Question 1187605: A mutual fund has a growth rate recorded each year, expressed by the function
๐โฒ(๐ก)=๐(๐ก)=200๐0.04๐ก, 0โค๐กโค10 (in thousands of dollars per year), where ๐ก 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 ๐(๐ก) which represents the value of the fund ๐ก 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).
โซV'(t) dt = โซ200e^(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) โ 5000 * 1.4918 + 995,000
V(10) โ 7459 + 995,000
V(10) โ 1,002,459
Therefore, the value of the fund in 2007 is approximately $1,002,459.
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