# Step-by-Step Solution ## Given Data - **Concrete density,** \( \rho_{conc} = 208 \ \text{kg/m}^3 \) - **Tank inside dimensions:** \( 30\, \text{m} \) (length) \( \times 27\, \text{m} \) (width) \( \times 5\, \text{m} \) (depth) - **Wall and floor thickness:** \( t = 200 \ \text{mm} = .2 \ \text{m} \) - **Water density:** \( \rho_{w} = 100 \ \text{kg/m}^3 \) - **Tank is empty, except possibly for water up to the check valve** - **Objective:** Find the **maximum height \( h \)** (from the bottom) at which the valve should be installed **so that the tank does not float** (buoyant force does not exceed weight). --- ## 1. **Calculate the Weight of the Tank (Concrete Only)** ### Step 1.1: Find the volume of concrete used in the tank #### **Floor** - Area = \( 30 \times 27 = 810 \ \text{m}^2 \) - Volume = \( 810 \times .2 = 162 \ \text{m}^3 \) #### **Walls** - **Two long sides (30 m):** - Each wall: \( 30 \ \text{m} \) (length) \( \times 5 \ \text{m} \) (height) \( \times .2 \ \text{m} \) (thick) - Volume per wall: \( 30 \times 5 \times .2 = 30 \ \text{m}^3 \) - Both walls: \( 2 \times 30 = 60 \ \text{m}^3 \) - **Two short sides (27 m):** - Each wall: \( 27 \ \text{m} \) (length) \( \times 5 \ \text{m} \) (height) \( \times .2 \ \text{m} \) (thick) - Volume per wall: \( 27 \times 5 \times .2 = 27 \ \text{m}^3 \) - Both walls: \( 2 \times 27 = 54 \ \text{m}^3 \) #### **Total wall volume:** - \( V_\text{walls} = 60 + 54 = 114 \ \text{m}^3 \) #### **Total tank concrete volume:** - \( V_\text{total} = V_\text{floor} + V_\text{walls} = 162 + 114 = 276 \ \text{m}^3 \) ### Step 1.2: Calculate the weight \[ W_\text{tank} = V_\text{total} \cdot \rho_{conc} \cdot g \] \[ = 276 \ \text{m}^3 \times 208 \ \text{kg/m}^3 \times 9.81 \ \text{m/s}^2 \] \[ = 574,080 \ \text{kg} \times 9.81 \ \text{m/s}^2 \] \[ = 5,634,713 \ \text{N} \] --- ## 2. **Calculate Buoyant Force When Tank is Empty** - When the tank is empty and water table rises above the top, the tank is submerged. - **Buoyant force = Weight of water displaced by outer volume of tank** ### Step 2.1: Outer dimensions - **Length:** \( 30 + 2 \times .2 = 30.4 \ \text{m} \) - **Width:** \( 27 + 2 \times .2 = 27.4 \ \text{m} \) - **Depth:** \( 5 + .2 = 5.2 \ \text{m} \) - **Outer volume:** \( V_\text{out} = 30.4 \times 27.4 \times 5.2 \) \[ V_\text{out} = 30.4 \times 27.4 = 832.96 \] \[ V_\text{out} = 832.96 \times 5.2 = 4,331.392 \ \text{m}^3 \] ### Step 2.2: Buoyant force \[ F_b = V_\text{out} \cdot \rho_{w} \cdot g \] \[ = 4,331.392 \ \text{m}^3 \times 100 \ \text{kg/m}^3 \times 9.81 \ \text{m/s}^2 \] \[ = 4,331,392 \ \text{kg} \times 9.81 \ \text{m/s}^2 \] \[ = 42,513,973 \ \text{N} \] --- ## 3. **Buoyant Force When Tank Is Filled to Height \( h \) with Water** - The check valve allows water in, filling the tank to height \( h \) (from bottom). - **The tank will not float if:** \[ \text{Weight of tank} + \text{Weight of water inside} \geq \text{Buoyant force (weight of water displaced)} \] ### Step 3.1: Water inside tank - **Volume inside to height \( h \):** \( V_\text{in} = 30 \times 27 \times h = 810h \ \text{m}^3 \) - **Weight:** \( W_{w,\text{in}} = 810h \times 100 \times 9.81 = 7,946,100 h \ \text{N} \) ### Step 3.2: Balance equation \[ W_\text{tank} + W_{w,\text{in}} = F_b \] \[ 5,634,713 + 7,946,100 h = 42,513,973 \] --- ## 4. **Solve for \( h \) (Height of Water Needed to Prevent Floating)** \[ 7,946,100 h = 42,513,973 - 5,634,713 = 36,879,260 \] \[ h = \frac{36,879,260}{7,946,100} \] \[ h \approx 4.64 \ \text{m} \] --- ## 5. **Final Answer** ### **Maximum height at which the valve should be installed:** \[ \boxed{4.64 \ \text{m}} \] **The check valve should be installed no higher than \( 4.64 \ \text{m} \) above the bottom of the tank.** --- ### **Summary Table** | Quantity | Value | |-----------------------------|---------------------| | Tank weight | \(5,634,713\,\text{N}\) | | Buoyant force (full subm.) | \(42,513,973\,\text{N}\) | | Required water height | \(4.64\,\text{m}\) | --- ### **Diagram (Schematic)**  *Figure: Cross-section of tank showing valve height \( h \) above floor*