SOLUTION: A mutual fund has a growth rate recorded each year, expressed by the function 𝑉′(𝑡)=𝑟(𝑡)=200𝑒0.04𝑡, 0≤𝑡≤10 (in thousands of dollars per year), where �

Question 1187605: A mutual fund has a growth rate recorded each year, expressed by the function
𝑉′(𝑡)=𝑟(𝑡)=200𝑒0.04𝑡, 0≤𝑡≤10 (in thousands of dollars per year), where 𝑡 is the number of years passed from year 1997. The value of the fund in year 1997 is $1,000,000.
a) Determine the function 𝑉(𝑡) which represents the value of the fund 𝑡 years after year 1997.

b) Determine the value of the fund in year 2007.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**a) Determining the function V(t):**
To find the function V(t) that represents the value of the fund *t* years after 1997, we need to integrate the growth rate function V'(t) = r(t).
∫V'(t) dt = ∫200e^(0.04t) dt
V(t) = (200/0.04)e^(0.04t) + C
V(t) = 5000e^(0.04t) + C
We know that in 1997 (t=0), the value of the fund is $1,000,000. We can use this information to find the constant of integration, C.
1,000,000 = 5000e^(0.04*0) + C
1,000,000 = 5000 + C
C = 995,000
Therefore, the function representing the value of the fund *t* years after 1997 is:
V(t) = 5000e^(0.04t) + 995,000
**b) Determining the value of the fund in 2007:**
The year 2007 is 10 years after 1997, so we need to find V(10):
V(10) = 5000e^(0.04*10) + 995,000
V(10) = 5000e^(0.4) + 995,000
V(10) ≈ 5000 * 1.4918 + 995,000
V(10) ≈ 7459 + 995,000
V(10) ≈ 1,002,459
Therefore, the value of the fund in 2007 is approximately $1,002,459.

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