A water tank is formed with the combination of cylinder and hemisphere. The total height of the tank is 14 m and base area is 38.5 squ m. If the tank is filled with water at the rate of 24 paisa per…

Solution

Step 1: Determine the radius of the tank

The base area of the tank is given as 38.5 m². Since the base is circular (common to both the cylinder and hemisphere), we use the formula for the area of a circle: $$\pi r^2=38.5$$ Solving for $r$: $$r^2=\frac{38.5}{\pi }\approx \frac{38.5}{3.1416}\approx 12.25$$ $$r=\sqrt{12.25}=3.5\text{ m}$$

Step 2: Determine the height of the cylindrical part

The total height of the tank is 14 m. The hemisphere has a radius of 3.5 m, so its height is also 3.5 m (since the height of a hemisphere is equal to its radius). Therefore, the height of the cylindrical part ($h_1$) is: $$h_1=14-3.5=10.5\text{ m}$$

Step 3: Calculate the volume of the cylindrical part

The volume of a cylinder is given by: $$V_{\text{cylinder}}=\pi r^2{h}_1$$ Substituting the values: $$V_{\text{cylinder}}=3.1416\times (3.5{)}^2\times 10.5$$ $$V_{\text{cylinder}}=3.1416\times 12.25\times 10.5\approx 404.39{\text{ m}}^3$$

Step 4: Calculate the volume of the hemispherical part

The volume of a hemisphere is given by: $$V_{\text{hemisphere}}=\frac{2}{3}\pi r^3$$ Substituting the values: $$V_{\text{hemisphere}}=\frac{2}{3}\times 3.1416\times (3.5{)}^3$$ $$V_{\text{hemisphere}}=\frac{2}{3}\times 3.1416\times 42.875\approx 91.63{\text{ m}}^3$$

Step 5: Calculate the total volume of the tank

$$V_{\text{total}}=V_{\text{cylinder}}+V_{\text{hemisphere}}$$ $$V_{\text{total}}=404.39+91.63=496.02{\text{ m}}^3$$

Step 6: Convert volume to liters

Since $1{\text{ m}}^3=1000\text{ liters}$: $$V_{\text{total}}=496.02\times 1000=496,020\text{ liters}$$

Step 7: Calculate the total cost to fill the tank

The cost to fill the tank is 24 paisa per liter. First, convert paisa to rupees: $$24\text{ paisa}=0.24\text{ rupees}$$ Now, calculate the total cost: $$\text{Total cost}=496,020\times 0.24=119,044.8\text{ rupees}$$

Final Answer: The total cost to fill the water tank is Rs. 119,044.80.