第22章测定III(右圆柱的表面积和体积)–练习22.2 |套装1
问题21:一口井的深度为20 m,直径为7 m。如此挖出的地球分布在22 m长,14 m宽的矩形图中。这样形成的平台的高度是多少?
解决方案:
Given that,
Depth of well = 20m
Diameter of well = 7 m
Radius of well = d/2 = 7/2 m
Dimension of rectangular field = 22m × 14m
As we know that Volume of earth dug out from well = πr2h
= 22/7 × 7/2 × 7/2 × 20 = 770 m3
When this earth will spread on rectangular field,
then the Height of platform formed on Rectangular Field = Volume of Earth dug out / Area of Field
= 770/ (22×14) = 2.5m
问题22:直径为14 m的一口井深挖了8 m。从中取出的泥土已均匀地分布在其周围,宽度为21 m,形成了路堤。找到路堤的高度。
解决方案:
Given that,
Diameter of well = 14 m,
Radius of well = d/2 = 14/2 = 7m,
Depth of well = 8 m.
We know that Volume of Earth dug out from well = πr2h
= 22/7 × 7 × 7 × 8 = 1232 m3.
This earth spread out on width of 21 m. (Given)
Area × h = 1232
π(R2 – r2) h = 1232
π(282 – 72) h = 1232
22/7 (735) h = 1232
h = 1232×7 / 22×735 = 8624/16170 = 0.533 m = 53.3 cm
Hence, the Height of embankment is 53.3 cm.
问题23.底部直径为56厘米的圆柱形容器装有足够的水,可以浸没尺寸为32厘米×22厘米×14厘米的矩形铁固体。当固体完全浸没时,求出水位的升高。
解决方案:
Given that,
Diameter of base of cylindrical vessel = 56 cm
Radius of base = d/2 = 56/2 = 28cm
Dimensions of rectangular Solid Vessel = 32cm × 22cm × 14cm
Volume of rectangular Solid Vessel = 32 × 22 × 14 = 9856 cm3
Let us assume that the rise of water level is ‘h’ cm.
Volume of cylindrical container = Volume of rectangular solid vessel (Given)
πr2h = 9856
22/7 × 28 × 28 × h = 9856
h = 9856×7 / 22×28×28 = 4cm
Hence, the Rise in Water Level is 4cm.
问题24.可以通过两种方法将30 cm×18 cm的矩形纸转换为直圆柱的曲面,即沿其长度滚动或沿其宽度滚动。求出由此形成的两个圆柱体的体积比。
解决方案:
Given that,
Dimensions of rectangular sheet = 30 cm × 18 cm
Case 1. When the paper is rolled along its length,
2 πr = 30
r = 30 / 2π cm
Height = 18 cm
We know that Volume of Cylinder V1 = πr2h
= π × (30/2π)2 × 18 cm3
Case 2. When paper is rolled along its breadth
2πr = 18
r = 18/2π cm
Height = 30 cm
We know that Volume of Cylinder V2 = πr2h
= π × (18/2π)2 × 30 cm3
Hence Volume of Cylinder V1 / Volume of Cylinder V2 = (π × (30/2π)2 × 18} / {π × (18/2π)2× 30)
= (π × (30/2π)2 × 18}×1/{π × (2π/18)2× 30)
= 302 × 18 / 182 × 30 = 5/3
Hence the ratio of two volumes is 5:3
问题25.允许将降雨落在18 m长,16.5 m宽的屋顶上,并将其存储在直径8 m的圆柱形水箱中。如果每天下雨10厘米,那么水箱中的水位上升是由于什么引起的?
解决方案:
鉴于,
Dimensions of roof = 18 m × 16.5 m,
Diameter of cylindrical tank = 8 m,
Radius of tank = d/2 = 8/2 = 4m,
It rains 10 cm a day
Let us assume that the rise in level of tank be ‘h’
Volume of tank = Volume of Roof
πr2h = lbh
22/7 × 4 × 4 × h = 18 × 16.5 × 0.1
h = (18 × 16.5 × 0.1 × 7) / 22×4×4
= 207.9/352 = 0.5906m = 59.06cm
Hence, Rise in water level is 59.06cm
问题26.一块易延展的金属是直径为1厘米,长度为5厘米的圆柱体的形式。将其拉出到直径为1 mm的电线中。这样形成的导线的长度是多少?
解决方案:
Given that,
Diameter of metallic cylinder = 1 cm
Radius of metallic cylinder = d/2 = 1/2 = 0.5 cm
Length of cylinder = 5 cm
Diameter of wire drawn from it = 1 mm = 0.1 cm
Radius of wire = 0.5mm = 0.05cm
Let us assume that the length of wire be ‘h’ cm
Length of wire drawn from metal = Volume of Metal / Volume of Wire (Given)
= πr2h / πr2
= (½)2 × 5 / (0.05)2 = 1.25/0.0025 = 500 cm or 5m
Hence, Length of the wire is 5m.
问题27.当1立方厘米的铜重8.4克时,找到直径为4毫米的13.2千克铜线的长度。
解决方案:
Given that,
Weight of copper wire = 13.2 kg = 13200 gm,
Diameter of wire = 4 mm,
Radius of wire = d/2 = 4/2 = 2mm = 0.2cm,
Let us assume the length of wire is ‘h’ cm
Weight of 1 cubic cm wire = 8.4 gm
As we know that Volume = Weight / Density
Volume × Density = Weight
πr2h × 8.4 = 13200
22/7 × 0.2 × 0.2 × h × 8.4 = 13200
h = 13200×7 / 22×0.2×0.2×8.4 = 12500 cm = 125m
Hence, the Length of 13.2kg of copper wire is 125 m.
问题28.将2.2立方分米的黄铜拉制成直径0.25厘米的圆柱线。找到电线的长度。
解决方案:
Given that,
Diameter of cylindrical wire = 0.25 cm,
Radius of wire = d/2 = 0.25/2 = 0.125cm,
Volume of brass wire = 2.2 dm3 = 2200 cm3.
Let us assume that the length of wire is ‘h’ cm
πr2h = 2200
22/7 × 0.125 × 0.125 × h = 2200
h = 2200×7 / 22×0.125×0.125 = 44800 cm = 448m
Hence, the Length of the wire is 448m.
问题29.长度为14厘米的圆柱管的内外表面之差为88平方厘米。如果管的体积为176立方厘米,请找到管的内半径和外半径。
解决方案:
Given that,
Length of Cylindrical tube = 14 cm
Let us assume that the outer radius of tube is = R cm
Let us assume that the inner radius of tube is = r cm
Difference between inside and outside surface of tube = 88 cm2
2π (R-r) h = 88 ——— (i)
Volume of Cylinder = 176 cm3 (Given)
π (R2 – r2) h = 176 ——— (ii)
Dividing equation (i) by equation (ii),
2π (R-r) h / π (R2 – r2) h = 88/176
2 / R + r = 1/2
R + r = 4 ——— (iii)
From equation (ii) we get,
π (R2 – r2) h = 176
π (R+r) (R-r) h = 176
22/7 × 4 × (R-r) × 14 = 176
R-r = 176×7 / 22×4×14 = 1232/1232
R-r = 1 ———— (iv)
By adding equation (iii) and (iv) we will get,
R+r = 4
R-r = 1
2R = 5
R = 5/2 = 2.5cm
R-r = 1
r = 2.5 – 1 = 1.5cm
Hence, the Inner and outer radii are 2.5cm and 1.5cm.
问题30.水通过一个内径为2厘米的圆形管道以每秒6米的速度流入圆柱形水箱,该水箱的底部半径为60厘米。在30分钟内发现水位上升?
解决方案:
Given that,
Internal diameter of pipe = 2 cm
Internal radius of pipe = d/2 = 2/2 = 1cm
Rate of flow of water = 6 m/s = 600 cm/s
Radius of base of cylindrical tank = 60 cm
Rise in height in Cylindrical tank = Rate of flow of water × Total time × Volume of pipe / Volume of cylindrical tank
= (600 × 30 × 60 × π × 1 × 1) / (π × 60 × 60) = 300 cm = 3m
Hence, Rise in water level is 3m.
问题31.两端开口的圆柱管由金属制成。管的内径为10.4厘米,长度为25厘米。各处的金属厚度均为8毫米。计算金属的体积。
解决方案:
Given that,
Internal diameter of cylindrical tube = 10.4 cm
Internal radius of tube = d/2 = 10.4/2 = 5.2cm
Length of tube = 25 cm
Thickness of metal = 8 mm = 0.8 cm
Outer radius of tube = R = 5.2+0.8 = 6 cm
We know that, Volume of metal = π(R2 – r2) × l
= 22/7 × (62 – 5.22) × 25
= 22/7 × (36 – 27.04) × 25
= 704 cm3
问题32.水从内径为0.75厘米的水龙头以每秒7 m的速度流动。在1小时内找到通过管道输送的水的升量。
解决方案:
Given that,
Inner radius of tap = 0.75 cm
Length of water flowing in 1s = 7m = 700 cm
Volume of water per second derived from tap = πr ‑ 2l
= 22/7 × 0.75 × 0.75 × 700 = 1237.5 cm3
Hence, Volume of water derived in 1 hour (3600 sec) = (1237.5 × 3600)/1000 = 4455 liters.
问题33.一个直径为1.4 m,高度为2.1 m的圆柱形水箱由直径为3.5 cm的管道供水,水以每秒2米的速度流过该管道。多少时间可以将水箱装满?
解决方案:
Given that,
Diameter of cylindrical tank = 1.4 m
Radius of tank = d/2 = 1.4/2 = 0.7m
Height of tank = 2.1 m
Diameter of pipe flowing water in tank = 3.5 cm
Radius of pipe = d/2 = 3.5/2 cm
Rate of flow of water = 2 m/s
Time taken to fill the tank = Volume of tank / Volume of pipe × Rate of flow
= πr2h/(πr2 × 2)
= (π × 0.7 × 0.7 × 2.1) / (π × 3.5/2 × 3.5/2 × 2) = 1.029 / 6.125 = 0.168= 1680 seconds = 28 minutes.
Hence, Time taken to fill the tank is 28 minutes.
问题34.将30 cm×18 cm的矩形纸片以两种方式转换为右圆柱的曲面,即沿其长度滚动或沿其宽度滚动。求出由此形成的两个圆柱体的体积比。
解决方案:
Given that,
Dimensions of rectangular sheet = 30 cm × 18 cm
Case 1. When paper is rolled along its length
2 πr = 30
r = 30 / 2π cm
Height = 18 cm
Volume of Cylinder V1 = πr2h
= π × (30/2π)2 × 18 cm3
Case 2. When paper is rolled along its breadth
2πr = 18
r = 18/2π cm
Height = 30 cm
Volume of cylinder V2 = πr2h
= π × (18/2π)2 × 30 cm3
Hence, the Volume of Cylinder V1 / Volume of Cylinder V2 = (π × (30/2π)2 × 18) / (π × (18/2π)2× 30)
= (π × (30/2π)2 × 18) × 1/(π × (2π/18)2× 30) = 302 × 18 / 182 × 30 = 5/3
Hence, the ratio of two volumes is 5:3
问题35.如果管道中水的速度为30厘米/秒,在一分钟内有多少升水从横截面为5厘米2的管道中流出?
解决方案:
Given that,
Cross-section area of pipe = 5 cm2,
Speed of water = 30 cm/s,
Time = 1 minute = 60 sec.
As we know that Volume of water flows through pipe = Area of cross-section × speed of flow × time
= 5 × 30 × 60 = 9000 cm3 = 9000/1000 = 9 liters
Hence, 9 liters of water flows out of pipe.
问题36:实心圆柱体的总表面积为231 cm 2 。它的弯曲表面积是总表面积的2/3。找到圆柱体的体积。
解决方案:
Given that,
Total surface area of cylinder = 231 cm2
Curved surface area = 2/3 total surface area = 2/3 × 231 = 154 cm2 (Given)
2πrh = 2/3 2πr(r + h)
3h = 2(r + h)
3h = 2h + 2r
h = 2r ———– (i)
2πr(h + r) = 231 (Given)
2 × 22/7 × r × (2r+r) =231
2 × 22/7 × r × 3r = 231
3r2 = 231×7 / 2×22 = 36.75
r2 = 36.75 / 3 = 12.25
r = √12.25 = 3.5 cm
Since, h = 2r = 2×3.5 = 7cm
We know that Volume of cylinder = πr2h
= 22/7 × 3.5 × 3.5 × 7 = 269.5 cm3
问题37.寻找下沉直径为3 m的深280 m的管井的成本,每立方米3.60卢比。还可以发现其内部曲面的粘合成本为每平方米2.50卢比。
解决方案:
Given that,
Depth of tube well = 280 m
Diameter of tube well = 3 m
Radius of well = d/2 = 3/2 = 1.5 m
As we know that Volume of Cylinder= πr2h
= 22/7 × 1.5 × 1.5 × 280 = 1980 m3
Cost of Sinking tube well at rate of Rs 3.60/m3 = 1980 × 3.60 = Rs 7128 (Given)
We know that Curved Surface Area = 2πrh
= 2 × 22/7 × 1.5 × 280 = 2640 m2
Hence, Cost of cementing its inner curved surface at rate Rs 2.50/m2 = 2.50 × 2640 = Rs 6600
问题38.当1立方厘米的铜重8.4克时,找到直径为4毫米的13.2千克铜线的长度。
解决方案:
Given that,
Weight of copper wire = 13.2 kg = 13200 gm
Diameter of wire = 4 mm
Radius of wire = d/2 = 4/2 = 2mm = 0.2cm
Let us assume that length of wire is ‘h’ cm.
Weight of 1 cubic cm wire = 8.4 gm (Given)
As we know that the Volume = Weight / Density
Volume × Density = Weight
πr2h × 8.4 = 13200
22/7 × 0.2 × 0.2 × h × 8.4 = 13200
h = 13200×7 / 22×0.2×0.2×8.4 = 12500 cm = 125m
Hence, the Length of 13.2kg of copper wire is 125m.
问题39.将2.2立方分米的黄铜拉成直径0.25厘米的圆柱线。找到电线的长度。
解决方案:
Given that,
Diameter of cylindrical wire = 0.25 cm
Radius of wire = d/2 = 0.25/2 = 0.125cm
Let us assume that length of wire is ‘h’ cm
Volume of brass wire = 2.2 dm3 = 2200 cm3 (Given)
πr2h = 2200
22/7 × 0.125 × 0.125 × h = 2200
h = 2200×7 / 22×0.125×0.125 = 44800 cm = 448m
Hence, the Length of the wire is 448m.
问题40.内径为10 m的一口井的深度为8.4 m。从中取出的土在其周围扩展至7.5 m的宽度,形成路堤。找到路堤的高度。
解决方案:
Given that,
Diameter of well = 10 m
Radius of well = d/2 = 10/2 = 5m
Depth of well = 8.4 m
Volume of earth dug out from well = πr2h
= 22/7 × 5 × 5 × 8.4 = 660 m3
This earth is spread out on width of 7.5 m.
Inner radii r = 5m and Outer radii R = (5+7.5) = 12.5cm
Area × h = 660
π(R2 – r2) h = 660
π(12.52 – 52) h = 660
22/7 (131.25) h = 660
h = 660×7 / 22×131.25 = 4620/2887.5 = 1.6 m
Hence, the Height of embankment is 1.6m.
问题41.一个空心的花园滚子,宽63厘米,周长440厘米,由4厘米厚的铁制成。找到铁的体积。
解决方案:
Given that,
Width of roller = 63 cm
Thickness of roller = 4 cm
Girth (Perimeter) = 440 cm
As we know that Perimeter = 2πR
2πR = 440
2 × 22/7 × R = 440
R = 440×7 / 2×22 = 70cm
Inner radius = R – Thickness = 70 – 4 = 66 cm
Volume of Cylindrical Iron = π(R2 – r2) l
= 22/7 × (702 – 662) × 63 = 107712 cm3
Hence, the Volume of Iron is 107712cm3.
问题42.必须将直径2厘米的实心圆柱体的长度重新铸造成长16厘米,外径20厘米,厚度2.5毫米的空心圆柱体?
解决方案:
Given that,
Length of Solid Cylinder = L
Diameter of Cylinder = 2 cm
Radius of Cylinder = d/2 = 2/2 = 1cm
Volume of Cylinder = πr2L ——— (i)
Length of Hollow Cylinder = 16 cm
External Diameter = 20 cm
External Radius = 20/2 = 10cm
Thickness = 2.5 mm = 0.25 cm
Inner Radius = 10 – 0.25 = 9.75 cm
Volume = π (R2 – r2) l ————– (ii)
From equation (i) and (ii)
πr2L = π (R2 – r2) l
π × 1 × 1 × L = π × (102 – 9.752) × 16
L = 79cm
Hence, the length of the solid cylinder should be 79cm.
问题43.在一个面积为30m×20m的矩形场的中间,挖了一个直径为7 m,深度为10 m的井。这样除去的土壤均匀地分布在田野的其余部分。找到通过其升高场高度的高度。
解决方案:
Given that,
Diameter of well = 7 m
Radius of well = d/2 = 7/2 = 3.5m
Depth of well = 10 m
As we know that Volume of well = πr2h
= 22/7 × 3.5 × 3.5 × 10 = 385 m3
Area of Embankment field = 30 × 20 – 22/7 × 7/2 × 7/2
= 600 – 38.5 = 561.5 m2
Volume of well = Area of Embankment field × Height of Embankment
385 = 561.5 × h
h = 385/561.5 = 0.6856m = 68.56 cm
Hence, the Height of Embankment is 68.56 cm.