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:

38(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.
The mushroom has: a hemispherical cap with radius $R = 3$ cm, and a cylindrical stem with height $h = 2$ cm and diameter $d = 1.4$ cm (so radius $r = 0.7$ cm). We need the total height, TSA, and volume.
Step 2: Find the Total Height.
The hemisphere sits on top of the cylinder. The height of the hemisphere = $R = 3$ cm (from its flat base to its top). The height of the cylinder = $h = 2$ cm. \[ \text{Total height} = R + h = 3 + 2 = 5 \text{ cm} \] Wait, but note that the hemisphere's flat face rests above the cylinder. The total height of the full mushroom = height of cylinder + height of hemisphere = $h + R = 2 + 3 = 5$ cm. Actually, if the mushroom cap (hemisphere radius 3) sits on top and the stem is cylindrical (radius 0.7), the total vertical height = stem height + hemisphere radius = $2 + 3 = 5$ cm.
Step 3: Compute the Volume of the Hemisphere.
\[ V_{\text{hemisphere}} = \frac{2}{3}\pi R^3 = \frac{2}{3} \times \frac{22}{7} \times 27 = \frac{2 \times 22 \times 27}{3 \times 7} = \frac{1188}{21} = \frac{396}{7} \approx 56.57 \text{ cm}^3 \]
Step 4: Compute the Volume of the Cylinder.
\[ V_{\text{cylinder}} = \pi r^2 h = \frac{22}{7} \times (0.7)^2 \times 2 = \frac{22}{7} \times 0.49 \times 2 = \frac{22 \times 0.98}{7} = \frac{21.56}{7} = 3.08 \text{ cm}^3 \]
Step 5: Compute the Total Volume.
\[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} = \frac{396}{7} + \frac{21.56}{7} \approx 56.57 + 3.08 = 59.65 \text{ cm}^3 \]
Step 6: Compute the Total Surface Area and State Final Answers.
TSA = Curved Surface of hemisphere + Curved Surface of cylinder + Base of cylinder (the exposed bottom of the stem): \[ \text{TSA} = 2\pi R^2 + 2\pi r h + \pi r^2 \] \[ = 2 \times \frac{22}{7} \times 9 + 2 \times \frac{22}{7} \times 0.7 \times 2 + \frac{22}{7} \times 0.49 \] \[ = \frac{396}{7} + \frac{61.6}{7} + \frac{10.78}{7} = \frac{468.38}{7} \approx 66.91 \text{ cm}^2 \] \[ \boxed{\text{Total height} = 5 \text{ cm},\ V \approx 59.65 \text{ cm}^3,\ \text{TSA} \approx 66.91 \text{ cm}^2} \]