# Ncert Solutions For Class 10 Maths Ex 13.1

## Ncert Solutions For Class 10 Maths Chapter 13 Ex 13.1

1. Two cubes each having volume 64 cm3$cm^{3}$ are connected end to end. Calculate the surface area of the cuboid formed.

Ans- Volume of each cube=64cm3$64cm^{3}$

If “x” is the side of cube, x3=64$x^{3}=64$

x=643x=4cm.$\Rightarrow x=\sqrt[3]{64} \Rightarrow x=4cm.$

Thereforesurfaceareaofacube=6x2=6(4)2=6(16)=96cm2$\ Therefore surface area of a cube=6x^{2} =6(4)^{2} =6(16) =96cm^{2}$

thereforesurfaceareaofresultingcuboid$\ therefore\, surface\, area\, of\, resulting\, cuboid$

=2(surfaceareaofeachcube)(surfaceareaofoneside)$=2(surface area of each cube)-(surface area of one side)$

=(2×96)2x2=1922(4)2=19232=160cm2.$(2\times 96)-2x^{2} =192-2(4)^{2} =192-32 =160cm^{2}.$

1. A container which is in the shape of a hollow hemisphere is covered from the top by a hollow cylinder with diameter of 14 cm and the total height of the container is 13 cm. Calculate the inner surface area of the container.

Height of cylindrical portion = 13 – 7 = 6 cm

Curved surface are of cylindrical portion can be calculated as follows:

=2πrh$=2\pi rh$ =2×22÷7×7×6$=2\times 22\div 7\times 7\times 6$ =264cm2$=264cm^{2}$

Curved surface area of hemispherical portion can be calculated as follows:

=2πr2$=2\pi r^{2}$ =2×22÷7×7×7$=2\times 22\div 7\times 7\times 7$ =308cm2$=308cm^{2}$

Total surface are = 308 + 264 = 572 sq cm

3) A toy in a shape of a cone having radius 3.5 cm, mounted on a hemisphere with same radius. The total height of the toy is 15.5 cm. Calculate the total surface area of the toy.

Ans.-Radius of cone = 3.5 cm, height of cone = 15.5 – 3.5 = 12 cm

Slant height of cone can be calculated as follows:

a=h2+r2$a=\sqrt{h^{2}+r^{2}}$ a=122+3.52$a=\sqrt{12^{2}+3.5^{2}}$ a=144+12.5$a=\sqrt{144+12.5}$ a=156.25=12.5cm$a=\sqrt{156.25}=12.5cm$

Curved surface area of cone can be calculated as follows:

=πra$=\pi ra$ =22÷7×3.5×12.5$=22\div 7\times 3.5\times 12.5$ 137.5cm2$137.5cm^{2}$

Curved surface area of hemispherical portion can be calculated as follows:

=2πr2$=2\pi r^{2}$ =2×22÷7×3.5×3.5$=2\times 22\div 7\times 3.5\times 3.5$ 77cm2$77cm^{2}$

Hence, total surface area = 137.5 + 77 = 214.5 sq cm.

1. A cubical block with side 7 cm is surmounted by a hemisphere. Identify the greatest diameter the hemisphere which can be formed? Determine the surface area of the solid.

Ans.-The greatest diameter = side of the cube = 7 cm

Surface Area of Solid = Surface Area of Cube – Surface Area of Base of Hemisphere + Curved Surface Area of hemisphere

Surface Area of Cube = 6 x Side2$^{2}$

= 6 x 7 x 7 = 294 sq cm

Surface Area of Base of Hemisphere

=πr2$=\pi r^{2}$

22÷7×3.52$22\div 7\times 3.5^{2}$=38.5cm2$=38.5cm^{2}$

Curved Surface Area of Hemisphere = 2 x 38.5 = 77 sq cm

Total Surface Area = 294 – 38.5 + 77 = 332.5 sq cm

1. A hemispherical dint is removed by cutting out from one side of a cubical wooden block in such a way that the diameter ‘d’ of the hemisphere becomes equal to the edge of the cube. Calculate the surface area of the remaining solid.

Ans.- This question can be solved like previous question. Here the curved surface of the hemisphere is a dint, unlike a projection in the previous question-:

Total Surface Area-:

6a2π(a÷2)2π+2π(a÷2)2$6a^{2}-\pi \left ( a\div 2 \right )^{2}\pi+2\pi \left ( a\div 2 \right )^{2}$ =6a2+π(a2)2$=6a^{2}+\pi\left ( \frac{a}{2} \right )^{2}$ =14a2×(π+24)$=\frac{1}{4}a^{2}\times \left ( \pi +24 \right )$

1. A medicine capsule having the shape of a cylinder consisting two hemispheres stuck to each of its ends. The length of the total capsule is 14 mm with diameter of 5 mm. Calculate its surface area.Ans.- Height of Cylinder = 14 – 5 = 9 cm, radius = 2.5 cm

Curved Surface Area of Cylinder
2πrh$2\pi rh$

=2π×2.5×9$=2\pi\times 2.5\times 9$
=45πcm2$=45\pi cm^{2}$
Curved Surface Area of two Hemispheres

=4πr2$=4\pi r^{2}$=4π×2.52$=4\pi\times2.5^{2}$

=25πcm2$=25\pi cm^{2}$

Total Surface Area
=45π+25π$=45\pi +25\pi$
=70π=220cm2$=70\pi =220cm^{2}$

1. A tent having the shape of a cylinder is covered by a conical top. The height of the cylindrical part is 2.1 m with a diameter of 4 m having a slant height of 2.8 m, calculate the area of the fabric used for making the tent. Also, calculate the cost of the fabric of the tent at the rate of Rs 500 per m2. (Note that the bottom of the tent can’t be covered with fabric).

Ans.-  Radius of cylinder = 2 m, height = 2.1 m and slant height of conical top = 2.8 m

Curved Surface Area of cylindrical portion

=2πrh$=2\pi rh$ =2π×2×2.5$=2\pi \times 2\times 2.5$ =8.4πm2$=8.4\pi m^{2}$

Curved Surface Area of conical portion

=πrl$=\pi rl$ =π×2×2.8=5.6πm2$=\pi\times 2\times 2.8=5.6\pi m^{2}$

Total CSA

=8.4π+5.6π$=8.4\pi +5.6\pi$ =14×227=44m2$=14\times\frac{22}{7}=44m^{2}$

Cost of fabric = Rate x Surface Area

= 500 x 44 = Rs. 22000

1. From a solid cylinder having height and diameter of 2.4 cm and 1.4 cm respectively, a conical cavity with same height and same diameter is hollowed out. Calculate the total surface area of the remaining solid to the closest cm2$cm^{2}$.

Ans.- Radius = 0.7 cm and height = 2.4 cm

Total Surface Area of Structure = Curved Surface Area of Cylinder + Area of top of cylinder + Curved Surface Area of Cone

Curved Surface Area of Cylinder

= 2πrh$2\pi rh$

=2π×0.7×2.4=3.36πcm2$=2\pi\times 0.7\times 2.4=3.36\pi cm^{2}$

Area of top

=πr2$=\pi r^{2}$ =π×0.72$=\pi \times 0.7^{2}$ =0.49πcm2$=0.49\pi cm^{2}$

Slant height of cone can be calculated as follows:

=l=r2+h2$=l=\sqrt{r^{2}+h^{2}}$ =2.42+0.72$=\sqrt{2.4^{2}+0.7^{2}}$ =5.76+0.49$=\sqrt{5.76+0.49}$ =6.25=2.5cm$=\sqrt{6.25}=2.5cm$

Curved Surface Area of Cone

=πrl$=\pi rl$ =π×0.7×2.5$=\pi\times 0.7\times 2.5$ =1.75πcm2$=1.75\pi cm^{2}$

Hence, remaining surface area of structure

=3.36π+0.49π+1.75π$=3.36\pi +0.49\pi +1.75\pi$ =5.6π=17.6cm2$=5.6\pi =17.6cm^{2}$ =18cm2(approx)$=18cm^{2} \left ( approx \right )$

1. A wooden article has been made by removing out a hemisphere from each end of a solid cylinder, as given in figure. If the height and radius of base of the cylinder is 10 cm and 3.5 cm respectively, then calculate the complete surface area of the article

Ans.- Radius = 3.5 cm, height = 10 cm

Total Surface Area of Structure = CSA of Cylinder + CSA of two hemispheres

Curved Surface Area of Cylinder

=2πrh$=2\pi rh$ =2π×3.5×10$=2\pi \times 3.5\times 10$ =70πcm2$=70\pi cm^{2}$

Surface Area of Sphere

=4πr2$=4\pi r^{2}$ =4π×3.52$=4\pi\times 3.5^{2}$ =49π$=49\pi$

Total Surface Area

=70π+49π=119π$=70\pi +49\pi=119\pi$ =119×227=374cm2$=119\times \frac{22}{7}=374cm^{2}$