Volume And Surface Area - Study Mode

[#151] The external and internal diameters of a hemispherical bowl are 10 cm and 8 cm respectively. What is the total surface area of the bowl ?
Correct Answer

(B) 286 cm 2

Explanation

Solution: Internal radius, r = 4 cm External radius, R = 5 cm Total surface area : $$eqalign{
& = 2pi {R^2} + 2pi {r^2} + pi left( {{R^2} - {r^2}}
ight) cr
& = 3pi {R^2} + pi {r^2} cr
& = left[ {pi left( {3 imes 25 + 16}
ight)}
ight]{ ext{ c}}{{ ext{m}}^2} cr
& = left( {frac{{22}}{7} imes 91}
ight){ ext{c}}{{ ext{m}}^2} cr
& = 286{ ext{ c}}{{ ext{m}}^2} cr} $$

[#152] A pyramid has an equilateral triangle as its base of which each side is 1 m. Its slant edge is 3 m. The whole surface are of the pyramid is equal to :
Correct Answer

(C) $$frac{{sqrt 3 + 3sqrt {35} }}{4}sq.m$$

Explanation

Solution: Area of base : $$eqalign{
& = left( {frac{{sqrt 3 }}{4} imes {1^2}}
ight){m^2} cr
& = frac{{sqrt 3 }}{4}{m^2} cr} $$ Clearly, the pyramid has 3 triangular faces each with sides 3m, 3m and 1 m So, area of each lateral face : $$eqalign{
& = sqrt {frac{7}{2} imes left( {frac{7}{2} - 3}
ight)left( {frac{7}{2} - 3}
ight)left( {frac{7}{2} - 1}
ight)} {m^2} cr
& left[ {x08ecause s = frac{{3 + 3 + 1}}{2} = frac{7}{2}}
ight] cr
& = sqrt {frac{7}{2} imes frac{1}{2} imes frac{1}{2} imes frac{5}{2}} {m^2} cr
& = frac{{sqrt {35} }}{4}{m^2} cr} $$ ∴ Whole surface area of the pyramid : $$eqalign{
& = left( {frac{{sqrt 3 }}{4} + 3 imes frac{{sqrt {35} }}{4}}
ight){m^2} cr
& = frac{{sqrt 3 + 3sqrt {35} }}{4}{m^2} cr} $$

[#153] A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surface will be :
Correct Answer

(A) $$sqrt 2 :1$$

Explanation

Solution: Let, $$eqalign{
& OP = OQ = OR = r cr
& herefore OR = h = r cr} $$ ∴ Curved surface area of the hemisphere = $$2pi {r^2}$$ Curved surface area of a cone = $$pi rl$$ Where, $$eqalign{
& l = sqrt {{h^2} + {r^2}} cr
& ,,,,, = sqrt {{r^2} + {r^2}} cr
& ,,,,, = rsqrt 2 cr} $$ ∴ Required ratio : $$eqalign{
& = frac{{2pi {r^2}}}{{pi rl}} cr
& = frac{{2pi {r^2}}}{{pi r imes rsqrt 2 }} cr
& = frac{2}{{sqrt 2 }} cr
& = frac{{2 imes sqrt 2 }}{{sqrt 2 imes sqrt 2 }} cr
& = frac{{2sqrt 2 }}{2} cr
& = frac{{sqrt 2 }}{1},Or,sqrt 2 :1 cr} $$

[#154] A swimming pool 9 m wide and 12 m long and 1 m deep on the shallow side and 4 m deep on the deeper side. Its volume is :
Correct Answer

(B) 270 m 3

Explanation

Solution: Given length of width of swimming pool is 9 m and 12 m respectively Volume of swimming pool : $$eqalign{
& = 9 imes 12 imes left( {frac{{1 + 4}}{2}}
ight) cr
& = 9 imes 12 imes frac{5}{2} cr
& = 270, ext{cu. metre} cr} $$

[#155] A closed aquarium of dimensions 30 cm × 25 cm × 20 cm is made up entirely of glass plates held together with tapes. The total length of tape required to hold the plates together (ignore the overlapping tapes) is :
Correct Answer

(D) 300 cm

Explanation

Solution: Total length of tape required : = Sum of lengths of edges = (30 × 4 + 25 × 4 + 20 × 4) cm = 300 cm