Question

hw 2.4
score: 2/8 answered: 2/8
question 3
the total cost (in dollars) to produce q units of a good is given by the function:
$c(q) = 4.9q + 47000$
a. what is the total cost to produce $q = 7900$ units?
total cost = $

b. what is the cost of the 7901st item?
cost of the 7901st item = $
question help: video

Explanation:

Part A

Step1: Substitute \( q = 7900 \) into \( C(q) \)

We have the cost function \( C(q)=4.9q + 47000 \). Substitute \( q = 7900 \) into this function:
\( C(7900)=4.9\times7900 + 47000 \)

Step2: Calculate \( 4.9\times7900 \)

First, calculate \( 4.9\times7900 \). \( 4.9\times7900=(5 - 0.1)\times7900 = 5\times7900-0.1\times7900=39500 - 790 = 38710 \)

Step3: Add 47000 to the result

Now, add 47000 to 38710: \( 38710+47000 = 85710 \)

Step1: Understand the cost of the \( n \)-th item

The cost of the \( n \)-th item is given by the difference \( C(n)-C(n - 1) \). For a linear cost function \( C(q)=mq + b \) (where \( m \) is the slope and \( b \) is the y - intercept), the marginal cost (cost of each additional item) is constant and equal to the slope \( m \). In our function \( C(q)=4.9q + 47000 \), the slope \( m = 4.9 \). So the cost of the 7901st item is \( C(7901)-C(7900) \)

Step2: Calculate \( C(7901)-C(7900) \)

\( C(7901)=4.9\times7901+47000 \) and \( C(7900)=4.9\times7900 + 47000 \)
\( C(7901)-C(7900)=(4.9\times7901 + 47000)-(4.9\times7900 + 47000)=4.9\times(7901 - 7900)=4.9\times1=4.9 \)

Answer:

For part A, the total cost is \(\$85710\)

Part B