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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?
(ii) Find the volume of the stem.
(iii) (a) Determine the volume of $7$ such mushrooms.
OR
(b) Find the total surface area of $7$ such mushrooms.
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?
(ii) Find the volume of the stem.
(iii) (a) Determine the volume of $7$ such mushrooms.
OR
(b) Find the total surface area of $7$ such mushrooms.
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Solution:
(i) Height of a mushroom = $2 + 3 = 5$ cm (1 Mark)
(ii) Volume of the stem = $\frac{22}{7} \times (\frac{1.4}{2})^2 \times 2$
$= \frac{22}{7} \times 0.7 \times 0.7 \times 2 = 3.08$ cm$^3$ (1 Mark)
(iii) (a) Volume of $7$ mushrooms = $7 \times (\frac{2}{3}\pi R^3 + \pi r^2 h)$, where $R = 3$ cm, $r = 0.7$ cm, $h = 2$ cm
$= 7 \times (\frac{2}{3} \times \frac{22}{7} \times 3^3 + \frac{22}{7} \times (0.7)^2 \times 2)$ (1 Mark)
$= 7 \times (\frac{2}{3} \times \frac{22}{7} \times 27 + 3.08) = 7 \times (\frac{396}{7} + 3.08) = 7 \times (56.57 + 3.08) = 7 \times 59.65 = 417.55$ cm$^3$ (1 Mark)
OR
(b) TSA = $7\{2\pi R^2 + (\pi R^2 - \pi r^2) + 2\pi r h + \pi r^2\} = 7(3\pi R^2 + 2\pi r h)$
$= 7 \times (3 \times \frac{22}{7} \times 3 \times 3 + 2 \times \frac{22}{7} \times \frac{7}{10} \times 2)$ (1 Mark)
$= 7 \times (\frac{594}{7} + \frac{88}{10}) = 7 \times (84.857 + 8.8) = 7 \times 93.657 = 655.599$ cm$^2$ or $655.6$ cm$^2$ (1 Mark)
(i) Height of a mushroom = $2 + 3 = 5$ cm (1 Mark)
(ii) Volume of the stem = $\frac{22}{7} \times (\frac{1.4}{2})^2 \times 2$
$= \frac{22}{7} \times 0.7 \times 0.7 \times 2 = 3.08$ cm$^3$ (1 Mark)
(iii) (a) Volume of $7$ mushrooms = $7 \times (\frac{2}{3}\pi R^3 + \pi r^2 h)$, where $R = 3$ cm, $r = 0.7$ cm, $h = 2$ cm
$= 7 \times (\frac{2}{3} \times \frac{22}{7} \times 3^3 + \frac{22}{7} \times (0.7)^2 \times 2)$ (1 Mark)
$= 7 \times (\frac{2}{3} \times \frac{22}{7} \times 27 + 3.08) = 7 \times (\frac{396}{7} + 3.08) = 7 \times (56.57 + 3.08) = 7 \times 59.65 = 417.55$ cm$^3$ (1 Mark)
OR
(b) TSA = $7\{2\pi R^2 + (\pi R^2 - \pi r^2) + 2\pi r h + \pi r^2\} = 7(3\pi R^2 + 2\pi r h)$
$= 7 \times (3 \times \frac{22}{7} \times 3 \times 3 + 2 \times \frac{22}{7} \times \frac{7}{10} \times 2)$ (1 Mark)
$= 7 \times (\frac{594}{7} + \frac{88}{10}) = 7 \times (84.857 + 8.8) = 7 \times 93.657 = 655.599$ cm$^2$ or $655.6$ cm$^2$ (1 Mark)