Need help on calculus problem

Question

A company deposits $5,000 into an account that earns interest. The rate at which the account value changes is given by , where A(t) is the account value, in dollars, after t years. What will be the value of the account after 10 years if no additional deposits or withdrawals are made?

a. $5,112.50

b. $5,113.78

c. $6,246.02

d. $6,261.61

Answer 1

A = Pe^.0225t

A =5000e^.225

A = 5000(1.2523)

A = $6,261.61 after 10 years

so d "seems" correct

but there's more than one way to solve this problem

5000(1.0225)^10 = $6246.01 after 10 years

then c is correct

or y'(t) = dy/dt = .0225t

integrate to get 0.0225t^2/2 = 0.01125(10^2) = 1.125

add 5000 + 1125 gives answer a. with $5,1125,

but that's really the interest added after just one year

that confuses 1.125 with 112.5 and confuses A with t

the interest rate = 0.0225 = 2 1/4 % per year

1st year 5,000 grows to 5,000 + 112.5 = 5112.5

2nd year 5227.5

3rd year 5345

4th year 5,5465.4

5th year 5,588.4

6th year 5,714

7th year 5843

8th year 5,974

9th year 6,109

10th year 6246.02

go with answer c for interest compounded annually

Unless interest is compounded continuously, then answer d is correct

as this is a calculus question, apparently, calculus involves continuous variables,

so then d is the correct answer