Question: The value of a house at the time of its completion was Ghc60,000. If the rate of change of the value of a house can be modelled by $\frac{dV}{dt}=150e^{0.1t}$, where $t$ is the time in years since the house was built and $V$ is the value of the house (in thousand of Ghana cedis), find the value $V(t)$ of the house 7 years from now.
1. **State the problem:** We are given the rate of change of the value of a house as $\frac{dV}{dt}=150e^{0.1t}$ and the initial value $V(0)=60$ (since 60,000 Ghana cedis is 60 thousand). We need to find the value $V(7)$, the value of the house 7 years after completion.
2. **Formula and rules:** To find $V(t)$ from $\frac{dV}{dt}$, we integrate the rate function with respect to $t$:
$$V(t) = \int \frac{dV}{dt} dt = \int 150e^{0.1t} dt + C$$
where $C$ is the constant of integration.
3. **Integration:**
$$\int 150e^{0.1t} dt = 150 \int e^{0.1t} dt$$
Recall that $\int e^{kt} dt = \frac{1}{k} e^{kt} + C$, so:
$$150 \int e^{0.1t} dt = 150 \times \frac{1}{0.1} e^{0.1t} + C = 1500 e^{0.1t} + C$$
4. **General solution:**
$$V(t) = 1500 e^{0.1t} + C$$
5. **Use initial condition to find $C$:**
At $t=0$, $V(0) = 60$:
$$60 = 1500 e^{0} + C = 1500 + C$$
$$C = 60 - 1500 = -1440$$
6. **Final function for $V(t)$:**
$$V(t) = 1500 e^{0.1t} - 1440$$
7. **Calculate $V(7)$:**
$$V(7) = 1500 e^{0.7} - 1440$$
Calculate $e^{0.7} \approx 2.01375$:
$$V(7) = 1500 \times 2.01375 - 1440 = 3020.625 - 1440 = 1580.625$$
8. **Interpretation:** The value of the house 7 years from now is approximately 1580.625 thousand Ghana cedis, or Ghc1,580,625.
**Answer:**
$$\boxed{V(7) \approx 1580.63 \text{ thousand Ghana cedis}}$$