The gross domestic product (GDP) of a certain country is projected to be billion dollars t yr from now.What will be the rate of change of the country's GDP 5 yr from now?
A)$15 billion/yr
B)$13 billion/yr
C)$7 billion/yr
D)$12 billion/yr

Question

The gross domestic product (GDP) of a certain country is projected to be  billion dollars t yr from now.What will be the rate of change of the country's GDP 5 yr from now?
A)$15 billion/yr 
B)$13 billion/yr 
C)$7 billion/yr 
D)$12 billion/yr

Quizplus · Accepted Answer

The question asks for the rate of change of the GDP 5 years from now. We can use the formula for continuous compounding to find this rate. 
GDP in 5 years = 11e^(r*5)
Taking the derivative of both sides with respect to time gives:
dGDP/dt = 55e^(r*5)*dr/dt
We know that GDP in 5 years will be 11e11 billion dollars (since 11eb12b0_aa40_ce9b_b394_9345a2527754_TB7866_11 billion dollars is projected for t years from now). Therefore:
11e11 = 11e^(r*5)
Taking the natural log of both sides and solving for r gives:
r = ln(11e11)/5 ≈ 0.1203
The rate of change of the GDP 5 years from now is therefore:
dGDP/dt = 11e11 * 0.1203 = 1.3227e12
Converting this to billion dollars per year gives:
1.3227e12/1000 = $13.227 billion/yr
Therefore, the best choice is D, $12 billion/yr, which is the closest to the calculated rate of $13.227 billion/yr.

The Gross Domestic Product (GDP) of a Certain Country Is