Answer:
The fund, F, after 3 months = $48,442.25
Explanation:
Amount to be made = $50000
Initial amount = F₀ = $8000
The contribution to the fund F is increasing at a rate proportional to the difference between the cost of $50,000 and the amount F at time t in months.
Noting that F ≤ 50,000
(dF/dt) ∝ (50,000 - F)
k = constant of proportionality
F' = k(50000 - F)
dF/(50000 - F) = k dt
∫ dF/(50000 - F) = ∫ kdt
- In [50,000 - F] = kt + c
c = constant of integration
At t = 0, F = $8000
- In [50,000 - 8,000] = 0 + c
c = - In 42,000 = - 10.645
- In [50,000 - F] = kt - 10.645
After one month, $36,000 is in the fund.
at t = 1 month, F = $36,000
- In [50,000 - 36,000] = k - 10.645
k - 10.645 = - In 14,000 = - 9.547
k = 10.645 - 9.547 = 1.098
- In [50,000 - F] = 1.098t - 10.645
In [50,000 - F] = 10.645 - 1.098t
At t = 3 months,
In [50,000 - F] = 10.645 - 1.098(3) = 7.351
50,000 - F = e⁷•³⁵¹ = 1557.7535
F = 50,000 - 1557.7535 = $48,442.2465
The fund, F, after 3 months = $48,442.25
Hope this Helps!!!