Question

$\huge{\underline{\underline{\bf Question}}}$
A log of wood in the shape of a cylinder of radius 1 ft and height 8 ft is cut all around to form the biggest cuboid. What volume of cylinder is removed ?

No absurd or useless answers ❌

Step-by-step explanation ✅

Good luck !!

​

Answer 1

$\large\underline{\bold{Solution-}}$

Given that

• Radius of cylinder, r = 1 ft.

• Height of cylinder, h = 8 ft.

So,

Volume of cylinder is

$\rm :\longmapsto\:V_1 = \pi \: {r}^{2} h$

$\rm :\longmapsto\:V_1 = \pi \: \times {(1)}^{2} \times 8$

$\bf\implies \:V_1 = 8\pi \: {ft.}^{3} - - - (1)$

Now, we have have to form a largest cuboid,

So,

$\bf\implies \:Height_{(cuboid)} = Height_{(cylinder)} = 8 \: ft$

Now,

since cuboid has a rectangular base.

And

We know,

The area of rectangle is maximum when it is a square.

So, it implies the base of cuboid is square.

Let side of square be 'x' ft.

We know,

$\: \: \: \: \: \: \boxed{ \bf \: Diagonal_{(square)} = \sqrt{2} \times side}$

Now,

$\rm :\longmapsto\:Diagonal_{(square)} = Diameter_{(cylinder)}$

$\rm :\longmapsto\:Diagonal_{(square)} = 2 \: ft.$

$\rm :\longmapsto\: \sqrt{2} \times side = 2$

$\rm :\longmapsto\: \sqrt{2} \times x = 2$

$\bf\implies \:x = \sqrt{2} \: ft.$

Thus,

Dimensions of Cuboid are

$\rm :\longmapsto\:\sf \: Length, x = \sqrt{2} \: ft.$

$\rm :\longmapsto\: \sf \: Breadth, x = \sqrt{2} \: ft.$

$\rm :\longmapsto\: \sf \: Height, H =8 \: ft.$

Therefore,

Volume of Cuboid is

$\rm :\longmapsto\:V_2 = length \times breadth \times height$

$\rm :\longmapsto\:V_2 = \sqrt{2} \times \sqrt{2} \times 8$

$\bf :\longmapsto\:V_2 = 16 \: {ft.}^{3}$

Hence,

$\bf :\longmapsto\:Volume_{(removed \: cylinder)} = V_1 - V_2$

$\rm :\longmapsto\:Volume_{(removed \: cylinder)} = 8\pi \: - 16$

$\rm :\longmapsto\:Volume_{(removed \: cylinder)} = 8(\pi \: - 2)$

$\rm :\longmapsto\:Volume_{(removed \: cylinder)} = 8 \times \bigg(\dfrac{22}{7} - 2 \bigg)$

$\rm :\longmapsto\:Volume_{(removed \: cylinder)} = 8 \times \bigg( \dfrac{22 - 14}{7} \bigg)$

$\rm :\longmapsto\:Volume_{(removed \: cylinder)} = 8 \times \dfrac{8}{7}$

$\bf :\longmapsto\:Volume_{(removed \: cylinder)} = \dfrac{64}{7} \: {ft.}^{3}$

More information :-

Perimeter of rectangle = 2(length× breadth)

Diagonal of rectangle = √(length ²+breadth ²)

Area of square = side²

Perimeter of square = 4× side

Volume of cylinder = πr²h

T.S.A of cylinder = 2πrh + 2πr²

Volume of cone = ⅓ πr²h

C.S.A of cone = πrl

T.S.A of cone = πrl + πr²

Volume of cuboid = l × b × h

C.S.A of cuboid = 2(l + b)h

T.S.A of cuboid = 2(lb + bh + lh)

C.S.A of cube = 4a²

T.S.A of cube = 6a²

Volume of cube = a³

Volume of sphere = 4/3πr³

Surface area of sphere = 4πr²

Volume of hemisphere = ⅔ πr³

C.S.A of hemisphere = 2πr²

T.S.A of hemisphere = 3πr²

Answer 2

Here is Your Answer ⤴

... Have A Nice Day Ahead

Sorry sis I can't type answer coz idk why Brainly is not accept my Typing font