A-pe suppose you invest S1000 at an annual interest rate ofr %, compounded conti

Question

A-pe suppose you invest S1000 at an annual interest rate ofr %, compounded continuously. nt end of 10 years, you have a balance of B dollars, where B-rn) Ir rs) . 1649 17, Circle the and f ,(5)-165, which of the following statements are letter of your choices (there may be more than one valid statement) A. The balance in your account after 5 years is $1649. B. The balance in your C. If the interest rate D. If the interest rate account grows at a rate of 165 S per year. increases by 1%, you would expect about S165 more in your account. increases from 5% to 6%, you would expect about S 165 more in your account. If the interest rate increases from 5% to 6%, you would expect about $18 14 in your account. E. If the interest rate decreases from 5% to 4%, you would expect about S 165 less in your account. F. Ler Peo rgpresen the population of a city t years from now. Suppose PrO) -200,000 and P (0)- 5,000. a. Estimate the population of the city last year. b. Estimate the population of the city two years from now. c. Suppose P "(0)-1,000. i. Is your estimate for the population of the city two years from now an overestimate or underestimate? Explain. ii. Estimate the rate at which the population will change next year.

Explanation / Answer

1000$ is invested at r% that is compounded continuously for 10 years and after 10 years the amount is B= f(r)
here, f(r) signifies that it is a function of the interest rate at which it is being compounded continuously;
f(5) = 1649 and f'(5) = 165;
This means that if r=5% then after 10 years, 1000$ will become 1649 dollars
Also, f'(5) = 165 means that when r=5%, a unit change in 'r' will bring about 165 units of change in B or f(r);

Evaluating the options:
Thus, option A is incorrect as the balance in the account after 10 (not 5) years is 1649$

Option B is incorrect as the balance in the account does not follow a fixed growth as compounding takes place which means their will be interest on the interest accrued too, so there will be no fixed amount of growth each year;

Option C and D are correct as f'(5) = 165; which means that if r=5% increases by 1 unit that is it becomes r=5+1 = 6% then f(r) = B = amount of money in account after 10 years becomes = 1649+165 ; i.e. increases by 165$

Option E is also correct as 1649+165 = 1814$

Option F is also correct as f'(5) = 165; Thus, a unit change in r (can also be unit decrease) will bring a change (decrease) in B=f(r) by 165 units
Thus, you will receive 165$ fewer in your account if interest rate changes from 5% to 4%

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