Section 2.2 – Modeling with Linear Functions
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. An item initially purchased for $3425 has a scrap value of $250 after 8 years. Assuming the item is depreciating linearly with time, write the value of the item (in dollars) as a function of time (in years since purchase).
For \(0\leq t\leq 8\), \(V(t)=-\dfrac{3175}{8}t+3425.\) For \(t>8\), \(V(t)=250.\)
2. A bicycle shop manufacturers small bicycle pumps for riders to mount to their bikes. The shop sells each pump for $12. The cost of producing each unit is $4.80. The fixed costs incurred by the division are $50,000. Find:
- The shop’s linear cost function for producing \(x\) small bicycle pumps.
- The shop’s linear revenue function for selling \(x\) small bicycle pumps.
- The shop’s linear profit function for producing and selling \(x\) small bicycle pumps.
- \(C(x) = 4.8x + 50000\)
- \(R(x) = 12x\)
- \(P(x) = 7.2x − 50000\)
3. The total cost of producing 40 items for a certain company is $13,243. The revenue earned from selling 125 items is $15,406.25. If the company’s fixed costs are $12,225, find the company’s linear cost function.
\(C(x) = 25.45x + 12, 225\)
4. The total cost of producing 40 items for a certain company is $13,243. The revenue earned from selling 125 items is $15,406.25. Find the company’s linear revenue function.
\(R(x)=123.25x\)
5. The total cost of producing 40 items for a certain company is $13,243. The revenue earned from selling 125 items is $15,406.25. Find the company’s linear profit function.
\(P(x)=97.8x−12,225\)
6. The quantity demanded for a certain brand of portable CD players is 200 units when the unit price is set at $72. The quantity demanded is 1200 units when the unit price is $32. Find the demand equation, assuming the demand equation is linear.
\(p(x) = −0.04x + 80\)
7. A producer will not supply any items when the price is $65 or lower, but when the price per item is $100, the producer is willing to supply 850 items. Construct the linear supply function, p(x), where p is in dollars and x is the number of items supplied.
\(p(x) = \dfrac{7}{170}x+65\)