Question

the frustum is made up of metal sheet whose circular bases are covered . the upper and lower circular bases are 25 π cm ² & π cm² . if the height of a frustum is 3cm , then find the surface area of frustum and cost of metal sheet need if it cost Rs 500 per 100 cm ².​

Answer 1

Given :-

• Height of frustum, h = 3 cm

• Base Area of frustum of one end of radius, r = π cm²

• Base Area of frustum of other end of radius, R = 25 π cm² .

To Find :-

• Total Surface Area of Frustum

• Cost of metal sheet needed if it cost Rs 500 per 100 cm² .

Formula Used :-

$\rm :\longmapsto\:TSA_{(frustum)} = \pi\bigg( {R}^{2} + {r}^{2} + (R + r)l \bigg)$

$\rm :\longmapsto\:l = \sqrt{ {(R - r)}^{2} + {h}^{2} }$

where,

• R = radius of upper end of frustum

• r = radius of lower end of frustum

• h = height of frustum

• l = slant height is frustum

• TSA = Total Surface Area

Solution :-

Given that,

• Height of frustum, h = 3 cm

• Base Area of frustum of one end of radius, r = π cm²

• Base Area of frustum of other end of radius, R = 25 π cm² .

So,

$\rm :\longmapsto\:\pi {r}^{2} = \pi$

$\rm :\longmapsto\:{r}^{2} =1$

$\bf\implies \:r = 11 \: cm$

Also,

$\rm :\longmapsto\:\pi {R}^{2} = 25\pi$

$\rm :\longmapsto \: {R}^{2} = 25$

$\bf\implies \:R = 5 \: cm$

So,

Slant height of frustum is

$\rm :\longmapsto\:l = \sqrt{ {(R - r)}^{2} + {h}^{2} }$

$\rm :\longmapsto\:l = \sqrt{ {(5 - 1)}^{2} + {3}^{2} }$

$\rm :\longmapsto\:l = \sqrt{ {4}^{2} + {3}^{2} }$

$\rm :\longmapsto\:l = \sqrt{16 + 9 }$

$\rm :\longmapsto\:l = \sqrt{25}$

$\bf\implies \:l = 5 \: cm$

Hence,

$\rm :\longmapsto\:TSA_{(frustum)} = \pi\bigg( {R}^{2} + {r}^{2} + (R + r)l \bigg)$

$\rm  \:  =  \: \:\pi\bigg( {5}^{2} + {1}^{2} + (5 + 1) \times 5 \bigg)$

$\rm  \:  =  \: \:\pi\bigg(25 + 1 + 30\bigg)$

$\rm  \:  =  \: \:\dfrac{22}{7} \times 56$

$\rm  \:  =  \: \:22 \times 8$

$\rm  \:  =  \: \:176 \: {cm}^{2}$

$\bf\implies \:TSA_{(frustum)}=  \: \:176 \: {cm}^{2}$

Hence,

Cost of metal sheet used to make a frustum is

$\rm  \:  =  \: \:176 \times \dfrac{500}{100}$

$\rm  \:  =  \: \:176 \times 5$

$\rm  \:  =  \: \:880$

Therefore,

• Cost of metal sheet = Rs 880.

Additional Information :-

$\rm :\longmapsto\:Volume_{(frustum)} = \dfrac{\pi \: h}{3}( {R}^{2} + {r}^{2} + Rr)$

$\rm :\longmapsto\:CSA_{(frustum)} = \pi \: (R + r) \: l$

$\rm :\longmapsto\:l = \sqrt{ {(R - r)}^{2} + {h}^{2} }$

where,

• R = radius of upper end of frustum

• r = radius of lower end of frustum

• h = height of frustum

• l = slant height is frustum

• CSA = Curved Surface Area