Volume And Surface Area - Study Mode

[#106] The height of a closed cylinder of given volume and the minimum surface area is :
Correct Answer

(A) Equal to its diameter

Explanation

Solution: $$eqalign{
& V = pi {r^2}h{ ext{ and }} cr
& S = 2pi rh + 2pi {r^2} cr
& ,,,,,,, = 2pi rleft( {h + r}
ight) cr
& { ext{Where, }}h = frac{V}{{pi {r^2}}} cr
& Rightarrow S = 2pi rleft( {frac{V}{{pi {r^2}}} + r}
ight) cr
& Rightarrow S = frac{{2V}}{r} + 2pi {r^2} cr
& Rightarrow frac{{dS}}{{dr}} = frac{{ - 2V}}{{{r^2}}} + 4pi r{ ext{ and}} cr
& frac{{{d^2}S}}{{d{r^2}}} = left( {frac{{4V}}{{{r^3}}} + 4pi }
ight){ ext{ > 0}} cr} $$ ∴ S is minimum when : $$eqalign{
& frac{{dS}}{{dr}} = 0 cr
& Rightarrow frac{{ - 2V}}{{{r^2}}} + 4pi r = 0 cr
& Rightarrow V = 2pi {r^3} cr
& Rightarrow pi {r^2}h = 2pi {r^3} cr
& Rightarrow h = 2r cr} $$

[#107] Water is poured into an empty cylindrical tank at a constant rate for 5 minutes. After the water has been poured into the tank. the depth of the water is 7 feet. The radius of the tank is 100 feet. Which of the following is the best approximation for the rate at which the water was poured into the tank ?
Correct Answer

(C) 700 cubic feet/sec

Explanation

Solution: Volume of water flown into the tank in 5 min : $$eqalign{
& = left( {frac{{22}}{7} imes 100 imes 100 imes 7}
ight){ ext{cu}}{ ext{.feet}} cr
& = 220000,{ ext{cu}}{ ext{.feet}} cr} $$ ∴ Rate of flow of water : $$eqalign{
& = left( {frac{{220000}}{{5 imes 60}}}
ight){ ext{cu}}{ ext{.feet/sec}} cr
& = 733.3 approx 700,{ ext{cu}}{ ext{.feet/sec}} cr} $$

[#108] The curved surface of a right circular cone of height 15 cm and base diameter 16 cm is :
Correct Answer

(D) 136π cm 2

Explanation

Solution: h = 15 cm, r = 8 cm So, $$eqalign{
& l = sqrt {{r^2} + {h^2}} cr
& ,,,,,, = sqrt {{8^2} + {{left( {15}
ight)}^2}} cr
& ,,,,,, = 17,cm cr} $$ ∴ Curved surface area : $$eqalign{
& = pi rl cr
& = left( {pi imes 8 imes 17}
ight){ ext{ c}}{{ ext{m}}^2} cr
& = 136pi { ext{ c}}{{ ext{m}}^2} cr} $$

[#109] A right circular cone and a right circular cylinder have equal base and equal height. If the radius of the base and the height are in the ratio 5 : 12, then the ratio of the total surface area of the cylinder to that of the cone is :
Correct Answer

(C) 17 : 9

Explanation

Solution: Let their radius and height be 5x and 12x respectively Slant height of the cone, $$l = sqrt {{{left( {5x}
ight)}^2} + {{left( {12x}
ight)}^2}} = 13x$$ $$eqalign{
& frac{{{ ext{Total surface area of cylinder}}}}{{{ ext{Total surface area of cone}}}} cr
& = frac{{2pi rleft( {h + r}
ight)}}{{pi rleft( {l + r}
ight)}} cr
& = frac{{2left( {h + r}
ight)}}{{left( {l + r}
ight)}} cr
& = frac{{2 imes left( {12x + 5x}
ight)}}{{left( {13x + 5x}
ight)}} cr
& = frac{{34x}}{{18x}} cr
& = frac{{17}}{9},Or,17:9 cr} $$

[#110] The curved surface area of a sphere is 5544 sq.cm. Its volume is :
Correct Answer

(C) 38808 cm 3

Explanation

Solution: $$eqalign{
& 4pi {r^2} = 5544 cr
& Rightarrow {r^2} = left( {5544 imes frac{1}{4} imes frac{7}{{22}}}
ight) cr
& Rightarrow {r^2} = 441 cr
& Rightarrow r = 21 cr} $$ ∴ Volume : $$eqalign{
& = left( {frac{4}{3} imes frac{{22}}{7} imes 21 imes 21 imes 21}
ight){ ext{ c}}{{ ext{m}}^3} cr
& = 38808{ ext{ c}}{{ ext{m}}^3} cr} $$