2. Appreciation of value of asset. The value, $$ S V $$, of a flat can be modelled by $$ V=V_{e} e^{\pi \prime} $$, where $$ t $$ is the time in years since it was built and $$ p $$ is a constant. Calculate (i) the value of $$ p $$ if, after 5 years, the value of the flat is doubled, (ii) the age of the flat when the value of the flat is 5 times its original value.

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UpStudy AI · Accepted Answer

Let's analyze the problem step by step.

The value $$ V $$ of the flat is modeled by the equation:
$$
V = V_0 e^{pt}
$$
where:
- $$ V_0 $$ is the initial value of the flat,
- $$ p $$ is a constant,
- $$ t $$ is the time in years since the flat was built.

### Part (i): Calculating the Value of $$ p $$

**Given:** After 5 years, the value of the flat doubles.  
Mathematically, this means:
$$
V(5) = 2V_0
$$

Substitute $$ t = 5 $$ into the value equation:
$$
2V_0 = V_0 e^{5p}
$$

Divide both sides by $$ V_0 $$:
$$
2 = e^{5p}
$$

Take the natural logarithm of both sides:
$$
\ln(2) = 5p
$$

Solve for $$ p $$:
$$
p = \frac{\ln(2)}{5}
$$

**Numerical Value:**
$$
p \approx \frac{0.6931}{5} \approx 0.1386 \text{ per year}
$$

### Part (ii): Determining the Age of the Flat When Its Value is 5 Times the Original

**Given:** We need to find $$ t $$ when $$ V(t) = 5V_0 $$.

Set up the equation:
$$
5V_0 = V_0 e^{pt}
$$

Divide both sides by $$ V_0 $$:
$$
5 = e^{pt}
$$

Take the natural logarithm of both sides:
$$
\ln(5) = pt
$$

Solve for $$ t $$:
$$
t = \frac{\ln(5)}{p}
$$

Substitute the value of $$ p $$ from part (i):
$$
t = \frac{\ln(5)}{\ln(2)/5} = 5 \cdot \frac{\ln(5)}{\ln(2)}
$$

**Numerical Calculation:**
$$
t \approx 5 \cdot \frac{1.6094}{0.6931} \approx 5 \cdot 2.3219 \approx 11.61 \text{ years}
$$

### **Final Answers:**
1. **(i)** The constant $$ p $$ is $$ \frac{\ln 2}{5} $$ per year.
2. **(ii)** The flat will be approximately **11.61 years** old when its value is five times its original value.
$$ p = \frac{\ln 2}{5} $$ per year, and the flat will be approximately 11.61 years old when its value is five times its original value.

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Solución

Part (i): Calculating the Value of

Part (ii): Determining the Age of the Flat When Its Value is 5 Times the Original

Final Answers:

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