Question

A toy is made in the form of hemisphere surmounted by a right cone whose circular base is joined with the plane surface of the hemisphere. The radius of the base of the cone is 7cm. and it's volume is 3/2 of the hemisphere. calculate the height of the cone and the surface area of the toy.​

Answer 1

Step-by-step explanation:

Radius of cone and hemisphere = 7 cm

Height of cone be h cm

Now, volume of hemisphere = (2/3)πr³

Volume of cone = (1/3)πr²h

A/q

(1/3)πr²h = (3/2)×[(2/3)πr³]

⇒h = 3r = 3×7 = 21 cm

Now, surface area

Slant height, l = √[(21)² +(7)²] = 7√10 cm = 22.13

Total surface area = (curved surface area of cone + hemisphere)

= (πrl + 2πr²)

=[(22/7)×7×22.13) + 2(22/7)×(7)²]

= 486.86 +308

=794.86 cm²

Answer 2

Given:

The toy is made in the form of hemisphere with the right cone surmounted such that the circular base coincides with the plane surface of the hemisphere.

The radius of the base of the cone is the same as the radius of the hemisphere ( as clear from the figure ) .

$The \: value \: of \: radius \: is \: given \: as \: r \: = \: 7 \: cm$

$Given\: : \:the \: value \: of \: the \: cone \: is \: \frac{3}{2} \: times \: th e\: volume \: of \: the \: hemisphere.$

$Therefore,\\ \\\pi {r}^{2} h \: = \: \frac{3}{2} \: \times \: \frac{2}{3} \: \pi {r}^{3}$

Simplifying by removing the common terms :-

Therefore,

$h = 3r = 3 \times 7 = 21 \: cm.$

The surface area of the toy is

$S = \pi \: r \: \sqrt{ {r}^{2} + {h}^{2} } + 2\pi \: {r}^{2}$

Substituting the values :-

$S = \frac{22}{7} \times 7 \sqrt{ {7}^{2} + {21}^{2} } + 2 \: \times \: \frac{22}{7} \: \times \: {7}^{2} \: \: \: \: \: \: ( \: taking \: \pi \: as \: \frac{22}{7 \: } \: )$

After simplifying we get

$794.99 \: {cm}^{2}$