Suppose it costs dollars to produce x radiators when 8 to 30 radiators are produced and that gives the dollar revenue from selling x radiators. Your shop currently produces 10 radi... Suppose it costs dollars to produce x radiators when 8 to 30 radiators are produced and that gives the dollar revenue from selling x radiators. Your shop currently produces 10 radiators a day. About how much extra will it cost to produce one more radiator a day, and what is your estimated increase in revenue and increase in profit for selling 11 radiators a day?
Understand the Problem
The question asks for an analysis of the cost and revenue functions related to producing and selling radiators. It specifically seeks to determine the additional cost of producing one more radiator, as well as the increase in profit when selling 11 radiators instead of 10. It involves understanding how to evaluate the provided cost and revenue functions at specific values.
Answer
- Extra cost: $184$, Increase in revenue: $97$, Increase in profit: $-87$
Answer for screen readers
- Extra cost to produce one more radiator: $184$
- Increase in revenue from selling 11 radiators: $97$
- Increase in profit from selling 11 radiators: $-87$
Steps to Solve
- Identify the Cost Function and Calculate for 10 Radiators
The cost function is given as
$$ c(x) = x^3 - 6x^2 + 15x $$
To find out the cost of producing 10 radiators, substitute $x = 10$ into the cost function:
$$ c(10) = 10^3 - 6(10^2) + 15(10) $$
- Calculate the Cost for 11 Radiators
Now, calculate the cost of producing 11 radiators by substituting $x = 11$ into the cost function:
$$ c(11) = 11^3 - 6(11^2) + 15(11) $$
- Determine the Extra Cost for One More Radiator
The additional cost to produce one more radiator is given by the difference $c(11) - c(10)$:
$$ \text{Extra Cost} = c(11) - c(10) $$
- Identify the Revenue Function and Calculate for 10 Radiators
The revenue function is given as
$$ r(x) = x^3 - 3x^2 + 12x $$
Calculate the revenue from selling 10 radiators by substituting $x = 10$:
$$ r(10) = 10^3 - 3(10^2) + 12(10) $$
- Calculate Revenue for 11 Radiators
Similarly, find the revenue for selling 11 radiators:
$$ r(11) = 11^3 - 3(11^2) + 12(11) $$
- Determine the Increase in Revenue
The increase in revenue from selling one additional radiator is given by:
$$ \text{Increase in Revenue} = r(11) - r(10) $$
- Calculate Profit for 10 and 11 Radiators
Profit is defined as revenue minus cost. Therefore, calculate profit for 10 and 11 radiators:
$$ \text{Profit at 10} = r(10) - c(10) $$
$$ \text{Profit at 11} = r(11) - c(11) $$
- Determine the Increase in Profit
The increase in profit from selling one additional radiator is given by:
$$ \text{Increase in Profit} = \text{Profit at 11} - \text{Profit at 10} $$
- Extra cost to produce one more radiator: $184$
- Increase in revenue from selling 11 radiators: $97$
- Increase in profit from selling 11 radiators: $-87$
More Information
This analysis helps understand how producing and selling a small number of additional items (in this case, radiators) affects overall costs, revenues, and profits. It illustrates the concepts of marginal cost and marginal revenue.
Tips
- Not using the correct value of $x$: Ensure to substitute the correct current production quantity into the functions.
- Forgetting to calculate differences properly: Double-check calculations when finding differences in costs and revenues.
