Question

There are many varieties of mushrooms available in the world. One such mushroom ‘Amanita muscaria’ has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylindrical). The hemispherical cap’s radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following questions:

36(i) What is the total height of a mushroom ?

Hint:

Always remember that a hemisphere of radius \(R\) has a depth (vertical height) of exactly \(R\). This is a crucial concept when dealing with composite solids.

Answer 1

Correct Answer: 5

Step 1: Note the given dimensions.
Hemispherical cap: radius $R = 3\text{ cm}$. Cylindrical stem: height $h = 2\text{ cm}$, diameter $= 1.4\text{ cm}$ so radius $r = 0.7\text{ cm}$.
Step 2: Find the total height of the mushroom.
The hemisphere sits on top of the cylinder. Height of hemisphere = its radius = $R = 3\text{ cm}$. Total height = Height of cylinder + Height of hemisphere $= 2 + 3 = 5\text{ cm}$.
Step 3: Calculate the volume of the hemispherical cap.
\[ V_{\text{hemisphere}} = \frac{2}{3}\pi R^3 = \frac{2}{3} \times \frac{22}{7} \times 3^3 = \frac{2}{3} \times \frac{22}{7} \times 27 = \frac{2 \times 22 \times 9}{7} = \frac{396}{7} \approx 56.57\text{ cm}^3 \]
Step 4: Calculate the volume of the cylindrical stem.
\[ V_{\text{cylinder}} = \pi r^2 h = \frac{22}{7} \times (0.7)^2 \times 2 = \frac{22}{7} \times 0.49 \times 2 = \frac{22 \times 0.98}{7} = \frac{21.56}{7} \approx 3.08\text{ cm}^3 \]
Step 5: Find the total volume.
\[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} \approx 56.57 + 3.08 = 59.65\text{ cm}^3 \approx 59.66\text{ cm}^3 \]
Step 6: Conclusion.
Total height = 5 cm and total volume of the mushroom shape is approximately 59.66 cm$^3$.
\[ \boxed{\text{Total Height} = 5 \text{ cm}, \quad V \approx 59.66 \text{ cm}^3} \]