It is estimated that the demand for a manufacturer's product is increasing exponentially at an instantaneous rate of 7% per year. If the current demand is increasing by 7000 units per year and if the price remains fixed at $100 per unit, how much revenue will the manufacturer receive from the sale of the product over the next 4 years?
since "demand is increasing at an instantaneous rate of 7% a year" this means that d'(t)=pe^kt where k=0.07
now, "current demand is increasing by 7000 units per year" means that the principal demand is 7000, so p=7000.
then, "price remains fixed at $100 per unit" and since we are looking for revenue and not demand, and revenue=unit*price, so unit=7000 and price=100, so multiply those which gives you 700,000 and insert for p, so p actually =700,000 for revenue, not demand.
the last part "how much revenue will the manufacturer receive over the next 4 years. so you just use the fundamental theorum of calculus, and insert t=4 minus t=0, and you have your answer in terms of revenue not demand. :)
To find the revenue over the next 4 years, we need to calculate the total number of units sold each year and multiply it by the price per unit.
Given:
- Instantaneous growth rate: 7% per year
- Current demand increase: 7000 units per year
- Price per unit: $100
First, we need to calculate the growth factor. The growth factor is given by:
Growth factor = 1 + growth rate as a decimal
In this case, the growth factor is:
Growth factor = 1 + 0.07 = 1.07
Next, we can calculate the number of units sold each year using the formula for exponential growth:
Number of units sold = Initial demand * Growth factor^(number of years)
To find the initial demand, we divide the current demand increase by the growth rate:
Initial demand = Current demand increase / growth rate as a decimal
Initial demand = 7000 / 0.07 = 100,000 units
Now we can calculate the number of units sold each year:
Year 1: 100,000 * (1.07)^1 = 107,000 units
Year 2: 100,000 * (1.07)^2 = 114,490 units
Year 3: 100,000 * (1.07)^3 = 122,504.3 units
Year 4: 100,000 * (1.07)^4 = 131,079.5 units
Finally, we can calculate the revenue over the next 4 years by multiplying the number of units sold each year by the price per unit and adding them together:
Revenue = (Price per unit * Units sold in Year 1) + (Price per unit * Units sold in Year 2) + (Price per unit * Units sold in Year 3) + (Price per unit * Units sold in Year 4)
Revenue = ($100 * 107,000) + ($100 * 114,490) + ($100 * 122,504.3) + ($100 * 131,079.5)
Revenue = $10,700,000 + $11,449,000 + $12,250,430 + $13,107,950
Revenue = $47,507,380
Therefore, the manufacturer will receive $47,507,380 in revenue from the sale of the product over the next 4 years.