If two cylinders of equal volumes have their heights in the ratio 2 : 3, then the ratio if their radii is :
(A) $$\sqrt 6 :\sqrt 3 $$
(B) $$\sqrt 5 :\sqrt 3 $$
(C) $$2 : 3 $$
(D) $$\sqrt 3 :\sqrt 2 $$
Solution:
Let their heights be 2h and 3h and radii be r and R respectively Then, $$\eqalign{ & \pi {r^2}\left( {2h} \right) = \pi {R^2}\left( {3h} \right) \cr & \Rightarrow \frac{{{r^2}}}{{{R^2}}} = \frac{3}{2} \cr & \Rightarrow \frac{r}{R} = \frac{{\sqrt 3 }}{{\sqrt 2 }}\,i.e.,\sqrt 3 :\sqrt 2 \cr} $$
23.
Two cans have the same height equal to 21 cm. One can is cylindrical, the diameter of whose base is 10 cm. The other can has square base of side 10 cm. What is the difference in their capacities ?
A sphere of maximum volume is cut out from a solid hemisphere of radius r. The ratio of the volume of the hemisphere to that of the cut out sphere is :
(A) 3 : 2
(B) 4 : 1
(C) 4 : 3
(D) 7 : 4
Solution:
Volume of hemisphere = $$\frac{2}{3}\pi {r^3}$$ Volume of biggest sphere : = Volume of sphere with diameter r $$\eqalign{ & = \frac{4}{3}\pi {\left( {\frac{r}{2}} \right)^3} \cr & = \frac{1}{6}\pi {r^3} \cr} $$ ∴ Required ratio : $$\eqalign{ & = \frac{{\frac{2}{3}\pi {r^3}}}{{\frac{1}{6}\pi {r^3}}} \cr & = \frac{4}{1}i.e.,4:1 \cr} $$
25.
A water tank is 30 m long, 20 m wide and 12 m deep. It is made of iron sheet which is 3 m wide. The tank is open at the top. If the cost of the iron sheet is Rs. 10 per metre, then the total cost of the iron sheet required to build the tank is :
(A) Rs. 6000
(B) Rs. 8000
(C) Rs. 9000
(D) Rs. 10000
Solution:
Since the tank is open at the top, we have : Area of sheet required = Surface area of the tank $$\eqalign{ & = lb + 2\left( {bh + lh} \right) \cr & = \left[ {30 \times 20 + 2\left( {20 \times 12 + 30 \times 12} \right)} \right]{{\text{m}}^2} \cr & = \left( {600 + 1200} \right){{\text{m}}^2} \cr & = 1800{\text{ }}{{\text{m}}^2} \cr} $$ Length of sheet required : $$\eqalign{ & = \left( {\frac{{{\text{Area}}}}{{{\text{Width}}}}} \right) \cr & = \frac{{1800}}{3}m \cr & = 600\,m \cr} $$ ∴ Cost of the sheet = Rs. (600 × 10) = Rs. 6000
26.
V1, V2, V3 and V4 are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4 cm respectively. Some statements regarding these volumes are given below :
(i) V1 + V2 + 2V3 V4
(ii) V1 + 4V2 + V3 V4
(iii) 2(V1 + V3) + V2 = V4
Which of these statements area correct ?
A water tank is 30 m long, 20 m wide and 12 m deep. It is made of iron sheet which is 3 m wide. The tank is open at the top. If the cost of the iron sheet is Rs. 10 per metre, then the total cost of the iron sheet required to build the tank is :
(A) Rs. 6000
(B) Rs. 8000
(C) Rs. 9000
(D) Rs. 10000
Solution:
Since the tank is open at the top, we have : Area of sheet required = Surface area of the tank $$\eqalign{ & = lb + 2\left( {bh + lh} \right) \cr & = \left[ {30 \times 20 + 2\left( {20 \times 12 + 30 \times 12} \right)} \right]{{\text{m}}^2} \cr & = \left( {600 + 1200} \right){{\text{m}}^2} \cr & = 1800{\text{ }}{{\text{m}}^2} \cr} $$ Length of sheet required : $$\eqalign{ & = \left( {\frac{{{\text{Area}}}}{{{\text{Width}}}}} \right) \cr & = \frac{{1800}}{3}m \cr & = 600\,m \cr} $$ ∴ Cost of the sheet = Rs. (600 × 10) = Rs. 6000
30.
The height of a closed cylinder of given volume and the minimum surface area is :