8类RD Sharma解决方案-第21章，确定II(长方体和立方体的体积和表面积)-练习21.4 |套装2

问题11：一个字段长150m，宽100m。将一块长50m，宽30m的地块(在田间外)挖到8m的深度，从该地块中取出的土壤均匀地分布在田野中。场水平提高了多少?

解决方案：

Given, the length of field = 150 m

The width of the field = 100 m

So, the area of field = Length × breadth

= 150 × 100 = 15000 m²

Also, the length of the plot = 50 m

The width of the plot = 30 m

The depth of the plot = 8 m

We can now find the volume of the plot,

The volume of plot = l × b × h 

= 50 × 30 × 8 = 12000 m³

Let h be the height of earth spread on the field

We know, volume = l × b × h 

So, h = (volume)/l × b

h = 12000/(150 × 100)

So, h = 0.8 m = 80 cm

Hence, the level of field raised by 80 cm

问题12.将两个立方体(每个体积为512 cm ³⁾首尾相连。找到所得长方体的表面积。

解决方案：

Given, the volume of each cube is 512 cm³

Let a be the edge of each cube

As we know, the Volume of cube = (edge)³

So, 512 = a³

a = 8 cm

When these two cubes are joined end to end a cuboid is formed,

So, the length of cuboid = 8 + 8 = 16 cm

The breadth of cuboid = 8 cm

The height of cuboid = 8 cm

So, the surface area of the resulting cuboid will be 2 (l × b + b × h + h × l)

= 2 (16 × 8 + 8 × 8 + 8 × 16) = 640 cm²

Hence, the surface area of the resulting cuboid is 640 cm²

问题13.将三个分别长为3 cm，4 cm和5 cm的立方体熔化，以形成一个新的立方体。找到形成的新立方体的表面积。

解决方案：

Given, the edge of three cubes are 3 cm, 4 cm, and 5 cm respectively.

As we know, the Volume of cube = (edge)³

The volume of the first cube = 3³ = 27 cm³

The volume of the second cube = 4³ = 64 cm³

The volume of the third cube = 5³ = 125 cm³

Now, the sum of volumes of given three cubes = 27 + 64 + 125 cm³

= 216 cm³

When these three cubes are melted, a new cube is formed,

Let a be the edge of the new cube

As we know, the Volume of cube = (edge)³

So, 216 = a³

a = 6 cm

So, the surface area of the new cube so formed = 6 x (edge)²

= 6 × 6² = 216 cm²

Hence, the surface area of the new cube formed is 216 cm²

问题14.以每平方米1.35卢比的价格准备12m长的房间墙壁的成本为340.20卢比，以每平方米85帕的价格铺垫地板的成本为91.80卢比。找到房间的高度

解决方案：

Given, the length of room is 12 m

Let b be the breadth of room

Let h be the breadth of room

As we know, the area of floor = l × b

= 12b m²

Given that cost of matting floor at 85 paise per square metre is Rs 91.80

So, we can write 12b × 0.85 = 91.80

b = 9 m

The breadth of room is 9 m

We know area of 4 walls = 2(l × h + b × h)

= 2(12 × h + 9 × h) = 42h m²

Also, the cost of preparing the walls at the rate of Rs 1.35 per square metre is Rs 340.20

So, we can write 42h × 1.35 = 340.20

h = 6 m

Hence, the height of room is 6 m and breadth is 9 m

问题15.大厅的长度为18 m，宽度为12 m。地板和平屋顶的面积之和等于四壁的面积之和。找到墙的高度。

解决方案：

Given, the length of hall is 18 m

The width of hall is 12 m

Let h be the height of hall

We know, sum of the areas of the floor and the flat roof = (l × b + l × b)

= (18 × 12 + 18 × 12) = 432 m²

Also, the areas of the four walls = 2(l × h + b × h)

= 2(18 × h + 12 × h) = 60h m²

It is given that sum of the areas of the floor and the flat roof is equal to the sum of the areas of the four walls,

So, 60h = 432

h = 7.2 m

Hence, the height of hall is 7.2 m

问题16：熔化一个边缘为12 cm的金属立方体，并成型为三个较小的立方体。如果两个较小的多维数据集的边缘分别为6厘米和8厘米，请找到第三个较小的多维数据集的边缘。

解决方案：

Given, the edge of metal cube is 12 cm

The edge of two smaller cubes are 6 cm and 8 cm

Let a be the edge of third smaller cube

We know that volume of metal cube will be equal to sum of volume of three smaller cubes

So, 12³ = 6³ + 8³ + a³

a = 10 cm

Hence, the edge of third smaller cube is 10 cm

问题17.电影院大厅的尺寸为100 m，50 m和18 m。^{如果每个人需要150 m 3}的空气，那么多少人可以坐在大厅里?

解决方案：

Given, the length of hall = 100 m

The breadth of hall = 50 m

The height of hall = 18 m

So, the volume of cinema hall = l × b × h

= 100 × 50 × 18 = 90000 m³

It is given that each person require 150 m³ of air

The number of person that can sit in hall = Volume of hall/Volume of each person

= 90000/150 = 600

Hence, 600 can persons can sit in the hall

问题18：封闭的木箱的外部尺寸为48厘米，36厘米和30厘米。盒子是用1.5厘米厚的木头制成的。这个盒子可以放几块6厘米×3厘米×0.75厘米的砖?

解决方案：

Given, the length of box = 48 cm

The breadth of box = 36 cm

The height of box = 30 cm

The dimensions of bricks are 6 cm× 3 cm× 0.75 cm

So, the volume of 1 brick = l × b × h

= 6 × 3 × 0.75 = 13.5 cm³

Also, the thickness of wood is 1.5 cm

So, the internal dimensions of box can be given as (48 – 2 × 1.5) × (36 – 2 × 1.5) × (30 – 2 × 1.5)

= (45 × 33 × 27) cm

The internal volume of box = l × b × h

= 45 × 33 × 27 = 40095 cm³

So, the number of bricks that can be put inside the box = Internal volume of box/Volume of 1 brick

= 40095/13.5 = 2970 bricks

Hence, 2970 bricks can be put inside the given wooden box

问题19：一个矩形盒子的尺寸之比为2：3：4，以每米²卢比8和9.50卢比的比率用纸覆盖它的成本之间的差额为1248卢比。盒子的尺寸。

解决方案：

Given, the dimensions of a rectangular box are in the ratio of 2: 3: 4 

So, the length of box be 2a

The breadth will be 3a

And, the height be 4a

The area of sheet that is required to cover the box will be the total surface area of cuboid,

= 2 (l × b + b × h + h × l)

= 2 (2a × 3a + 3a × 4a + 4a × 2a) = 52a² m²

Also, the cost of covering it with sheet of paper at the rates of Rs 8 is 52a² × 8

= Rs 416a²

And, the cost of covering it with sheet of paper at the rates of Rs 9.50 is 52a² × 9.50

= Rs 494a²

We know that difference between the cost of covering it with sheet of paper at the rates of Rs 8 and Rs 9.50 per m² is Rs 1248

So, 494a² – 416a² = 1248

a = 4

So, the length is 2a = 8 m

The breadth = 3a = 15 m

The height = 4a = 16 m

Hence, the dimensions of box are 8 m × 12 m × 16 m