A juice seller was serving his customers using glasses as shown in Fig. 12.13. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm, find the apparent capacity of the glass and its actual capacity. (Use $\pi = 3.14$)

Step 1: Identify Given Values

We are given the inner diameter of the cylindrical glass as $D = 5 \text{ cm}$. From this, we calculate the radius $r$ by dividing the diameter by 2. The height of the glass is given as $H = 10 \text{ cm}$. The hemispherical raised portion at the bottom has the same radius as the cylinder.

Step 2: Calculate Apparent Capacity of the Glass

The apparent capacity of the glass is the volume of the cylinder as if there were no raised portion at the bottom. The formula for the volume of a cylinder is $V = \pi r^2 H$. We substitute the given values for $\pi$, $r$, and $H$ into this formula.

Step 3: Compute Apparent Capacity

First, we calculate $2.5^2 = 6.25$. Then, we multiply $6.25$ by $10$ to get $62.5$. Finally, we multiply $3.14$ by $62.5$ to find the apparent capacity of the glass.

Step 4: Calculate Volume of Hemispherical Raised Portion

The raised portion at the bottom is a hemisphere with the same radius as the cylindrical glass, $r = 2.5 \text{ cm}$. The formula for the volume of a hemisphere is $\frac{2}{3} \pi r^3$. We substitute the values for $\pi$ and $r$.

Step 5: Compute Volume of Hemispherical Raised Portion

First, we calculate $(2.5)^3 = 15.625$. Then, we multiply $2 \times 3.14 \times 15.625 = 98.125$. Finally, we divide $98.125$ by $3$ to get the volume of the hemispherical portion, which is approximately $32.708 \text{ cm}^3$.

Step 6: Calculate Actual Capacity of the Glass

The actual capacity of the glass is obtained by subtracting the volume of the hemispherical raised portion from the apparent capacity of the cylindrical glass. We subtract the calculated volume of the hemisphere from the apparent capacity.

Step 7: Compute Actual Capacity

Subtracting $32.708$ from $196.25$ gives us $163.542 \text{ cm}^3$. This is the actual amount of juice the glass can hold.

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