Question

A toy is in the form of cone mounted on a hemisphere of diameter 7 cm .the total height is 15.5cm .find the volume and total surface area of toy

Answer 1

$\textsf {Question}$ :

A toy is in the form of a cone mounted on a hemisphere of diameter 7 cm. The total height is 15.5 cm. Find the volume and total surface area of the toy.

$\textsf {Solution}$ :

The diameter of the cone as well as the hemisphere is 7 cm. Diameter is double the Radius. Thus, the Radius of the hemisphere and the cone will be 7/2 cm or 3.5 cm.

The total height of the toy is 15.5 cm. The hemispherical Radius is 3.5 cm. So, the height of the cone will be:

$\mathsf{ \: h \: of \: the \: toy-r \: of \: the \: hemisphere} \\ \textsf{15.5 - 3.5 = 12 cm}$
So, we get the height of the cone as 12 cm.

Firstly, we are asked to find the volume of the toy.

$\mathsf{V(toy)=V(cone) + V(hemisphere)} \\ \\ \mathsf{ = \frac{1}{3} \pi \: {r}^{2}h + \frac{2}{3} \pi \: {r}^{3} } \\ \\ \mathsf{ = \frac{1}{3} \pi \: {r}^{2}(h + 2r) } \\ \\ \mathsf{ = \frac{1}{3} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} (12 + 2 \times \frac{7}{2} ) } \\ \\ \mathsf{ = \frac{1}{3} \times \frac{77}{2}(12 + 7) } \\ \\ \mathsf{ = \frac{1}{3} \times \frac{77}{2} \times 19} \\ \\ \mathsf{ = 975.34 \: {cm}^{3} }$

So, we get the volume as 975.34 cubic cms.

Secondly, we are asked to find the surface area of the given toy.

$\mathsf{SA(toy) = CSA(cone) + CSA(hemisphere)} \\ \\ \mathsf{ = \pi\:r\:l + 2\pi \: {r}^{2} } \\ \\ \mathsf{ = \pi \: r(l + 2r)}$

But here, we find a new variable l. This 'l' is the slant height of the cone. The measure of l is given by the formula:

$\mathsf{l = \sqrt{ {h}^{2} + {r}^{2} }} \\ \\ \mathsf{ = \sqrt{ {12}^{2} + {3.5}^{2} } } \\ \\ \mathsf{ = \sqrt{144 + 12.25} } \\ \\ \mathsf{ = \sqrt{156.25} } \\ \\ \mathsf{ = 12.5 \: cm}$

By this formula, we get the slant height of the cone as 12.5 cm.

Putting the value of l in the area equation,

$\mathsf{ SA(toy)= \pi \: r(l + 2r)} \\ \\ \mathsf{ = \frac{22}{7 } \times \frac{7}{2} \times \frac{7}{2} (12.5 + 2 \times \frac{7}{2})} \\ \\ \mathsf{ = \frac{77}{2} (12.5 + 7)} \\ \\ \mathsf{ = \frac{77}{2} \times 19.5 } \\ \\ \mathsf{ = 750.75 \: {cm}^{2} }$

By this formula, we get the surface area of the toy being 750.75 sq. Cms.

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