Question

The ratio of height and slant height of a square based pyramid
is 4 : 5 and its total surface area is 96 cm. Find the volume of
the pyramid.​

Answer 1

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The ratio of height and slant height of a square based pyramid is 4 : 5 and its total surface area is 96 cm. Find the volume of the pyramid.

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given ratio of height and slant height a square

pyramid = 4:5

let the height be 4x and slant height be 5x

given TSA of square pyramid = 96cm

TSA = LSA + area of base

LSA = perimeter of base x slant height/2 -----------(1)

here radius is side of base / 2

we have a relation,

$\bf1^2=r^2+h^2 \\ \\ \bf \dashrightarrow(5x)^2 = ( \frac{s}{2} )^2 + (4x)^2 \\ \\ \bf \dashrightarrow \: V[25x^2 - 16x^2 ] = \frac{s}{2} \\ \\ \bf \dashrightarrow√9x^2 = \frac{s}{2} \\ \\ \bf \dashrightarrow \red{ s = 6x \: cm}$

from eq(1),

$\bf96 = 4s \times \frac{5x}{2} + s^2 \\ \\ \bf \dashrightarrow96= 60x^2 + 36x^2 \\ \\\bf \dashrightarrow 96 = 96x^2 \\ \\ \bf \dashrightarrow \red{ x = 1 \: cm}$

so height = 4(1) = 4cm

slant height = 5(1) 5cm and side = 6cm

$\bf \: volume \: of \: pyramid = area \: of \: base \times \frac{height}{3} \\ \\ \bf \dashrightarrow s^2 \times \frac{4}{3} = 36 \times \frac{4}{3} \\ \\ \bf \dashrightarrow \red{48 \: cm^3}$

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