to produce. (h) Marginal cost, $120; 700 items cost $96,500 to produce.

Answer 1

Given:

Cost function is defined as,

`\begin{gathered} C(x)=mx+b \\ m=\text{marginal cost} \\ b=\text{fixed cost} \end{gathered}`

a) Fixed cost = $100, 50 items cost $1600.

The cost function is given as,

`\begin{gathered} C=\text{Fixed cost+}x(\text{ production cost)} \\ x\text{ is number of items produced} \\ \text{Given that, }50\text{ items costs \$1600} \\ 1600=100\text{+50}(\text{ production cost)} \\ \text{production cost=}(1600-100)/(50) \\ \text{production cost}=30 \end{gathered}`

So, the cost function is,

`C=30x+100`

b) Fixed cost = $400, 10 items cost $650.

`\begin{gathered} 650=400+10p \\ 650-400=10p \\ p=25 \\ \text{ Cost function is,} \\ C=25x+400 \end{gathered}`

c) Fixed cost= $1000, 40 items cost $2000 .

`\begin{gathered} 2000=1000+40p \\ p=25 \\ C=25x+1000 \end{gathered}`

d) Fixed cost = $8500, 75 items cost $11,875.

`\begin{gathered} 11875=8500+75p \\ 11875-8500=75p \\ p=45 \\ C=45x+8500 \end{gathered}`

e) Marginal cost= $50, 80 items cost $4500.

In this case we know the value of m = 50 .

Use the slope point form,

`\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(80,4500) \\ y-4500=50(x-80) \\ y=50x-4000+4500 \\ y=50x+500 \\ C=50x+500 \end{gathered}`

f) Marginal cost=$120, 100 items cost $15,800.

`\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(100,15800) \\ y-15800=120(x-100) \\ y=120x-12000+15800 \\ y=120x+3800 \\ C=120x+3800 \end{gathered}`

g) Marginal cost= $90,150 items cost $16,000.

`\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(150,16000) \\ y-16000=90(x-150) \\ y=90x-13500+16000 \\ y=90x+2500 \\ C=90x+2500 \end{gathered}`

h) Marginal cost = $120, 700 items cost $96,500

`\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(700,96500) \\ y-96500=120(x-700) \\ y=120x-84000+96500 \\ y=120x+12500 \\ C=120x+12500 \end{gathered}`