问题1:求出长80厘米,宽40厘米,高20厘米的长方体的侧面表面积和总表面积。
解决方案:
Given, dimensions of cuboid are:
Length(l) = 80 cm
Breadth(b) = 40 cm
Height(h) = 20 cm
Formula for total surface area of cuboid:
TSA(Cuboid) = 2[lb + bh + hl]
Now, substituting the given values of in the formula,
= 2[(80)(40) + (40)(20) + (20)(80)]
= 2[3200 + 800 + 1600]
= 2[5600]
= 11200
Therefore, Total Surface Area = 11200 cm2
Now, Formula for Lateral Surface area of cuboid:
LSA(Cuboid) = 2[l + b] * h
= 2[80 + 40]20
= 2[120]20
= 40[120]
= 4800
Thus, Lateral Surface Area is 4800 cm2.
问题2:求出一个10厘米长的立方体的侧面表面积和总表面积。
解决方案:
Given,
Side of the Cube = 10 cm
Formula for Lateral Surface Area of Cube :
LSA(cube) = 4 side2
= 4(10 × 10)
= 400 cm2
Formula for Total Surface Area of Cube :
TSA(cube) = 6 side2
= 6(10 × 10)
= 6(100)
= 600 cm2
问题3:求出一个立方体的总表面积与横向表面积之比。
解决方案:
Formula for Lateral Surface Area of Cube :
LSA(cube) = 4 side2
Formula for Total Surface Area of Cube :
TSA(cube) = 6 side2
Therefore,
Ratio of TSA and LSA = (6 side2)/(4 side2) = 6/4 = 3/2 or 3:2
问题4:玛丽想装饰她的圣诞树。她想将树放置在一个覆盖有彩色纸的木块上,上面有圣诞老人的照片。她必须知道为此目的购买的确切纸张数量。如果该框的长度,宽度和高度分别为80厘米,40厘米和20厘米。她需要多少张40厘米长的方形纸?
解决方案:
Given the dimensions of the wooden block are:
Length(l) = 80 cm
Breadth(b) = 40 cm
Height(h) = 20 cm
Surface area of the wooden box = 2[lb + bh + hl]
= 2[80 × 40 + 40 × 20 + 20 × 80]
= 2[5600]
= 11200
Therefore, Surface Area of wooden box is 11200 cm2 .
Now, the area of each sheet of paper = 40 × 40 = 1600 cm2.
So, the total number of sheets required = (Surface area of the box)/(Area of one sheet of paper)
= 11200/1600
= 7
Therefore, Mary would require 7 sheets.
问题5:房间的长度,宽度和高度分别为5m,4m和3m。找出以7.50 m 2 Rs的速率粉刷房间墙壁和天花板的成本。
解决方案:
Total area to be whitewashed = lb + 2(l + b)h —-(1)
Given,
Length(l) = 5m
Breadth(b) = 4m
Height(h) = 3m
Total area to be whitewashed = (5 × 4) + 2(5 + 4)3
= 20 + 54
= 74
Total area to be whitewashed is 74 m2.
Now, cost of white washing1m2 is Rs. 7.50 (given)
Therefore, the cost of white washing 74 m2 = 74 × 7.50
= Rs. 555
问题6:三个相等的立方体连续相邻放置。求出新长方体的总表面积与三个立方体的总表面积之比。
解决方案:
Let the breadth of the cuboid = a
Then, length of the new cuboid = 3a and
Height of the new cuboid = a
Now,
Total surface Area of the new cuboid (TSA) = 2(lb + bh + hl)
= 2(3a × a + a × a + a × 3a)
= 14 a2
Again,
Total Surface area of three cubes = 3 × (6 side2)
= 3 × 6a2
= 18a2
Therefore, ratio of a total surface area of the new cuboid to that of the sum of the areas of three cubes = 14a2/18a2
= 7/9 or 7:9
问题7:将4厘米的立方体切成1厘米的立方体。计算小立方体的总表面积。
解决方案:
Given,
Edge of the cube = 4 cm
We know that,
Volume of the cube = Side3
= 43
= 4 × 4 × 4 = 64
Therefore, volume of the cube is 64cm3
Now,
Edge of the cube = 1 cm3
So, Total number of small cubes = 64cm3/1cm3 = 64
And, the total surface area of all the cubes = 64 × 6 × 1 = 384 cm2.
问题8:大厅的长度为18m,宽度为12m。地板和平屋顶的面积之和等于四壁的面积之和。找到大厅的高度。
解决方案:
Given, dimensions of the hall are:
Length(l) = 18 m
Breadth(b)= 12m
Let the height of the walls be h
From the given statement,
Area of floor and flat roof = sum of area of four walls —-(1)
On applying respective formulas,
Area of floor and flat roof = 2lb = 2 × 18 × 12 = 432 sq/ft —-(2)
Sum of area of four walls = (2 × 18h + 2 × 12h) —-(3)
On using equations (2) and (3) in (1), we get
432 =2 × 18h + 2 × 12h
18h + 12h = 216
or h = 7.2
Therefore, height of the hall is 7.2 m.
问题9:Hameed为自己的房屋建造了一个带盖的立方体水箱,彼此的边长1.5 m。他得到除底部外的水箱外表面,上面覆盖着25厘米长的正方形瓷砖。找出如果瓷砖成本为Rs,他将为瓷砖花费多少。每打360个。
解决方案:
Edge of the cubical tank = 1.5m or 150 cm
Surface area of the cubical tank(5 faces)= 5 x Area of one Face
= 5 × (150 × 150)cm2 —-(1)
Find area of each square tile:
Side of tile = 25 cm(Given)
Area of one tile = 25 × 25 cm2 —–(2)
Now,
Number of tiles required = (Surface Area of Tank)/(Area of each Tile)
= (5 × 150 × 150) / 25 × 25
= 180
Find cost of tiles:
Cost of 1 dozen tiles, i.e., cost of 12 tiles = Rs. 360
Therefore, cost of one tile = Rs.360/12 = Rs. 30
So, the cost of 180 tiles = 180 × 30 = Rs. 5400
问题10:立方体的每个边都增加了50%。找到立方体表面积增加的百分比。
解决方案:
Let ‘a’ be the edge of a cube.
Surface area of the cube having edge ‘a’ = 6a2 —-(1)
After increasing an edge by 50%, we get,
The new edge = a + 50a/100
= 3a/2
Surface area of the cube having edge ‘3a/ 2’ = 6 × (3a/2)2 = (27/2) a2 —-(2)
Subtracting equation(1) from (2) to find the increase in the Surface Area :
Increase in the Surface Area = (27/2) a2 – 6a2
= (15/2)a2
Now, Percentage increase in the surface area = ((15/2)a2/6a2) × 100
= 15/12 × 100
= 125%
= Therefore, percentage increase in the surface area of a cube is 125.