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 (cylinderical). 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 :

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

Hint:

Remember that for any hemisphere, the height from the flat base to the top of the dome is exactly equal to its radius!

Answer 1

Correct Answer: 5

Step 1: Understanding the Question:
This question is from the chapter Surface Areas and Volumes.
The mushroom is a solid composite object consisting of a hemispherical cap placed on top of a cylindrical stem.
We need to calculate the total vertical height of the mushroom from the base of the stem to the top of the cap.

Step 2: Key Formula or Approach:
1. Let \(R\) be the radius of the hemispherical cap, where \(R = 3\text{ cm}\).
2. Let \(h\) be the height of the cylindrical stem, where \(h = 2\text{ cm}\).
3. The vertical height of a hemisphere is equal to its radius \(R\).
4. Therefore, the total height of the mushroom is: \[ \text{Total Height} = \text{Height of stem } (h) + \text{Radius of cap } (R) \]

Step 3: Detailed Explanation:
1. Write down the given dimensions: - Radius of the hemispherical cap, \(R = 3\text{ cm}\)
- Height of the cylindrical stem, \(h = 2\text{ cm}\)
2. The distance from the center of the hemisphere to the highest point on top of the dome is equal to the radius \(R = 3\text{ cm}\).
3. Calculate the total height: \[ \text{Total Height} = h + R \] \[ \text{Total Height} = 2\text{ cm} + 3\text{ cm} = 5\text{ cm} \]

Step 4: Final Answer:
The total height of the mushroom is 5 cm.