Volume And Surface Area - Study Mode

[#141] The sum of perimeters of the six faces of a cuboid is 72 cm and the total surface area of the cuboid is 16 cm 2 . Find the longest possible length that can be kept inside the cuboid :
Correct Answer

(C) 8.05 cm

Explanation

Solution: Sum of perimeters of the six faces : $$eqalign{
& = 2left[ {2left( {l + b}
ight) + 2left( {b + h}
ight) + 2left( {l + h}
ight)}
ight] cr
& = 4left( {2l + 2b + 2h}
ight) cr
& = 8left( {l + b + h}
ight) cr} $$ Total surface area $$ = 2left( {lb + bh + lh}
ight)$$ $$eqalign{
& herefore 8left( {l + b + h}
ight) = 72 cr
& Rightarrow l + b + h = 9 cr
& 2left( {lb + bh + lh}
ight) = 16 cr
& Rightarrow lb + bh + lh = 8 cr} $$ Now, $${left( {l + b + h}
ight)^2} = {l^2} + {b^2} + {h^2} + 2$$ xa0 xa0 $$left( {lb + bh + lh}
ight)$$ $$eqalign{
& Rightarrow {left( 9
ight)^2} = {l^2} + {b^2} + {h^2} + 16 cr
& Rightarrow {l^2} + {b^2} + {h^2} = 81 - 16 cr
& Rightarrow {l^2} + {b^2} + {h^2} = 65 cr} $$ Required length : $$eqalign{
& = sqrt {{l^2} + {b^2} + {h^2}} cr
& = sqrt {65} cr
& = 8.05,cm cr} $$

[#142] The surface area of a cube is 150 cm 2 . Its volume is :
Correct Answer

(B) 125 $${ ext{c}}{{ ext{m}}^3}$$

Explanation

Solution: $$eqalign{
& 6{a^2} = 150 cr
& Rightarrow {a^2} = 25 cr
& Rightarrow a = 5 cr} $$ ∴ Volume : $${a^3} = {5^3} = 125,{ ext{ c}}{{ ext{m}}^3}$$

[#143] If three equal cubes are placed adjacently in a row, then the ratio of the total surface area of the new cuboid to the sum of the surface areas of the three cubes will be ?
Correct Answer

(D) 7 : 9

Explanation

Solution: Let the length of each edge of each cube be a Then, the cuboid formed by placing 3 cubes adjacently has the dimensions 3a , a and a Surface area of the cuboid : $$eqalign{
& = 2left[ {3a imes a + a imes a + 3a imes a}
ight] cr
& = 2left[ {3{a^2} + {a^2} + 3{a^2}}
ight] cr
& = 14{a^2} cr} $$ Sum of surface area of 3 cubes : $$eqalign{
& = left( {3 imes 6{a^2}}
ight) cr
& = 18{a^2} cr} $$ ∴ Required ratio : $$eqalign{
& = 14{a^2}:18{a^2} cr
& = 7:9 cr} $$

[#144] The curved surface area of a cylindrical pillar is 264 m 2 and its volume is 924 m 3 . Find the ratio of its diameter to its height.
Correct Answer

(B) 7 : 3

Explanation

Solution: $$eqalign{
& frac{{pi {r^2}h}}{{2pi rh}} = frac{{924}}{{264}} cr
& Rightarrow r = left( {frac{{924}}{{264}} imes 2}
ight) cr
& Rightarrow r = 7,m cr} $$ And, $$eqalign{
& herefore 2pi rh = 264 cr
& Rightarrow h = left( {264 imes frac{7}{{22}} imes frac{1}{2} imes frac{1}{7}}
ight) cr
& Rightarrow h = 6,m cr} $$ ∴ Required ratio : $$ = frac{{2r}}{h} = frac{{14}}{6} = 7:3$$

[#145] It is required to fix a pipe such that water flowing through it at a speed of 7 metres per minute fills a tank of capacity 440 cubic metres in 10 minutes. The inner radius of the pipe should be :
Correct Answer

(A) $$sqrt 2 ,m$$

Explanation

Solution: Let the inner radius of the pipe be r metres Then, Volume of water flowing through the pipe in 10 minutes : $$eqalign{
& = left[ {left( {frac{{22}}{7} imes {r^2} imes 7}
ight) imes 10}
ight]{m^3} cr
& = left( {220{r^2}}
ight){m^3} cr
& herefore 220{r^2} = 440 cr
& Rightarrow {r^2} = 2 cr
& Rightarrow r = sqrt 2 , m cr } $$