问题1.找出一个圆柱体的体积
(i)r = 3.5厘米,h = 40厘米
(ii)r = 2.8 m,h = 15 m
解决方案:
(i) Given that,
r = 3.5 cm,
h = 40 cm
As we know that Volume of a cylinder = πr2h
= 22/7 × 3.5 × 3.5 × 40 = 1540 cm3
(ii) Given that,
r = 2.8 m,
h =15 m
As we know that Volume of a cylinder = πr2h
= 22/7 × 2.8 × 2.8 × 15 = 369.6 m3
问题2。如果圆柱体的直径(d)和高度(h)为:,求出圆柱体的体积:
(i)d = 21厘米,h = 10厘米
(ii)d = 7 m,h = 24 m
解决方案:
(i) Given that,
d = 21cm,
r = d/2 = 21/2cm,
h = 10 cm.
As we know that Volume of a cylinder = πr2h
= 22/7 × 21/2 × 21/2 × 10 = 3465 cm3
(ii) Given that,
d = 7 m2
r = d/2 = 7/2m
h = 24 m
As we know that Volume of a cylinder = πr2h
= 22/7 × 7/2 × 7/2 × 24 = 924 m3
问题3.直圆柱的底部面积为616 cm 2 ,高度为25 cm。找到圆柱体的体积。
解决方案:
Given that,
Area of base of right circular cylinder = 616 cm2
Height of cylinder = 25 cm
Let us assume that radius of cylinder is ‘r’ cm
As we know that Area of base of right circular cylinder = πr2,
πr2 = 616
22/7 × r2 = 616
r2 = 616 × 7/22 = 196
r = √196 = 14cm
As we know that, Volume of cylinder = Area of base of right circular cylinder × height
= 616 × 25 = 15400 cm3
问题4.圆柱体底部的周长为88厘米,高度为15厘米。找到圆柱体的体积。
解决方案:
Given that,
Circumference of base of cylinder = 88 cm
Height of cylinder = 15 cm
Let us assume that ‘r’ be the radius of the cylinder.
As we know that Circumference of base of cylinder = 2πr,
2πr = 88
2 × 22/7 × r = 88
r = 88 × 7 / 2 × 22 = 616/44 = 14cm
Radius of cylinder = 14 cm
We know that Volume of cylinder = πr2h
= 22/7 × 14 × 14 × 15 = 9240 cm3
问题5.空心圆柱管长21 dm。它的外径和内径分别为10厘米和6厘米。找到用于制造管道的铜的体积。
解决方案:
Given that,
Length of cylinder = 21 dm = 210 cm
Outer diameter = 10 cm
Outer radius, R = 10/2 = 5cm
Inner diameter = 6 cm
Inner radius, r = 6/2 = 3cm
As we know that Volume of copper used in making the pipe(hollow) = π (R2 – r2)h2
= 22/7 (52 – 32) 210
= 22/7 (25-9) 210 = 10560 cm3
问题6.找到高度为15 cm,底部半径为7 cm的直圆柱体的(i)弯曲表面积(ii)总表面积和(iii)体积。
解决方案:
Given that,
Height of cylinder = 15 cm
Radius of base = 7 cm
(i) We know that formula of Curved surface area = 2πrh
= 2 × 22/7 × 7 × 15 = 660 cm2
(ii) We know that formula of Total surface area = 2πr(h+r)
= 2 × 22/7 × 7 (15+7) = 968 cm2
(iii) We know that formula of Volume of cylinder = πr2h
= 22/7 × 7 × 7 × 15 = 2310 cm3
问题7.直圆柱体的底部直径为42厘米,高度为10厘米。找到圆柱体的体积。
解决方案:
Given that,
Diameter of base of cylinder = 42 cm
Radius of base = d/2 = 42/2 = 21cm
Height = 10 cm
We know that Volume of cylinder = πr2h
= 22/7 × 21 × 21 × 10 = 13860 cm3
问题8.找到圆柱体的体积,圆柱体的直径为7厘米,高度为60厘米。另外,找到以升为单位的气瓶容量。
解决方案:
Given that,
Diameter of base = 7 cm
Radius of base = d/2 = 7/2 cm
Height of cylinder = 60 cm
As we know that Volume of cylinder = πr2h
= 22/7 × 7/2 × 7/2 × 60 = 2310 cm3
Capacity of cylinder in liters = 2310 / 1000 = 2.31 liters.
问题9. 25厘米×7厘米的矩形带绕着较长的一侧旋转。查找由此产生的固体体积。
解决方案:
Given that,
Dimensions of Rectangular Strip = 25 cm × 7 cm
When it rotated about its longer side then it will become,
Radius of base = 7 cm
Height of cylinder = 25 cm
As we know that Volume of cylinder = πr2h
= 22/7 × 7 × 7 × 25 = 3850 cm3
问题10.将一张44厘米×20厘米的矩形纸沿着其长度滚动以形成一个圆柱体。查找如此形成的圆柱体的体积。
解决方案:
Given that,
Dimensions of rectangular sheet = 44cm × 20cm
When it rolled along its length it will become,
Radius of base = length/2π
= 44×7 / 2×22 = 7cm
Height of cylinder = 20 cm
We know that Volume of cylinder = πr2h
= 22/7 × 7 × 7 × 20 = 3080 cm3
问题11.圆柱体的体积和弯曲表面积分别为1650 cm 3和660 cm 2 。找到圆柱体的半径和高度。
解决方案:
Given that,
Volume of cylinder = 1650 cm3
Curved surface area = 660 cm2
Volume of Cylinder/Curved Surface Area = 1650/660
πr2h/ 2πrh = 1650/660
r/ 2 = 5/2
r = 5cm
Surface area = 660 cm2 (Given)
As we know that Surface Area of Cylinder = 2πrh
2πrh = 660
2 × 22/7 × 5 × h = 660
h = 660×7 / 2×22×5
= 4620/220 = 21cm
Hence, Radius is 5cm and height is 21cm.
问题12:两个圆柱体的半径之比为2:3,其高度之比为5:3。计算它们的体积比。
解决方案:
Given that,
Ratio of radii of two cylinder = 2:3
Radius of 1st Cylinder = r1
Radius of 2nd Cylinder = r2
r1/r2 = 2/3
Ratio of their heights = 5:3
Height of 1st Cylinder = h1
Height of 2nd Cylinder = h2
h1/h2 = 5/3
Volume of 1st Cylinder = v1
Volume of 2nd Cylinder = v2
v1 / v2 = π(r1)2h1 / π(r2)2h2
= 22 × 5 / 32 × 3
= 4×5 / 9×3 = 20/27
Hence, Ratio of Volumes of two Cylinder’s is 20 : 27.
问题13:直角圆柱的弯曲表面积与总表面积之比为1:2。如果圆柱体的总表面积为616 cm 2 ,则找到圆柱体的体积。
解决方案:
Given that,
Total surface area of cylinder = 616 cm2
Ratio between Curved Surface Area and Total Surface Area of Cylinder = 1:2
2πrh / 2πr (r+h) = 1/2
h / (r+h) = 1/2
2h =r+h
Hence, r = h
As we know, 2πr (h+r) = 616
2πr (r+r) = 616
2πr (2r) = 616
4πr2 = 616
r2 = 616/4π
= 616×7 / 4×22 = 49
r = √49 = 7
Hence, Radius = 7cm and Height = 7cm
As we know that Volume of cylinder = πr2h
= 22/7 × 7 × 7 × 7 = 1078 cm3
问题14.圆柱体的曲面面积为1320 cm 2 ,圆柱体的直径为21 cm。找到圆柱体的体积。
解决方案:
Given that,
Diameter of base = 21 cm
Radius of base = d/2 = 21/2 cm
Curved surface area = 1320 cm2
As we know that Curved surface Area of Cylinder = 2πrh
2πrh = 1320
2 × 22/7 × 21/2 × h = 1320
h = 1320×7×2 / 2×22×21
= 18480/924 = 20cm
As we know that Volume of Cylinder = πr2h
= 22/7 × 21/2 × 21/2 × 20 = 6930 cm3
问题15:底座的半径与圆柱体的高度之比为2:3。如果圆柱体的体积为1617 cm3,请找到圆柱体的总表面积。
解决方案:
Given that,
Ratio between radius and height of a cylinder = 2:3
r/h = 2/3
h = 3/2 r
Volume of cylinder = 1617 cm3
As we know that Volume of Cylinder = πr2h
πr2h = 1617
22/7 × r2 × 3/2r = 1617
r3 = 1617×7×2 / 22×3 = 343
r = 3√343
= 7cm
Hence, Radius = 7 cm
Height = 3/2r = 3/2 × 7 = 21/2 = 10.5cm
As we know that Total Surface Area of Cylinder = 2πr (r+h)
= 2 × 22/7 × 7 (10.5+7) = 770 cm2
问题16.圆柱柱的弯曲表面积为264 m 2 ,其体积为924 m 3 。找到支柱的直径和高度。
解决方案:
Given that,
Curved surface area of cylinder = 264 m2,
Volume = 924 m3,
Volume of Cylinder / Curved Surface Area of Cylinder
πr2h / 2πrh = 924 / 264
r/2 = 924 / 264
r = 924×2 / 264 = 7m
Hence, Radius = 7 m
Diameter of cylinder = 2 × radius = 2×7 = 14m
Curved surface area = 264 m2 (Given)
2πrh = 264
2 × 22/7 × 7 × h = 264
h = 264×7 / 2×22×7 = 6m
Hence, Height of cylinder is 6m & Diameter of cylinder is 14m.
问题17:两个等体积的圆柱体的高度比例为1:2。求出它们半径的比例。
解决方案:
Given that,
Ratio of their height = 1:2,
Height of 1st Cylinder = h1,
Height of 2nd Cylinder = h2
h1 / h2 = 1/2
Volume of 1st Cylinder, V1 = Volume of 2nd Cylinder, V2
V1 = V2
π(r12)h1 = π(r22 )h2
r12 / r22 = 2/1
r1 / r2 = √(2/1) = √2 / 1
Hence, the Ratio of their radii is √2:1
问题18:右圆柱的高度为10/5 m。其两个圆形表面的面积之和的三倍是曲面面积的两倍。找到圆柱体的体积。
解决方案:
Given that,
Height of cylinder = 10.5 m
3(A+A) = 2 Curved Surface Area (where, A = Circular Area of Box)
3×2A = 2(2πrh)
6A = 4πrh
6πr2 = 4πrh
r2/r = 4πh/6π
r = 2/3 h = 2×10.5 / 3 = 7m
As we know that Volume of Cylinder = πr2h
= 22/7 × 7 × 7 × 10.5 = 1617 m3
问题19.必须挖出多少立方米的土壤才能下沉21 m深,直径6m的井?
解决方案:
Given that,
Height of cylinder = 21m,
Diameter of well = 6m,
Radius of well = d/2 = 6/2 = 3m
We know that Volume of Earth that must be dug out from this well is = πr2h
= 22/7 × 3 × 3 × 21 = 594 m3
问题20:一棵树的树干是圆柱形的,其周长为176厘米。如果树干的长度为3 m,请找到可以从树干中获得的木材量。
解决方案:
Given that,
Length of the trunk = 3m = 300 cm,
Circumference of trunk of tree = 176 cm
As we know that Circumference of Trunk of tree = 2πr
2πr = 176
2 × 22/7 × r = 176
r = 176×7 / 2×22 = 28cm
Hence, Radius = 28cm
We know that Volume of Timber can be obtained from trunk of tree = πr2h
= 22/7 × 28 × 28 × 300 = 7392 cm3 = 0.74 m3