8级RD沙尔玛–第21章，测定II(长方体和立方体的体积和表面积)–练习21.3 |套装1

问题1.求长方体的表面积

(i)长度= 10厘米，宽度= 12厘米，高度= 14厘米
(ii)长度= 6 dm，宽度= 8 dm，高度= 10 dm
(iii)长度= 2 m，宽度= 4 m，高度= 5 m
(iv)长度= 3.2 m，宽度= 30 dm，高度= 250 cm。

解决方案 ：

i) length = 10 cm, breadth = 12 cm, height = 14 cm

Given, the Length of cuboid = 10 cm

The Breadth of cuboid = 12 cm

The Height of cuboid = 14 cm

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(10 × 12 + 12 × 14 + 14 × 10) = 856 cm²

Hence, the surface area of given cuboid is 856 cm²

ii) length = 6 dm, breadth = 8 dm, height = 10 dm

Given, the Length of cuboid = 6 dm

The Breadth of cuboid = 8 dm

The Height of cuboid = 10 dm

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(6 × 8 + 8 × 10 + 10 × 6) = 376 dm²

Hence, the surface area of given cuboid is 376 dm²

iii) length = 2 m, breadth = 4 m, height = 5 m

Given, the Length of cuboid = 2 m

The Breadth of cuboid = 4 m

The Height of cuboid = 5 m

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(2 × 4 + 4 × 5 + 5 × 2) = 76 m²

Hence, the surface area of given cuboid is 76 m²

iv) length = 3.2 m, breadth = 30 dm, height = 250 cm

Given, the Length of cuboid = 3.2 m = 32 dm

The Breadth of cuboid = 30 dm 

The Height of cuboid = 250 dm = 25 dm

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(32 × 30 + 30 × 25 + 25 × 32) = 5020 dm²

Hence, the surface area of given cuboid is 5020 dm²

问题2.求一个边为

(i)1.2米
(ii)27厘米
(iii)3厘米
(iv)6米
(v)2.1百万

解决方案：

i) 1.2 m

Given, the Edge of cube = 1.2 m

And the Surface area of cube = 6 × edge²

= 6 × (1.2)² = 8.64 m²

Hence, the surface area of a given cube is 8.64 m²

ii) 27 cm

Given, the Edge of cube = 27 cm

And the Surface area of cube = 6 × edge²

= 6 × (27)² = 4374 cm²

Hence, the surface area of a given cube is 4374 cm²

iii) 3 cm

Given, the Edge of cube = 3 cm

And the Surface area of cube = 6 × edge²

= 6 × (3)² = 54 cm²

Hence, the surface area of a given cube is 54 cm²

iv) 6 m

Given, the Edge of cube = 6 m

And the Surface area of cube = 6 × edge²

= 6 × (6)² = 216 m²

Hence, the surface area of a given cube is 216 m²
 

v) 2.1 m

Given, the Edge of cube = 2.1 m

And the Surface area of cube = 6 × edge²

= 6 × (2.1)² = 26.46 m²

Hence, the surface area of a given cube is 26.46 m²

问题3.一个长方体的盒子是5厘米x 5厘米x 4厘米。找到其表面积?

解决方案 ：

Given, the Length of cuboidal box = 5 cm

The Breadth of cuboidal box = 5 cm

The Height of the cuboidal box = 4 cm

And, Surface area of cuboidal box = 2(l × b + b × h + h × l)

= 2(5 × 5 + 5 × 4 + 4 × 5) = 130 cm²

Hence, the surface area of the given cuboidal box is 130 cm²

问题4.求体积为

(i)343 m ³
(ii)216立方米³

解决方案：

i) Given, the volume of cube = 343 m³

It means (side)³ = 343 

So, side = 7 m

And, the Surface area of cube = 6 × side²

= 6 × (7)² = 294 m²

Hence, the surface area of a given cube is 294 m²

ii) Given, the volume of cube = 216 dm³

It means (side)³ = 216 

So, side = 6 dm

And, the Surface area of cube = 6 × side²

= 6 × (6)² = 216 dm²

Hence, the surface area of a given cube is 216 dm²

问题5.求表面积为

(i)96厘米²
(ii)150 m ²

解决方案：

i) Given, the surface area of cube = 96 cm²

It means 6 × side² = 96

So, side = 4 cm 

And, the Volume of cube = (side)³

= (4)³ = 64 cm³

Hence, the volume of the given cube is 64 cm³

ii) Given, the surface area of cube = 150 m²

It means 6 × side² = 150

So, side = 5 m 

And, the Volume of cube = (side)³

= (5)³ = 125 m³

Hence, the volume of the given cube is 125 m³

问题6.长方体的尺寸比例为5：3：1，总表面积为414 m ² 。找到尺寸了吗?

解决方案：

Given, the dimensions of a cuboid are in the ratio 5 : 3 : 1

The total surface area is 414 m²

So, the Length of cuboid = 5a

The Breadth of cuboid = 3a

The Height of cuboid = 1a

And, Surface area of cuboid = 2(l × b + b × h + h × l)

414 = 2(5a × 3a + 3a × a + a × 5a) 

414 = 46a²

So, a = 3 m

We can conclude, Length = 5a = 15 m

The Breadth = 3a = 9 m

The Height = a = 3 m

Hence, the dimensions of a cuboid are 15 m, 9 m, and 3 m.

问题7.找到一个长度为25 cm，0.5 m和高度为15 cm的密闭盒所需的纸板面积吗?

解决方案：

Given, the Length of Cuboidal cardboard = 25 cm

The Breadth of Cuboidal cardboard = 0.5 m = 50 cm

The Height of Cuboidal cardboard = 15 cm

Area of cardboard needed = Area of Cuboid

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(25 × 50 + 50 × 15 + 15 × 25) = 4750 cm²

Hence, the area of cardboard required is 4750 cm²

问题8.找到一个形状为立方体的木箱的表面积，并且箱的边缘是否为12厘米?

解决方案 ：

Given, the Edge of cube = 12 cm

The surface area of a wooden box = Surface area of Cube

And, the Surface area of cube = 6 × edge²

= 6 × (12)² = 864 cm²

Hence, the surface area of a wooden box is 864 cm²

问题9.油罐的尺寸为26厘米×26厘米×45厘米。找到制作20罐这样的罐子所需的锡板面积。如果1平方米的锡纸成本为Rs。 10，找到用于这20罐锡的锡纸的成本吗?

解决方案 ：

Given, the Length of oil tin = 26 cm 

The Breadth of oil tin = 26 cm

The Height of oil tin = 45 cm

Area of tin sheet required to make 1 oil tin = Surface area of Cuboid

And, Surface area of cuboid = 2(l × b + b × h + h × l)

= 2(26 × 26 + 26 × 45 + 45 × 26) = 6032 cm²

And, Area of tin sheet required to make 20 oil tins = 20 × 6032 = 120640 cm²

= 12.064 m²

Since, 1 m² Tin cost = Rs 10 

So, 12.064 m² Tin cost = Rs 10 × 12.064

= Rs 120.64

Hence, the area of tin required to make 20 oil tin is 12.064 m² and the cost of making 20 oil tin is Rs 120.64