A national record distribution company sells records and tapes by mail only. A promotion program is planned for a major metropolitan area for a new country western album. The target audience is estimated at 800,000. Past experience indicates that for this city and this type of album the percentage of the target market R actually purchasing an album or tape is a function of the length of the advertising campaign,t. Specifically this sales responce funtion is: R=1-(e^(-0.025t)).
Profit margin is 2.50$ on each album and its advertising cost is 20000 as fixed cost and 2,500 as A variable cost. Determine how long the campaign should be conducted if the goal is to maximize net profit
To determine how long the campaign should be conducted to maximize net profit, we'll first need to define the net profit function. The net profit can be calculated by subtracting the total cost from the total revenue.
The total revenue (TR) is given by multiplying the number of albums sold (R) by the profit margin on each album ($2.50). Therefore, TR = R * $2.50.
The total cost (TC) is the sum of the fixed cost ($20,000) and the variable cost per album ($2,500) multiplied by the number of albums sold (R). Therefore, TC = $20,000 + ($2,500 * R).
The net profit (NP) is calculated as NP = TR - TC.
Substituting the values into the equations, we have NP = (R * $2.50) - ($20,000 + ($2,500 * R)).
To maximize the net profit, we must differentiate NP with respect to t and find the critical point where the derivative is zero.
Let's differentiate NP with respect to t:
dNP/dt = d(R * $2.50)/dt - d($20,000 + ($2,500 * R))/dt
= 2.50 * dR/dt - 2,500 * dR/dt
Now, we need to find the value of t that makes dNP/dt = 0:
2.50 * dR/dt - 2,500 * dR/dt = 0
Simplifying the equation: 2.50 - 2,500 * dR/dt = 0
Rearranging the equation: dR/dt = 1/2,500
Now, let's solve for t:
dR/dt = 1 - (e^(-0.025t))
Setting dR/dt = 1/2,500:
1 - (e^(-0.025t)) = 1/2,500
Simplifying the equation: e^(-0.025t) = 2,499/2,500
Taking the natural logarithm (ln) of both sides: -0.025t = ln(2,499/2,500)
Solving for t: t = (ln(2,499/2,500)) / -0.025
Using a calculator, we can find the value of t to determine how long the campaign should be conducted to maximize net profit.
To determine how long the campaign should be conducted to maximize net profit, we need to find the point where the net profit is maximized. Net profit is calculated by subtracting the total cost from the total revenue.
Revenue:
The total revenue is calculated by multiplying the number of albums sold by the selling price per album. In this case, the selling price per album is $2.50.
Cost:
The total cost is calculated by adding the fixed cost and the variable cost. The fixed cost is $20,000, and the variable cost per album is $2,500.
We can set up the equation for net profit as follows:
Net Profit = Revenue - Cost
Now, let's break down each component and determine their formulas.
Revenue Formula:
Revenue = Number of albums sold * Selling price per album
Number of albums sold = Target audience * Sales response rate (R)
Based on the information given, the sales response rate function is:
R = 1 - e^(-0.025t)
Cost Formula:
Cost = Fixed cost + Variable cost per album * Number of albums sold
Fixed cost = $20,000
Variable cost per album = $2,500
Now, substitute the formulas into the net profit equation:
Net Profit = (Target audience * Sales response rate) * Selling price per album - Fixed cost - Variable cost per album * (Target audience * Sales response rate)
To maximize net profit, you need to find the value of t (length of the advertising campaign) that gives the maximum value for Net Profit. You can achieve this by either graphing the function and finding the maximum point or by using calculus to find the derivative of the net profit function and setting it equal to zero.
Once you've identified the value of t that maximizes net profit, you can determine how long the campaign should be conducted.