1. **Problem statement:** We have an initial value of apartments and land equal at the end of construction, each 4,000,000. The values fluctuate yearly: apartments by 10%, land by 2%. After 3 years, the apartment value is 4,514,624. We want to find: 1) The initial values of apartments and land when apartment value is higher than land. 2) How many years since construction ended when the building value equals a certain value. 3) In how many years since construction ended the building value will be at a certain time. 2. **Formulas and rules:** Value after $n$ years with annual growth rate $r$ is: $$ V_n = V_0 \times (1 + r)^n $$ where $V_0$ is initial value, $r$ is growth rate (as decimal), $n$ is number of years. 3. **Step 1: Find initial values and verify given data** Given: - Initial apartment value $A_0 = 4,000,000$ - Initial land value $L_0 = 4,000,000$ - Apartment growth rate $r_A = 10\% = 0.10$ - Land growth rate $r_L = 2\% = 0.02$ - After 3 years, apartment value $A_3 = 4,514,624$ Calculate apartment value after 3 years using formula: $$ A_3 = A_0 \times (1 + r_A)^3 = 4,000,000 \times (1.10)^3 = 4,000,000 \times 1.331 = 5,324,000 $$ But given $A_3 = 4,514,624$, which is less than calculated, so initial apartment value might be different or growth rate is effective rate. 4. **Step 2: Calculate initial apartment value from given $A_3$** Rearranged formula: $$ A_0 = \frac{A_3}{(1 + r_A)^3} = \frac{4,514,624}{1.331} \approx 3,392,000 $$ 5. **Step 3: Calculate land value after 3 years** Using $L_0 = 4,000,000$ and $r_L = 0.02$: $$ L_3 = L_0 \times (1 + r_L)^3 = 4,000,000 \times (1.02)^3 = 4,000,000 \times 1.061208 = 4,244,832 $$ 6. **Step 4: Determine when apartment value exceeds land value** We want to find $n$ such that: $$ A_0 (1 + r_A)^n > L_0 (1 + r_L)^n $$ Divide both sides by $L_0 (1 + r_L)^n$: $$ \frac{A_0}{L_0} > \left( \frac{1 + r_L}{1 + r_A} \right)^n $$ Take natural logarithm: $$ \ln \left( \frac{A_0}{L_0} \right) > n \ln \left( \frac{1 + r_L}{1 + r_A} \right) $$ Solve for $n$: $$ n < \frac{\ln \left( \frac{A_0}{L_0} \right)}{\ln \left( \frac{1 + r_L}{1 + r_A} \right)} $$ Using $A_0 = 3,392,000$, $L_0 = 4,000,000$: $$ \frac{A_0}{L_0} = \frac{3,392,000}{4,000,000} = 0.848 $$ $$ \frac{1 + r_L}{1 + r_A} = \frac{1.02}{1.10} = 0.92727 $$ Calculate logarithms: $$ \ln(0.848) = -0.164 $$ $$ \ln(0.92727) = -0.0757 $$ Calculate $n$: $$ n < \frac{-0.164}{-0.0757} = 2.17 $$ So apartment value exceeds land value after approximately 2.17 years. 7. **Step 5: Calculate building value after 3 years** Building value is sum of apartment and land values: $$ B_3 = A_3 + L_3 = 4,514,624 + 4,244,832 = 8,759,456 $$ 8. **Step 6: Calculate when building value reaches a certain value** If we want to find $n$ such that building value $B_n = B$, we solve: $$ B_n = A_0 (1 + r_A)^n + L_0 (1 + r_L)^n = B $$ This requires numerical methods or trial for specific $B$. **Final answers:** - Initial apartment value from given data: approximately 3,392,000 - Land initial value: 4,000,000 - Apartment value exceeds land value after about 2.17 years - Building value after 3 years: 8,759,456