2. In 2000, the population of the United States was 280 million and the number o
ID: 3018914 • Letter: 2
Question
2. In 2000, the population of the United States was 280 million and the number of vehicles was 200 million. If the population of the United States is growing at a rate of 1%per year while the number of vehicles is growing at the rate of 3%, in what year will there an average of one vehicle per person? 3. How long will it take to double your m oney if it earns 6.5% compounded continuously? At what interest rate, compounded continuously, would your money double in 5 years? 4. A drug manufactured by a pharmaceutical company is sold in bulk at a price of $150 per unit. The total production cost (in dollars) for x units in one week is C(2) 0.02x2 + 100x + 3000, How many units of the drug must be manufactured and sold in a week to maximize the profit? What is the maximum profit?Explanation / Answer
2. let after x years from 2000 there is an average of one vehicle per person
=> (200 + 3%.x.200) / ( 280 + 1%.x.280)=1
=> 200 + 6x = 280 + 2.8x
=> 6x - 2.8x = 280 -200
=> 3.2x = 80
=> x = 25 years
3) let initial amount be $x
Continuous Compounding
A=Pert
where A is the Future amount,P is initial amount , t is time , r is interest rate
for amount to get double time required is : -
2x = x (e)rt
=> 2 = (e)0.065t
taking log both sithes
=> ln(2) = 0.065t ln(e)
=> t = 0.6931471/ 0.065
=> t ~10.6 years
t = 10 years and 7 months approximately
2nd part)
interest rate is r , time = 5 years , f=2x, P=x
=> 2x = x e5r
=> 2 = e5r
taking log both sides
ln(2)= 5r ln(e)
=> r = ln(2)/ 5
=> r = 0.1386
=> r= 13% approximately
4) selling price SP = $ 150 per unit
total SP for x products = 150x
total cost CP= 0.02x2 + 100x + 3000
profit = SP - CP
proft = 150x - 0.02x2 - 100x - 3000
profit = 50x - 0.02x2 - 3000
for maximum profit first derivative should be zero
=> dp/dx = d( 50x - 0.02x2 - 3000)/ dx=0
=> dp/dx = 50 - 0.04x= 0
=> 0.04x = 50
=> x = 1250
maximum profit = 50(1250) - 0.02(1250)2 - 3000
= 62500 - 34250
= $ 28,250
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