A supply store sells 140 scientific calculators per month at a price of $24 each. For each $4 increase in price, sales will decrease by 10 calculators.
a) determine a revenue function based on the number of price changes. write the function in standard form
b) determine the marginal revenue for the revenue function in part a).
c) when is this marginal revenue function equal to zero? why is this calculation so important?
d) what is the total revenue when the marginal revenue is zero? what is the price of each scientific calculator and how many will be sold when the marginal revenue is zero?
I think it would be A.
suppose there have been x $4 increases. Then we have
(a) r(x) = (24+4x)(140-10x)
(b) dr/dx = ____
(c) dr/dx=0 at x=____ This is where the next sale will not add revenue
(d) find r(x) at (c) value
a) To determine the revenue function, we need to consider the relationship between the price and the quantity sold.
Let's start by identifying the initial values:
- Price per calculator: $24
- Quantity sold: 140
Now, we know that for each $4 increase in price, sales will decrease by 10 calculators. This means that the price changes and the corresponding quantity sold are related by the equation:
Quantity sold = 140 - 10*(number of price changes)
To express this equation in terms of price instead of the number of price changes, we need to relate the price changes to the actual price. Since a price increase of $4 corresponds to a decrease of 10 calculators, each price change represents an increase of $4. Therefore, the number of price changes is equal to the price increase divided by $4.
Let's introduce a new variable, "x," to represent the number of price changes:
x = (Price - $24) / $4
Now, substituting the value of x back into the equation for quantity sold, we have:
Quantity sold = 140 - 10*x
= 140 - 10*((Price - $24) / $4)
To find the revenue function, we multiply the quantity sold by the price:
Revenue function = Price * Quantity sold
= Price * (140 - 10*((Price - $24) / $4))
Simplifying this expression gives us the revenue function in standard form.
b) The marginal revenue is the rate of change of revenue with respect to the price. To determine the marginal revenue, we need to differentiate the revenue function with respect to the price.
Let's differentiate the revenue function:
Revenue function = Price * (140 - 10*((Price - $24) / $4))
= Price * (140 - 2.5*(Price - $24))
To find the marginal revenue, differentiate with respect to the price:
Marginal Revenue = d(Revenue function) / d(Price)
= (140 - 2.5*(Price - $24)) + Price * (-2.5)
Simplifying this expression gives us the marginal revenue function.
c) To find when the marginal revenue is equal to zero, we set the marginal revenue function to zero and solve for the price. This is important because when the marginal revenue is zero, it indicates the maximum revenue, as it represents the point where the revenue is neither increasing nor decreasing with a change in price.
d) When the marginal revenue is zero, we can substitute the price obtained into the revenue function to calculate the total revenue. We can also determine the quantity sold by substituting the price into the equation we derived earlier.
By following these steps, we can find the total revenue, the price per calculator, and the quantity sold when the marginal revenue is zero.