问题1.找到半径为的球体的体积:
(i)2厘米(ii)3.5厘米(iii)10.5厘米。
解决方案:
As we know that,
Volume of a sphere = 4/3πr3 Cubic Units Where r is radius of a sphere
(i) Given that, Radius = 2 cm
put in formula and we get,
Volume = 4/3 × 22/7 × (2)3 = 33.52
Volume = 33.52 cm3
(ii) Given that, Radius = 3.5cm
putting value in formula and we get,
Volume = 4/3×22/7×(3.5)3 = 179.666
Volume = 179.666 cm3
(iii) Given that, Radius = 10.5 cm
putting this value in formula and we get,
Volume = 4/3×22/7×(10.5)3 = 4851
Volume = 4851 cm3
问题2.找到直径为的球体的体积:
(i)14厘米(ii)3.5 dm(iii)2.1 m
解决方案:
As we know that,
Volume of a sphere = 4/3πr3 Cubic Units Where r is radius of a sphere
(i) Given that, diameter = 14 cm
So, radius = diameter / 2 = 14/2 = 7cm
putting these value in formula and we get,
Volume = 4/3×22/7×(7)3 = 1437.33
Volume = 1437.33 cm3
(ii) Given that,
Diameter = 3.5 dm
So, radius = diameter/2 = 3.5/2 = 1.75 dm
putting these value in formula and we get,
Volume = 4/3×22/7×(1.75)3 = 22.46
Volume = 22.46 dm3
(iii) Given that,
Diameter = 2.1 m
So, radius = diameter/2 = 2.1/2 = 1.05 m
putting these value in formula and we get,
Volume = 4/3×22/7×(1.05)3 = 4.851
Volume = 4.851 m3
问题3.半球形水箱的内半径为2.8 m。找到它的容量(升)。
解决方案:
Given that,
Radius of hemispherical tank is 2.8 m
Capacity of hemispherical tank is 2/3 πr3 = 2/3×22/7×(2.8)3 m3 = 45.997 m3
[As we know that 1m3 = 1000 liters]
Therefore, capacity in liters = 45997 liters
问题4.半球形碗是由0.25厘米厚的钢制成的。碗的内半径为5厘米。查找用于制作碗的钢的体积。
解决方案:
Given that,
Inner radius of a hemispherical bowl is 5 cm
Outer radius of a hemispherical bowl is 5 cm + 0.25 cm = 5.25 cm
As we know that,
Volume of steel used = Outer volume – Inner volume
= 2/3×π×((5.25)3−(5)3) = 2/3×22/7×((5.25)3−(5)3) = 41.282
Hence Volume of steel used is 41.282 cm3
问题5.用一个铅块可以制造多少颗子弹,铅块的边长22厘米,每个子弹的直径为2厘米?
解决方案:
Given that,
Edge of a cube = 22 cm,
Diameter of bullet = 2 cm,
So, radius of bullet(r) = 1 cm,
Volume of the cube = (side)3 = (22)3 cm3 = 10648 cm3 and,
Volume of each bullet which will be in spherical in shape = 4/3πr3
= 4/3 × 22/7 × (1)3 = 4/3 × 22/7
= 88/21 cm3
As we know that,
Number of Bullets = (Volume of Cube) / (Volume of Bullet)
= 10648 / (88/21) = 2541
Hence, 2541 bullets can be made.
问题6.一位店主有一个半径为5厘米的梯子。使用相同的材料,可以制成几只半径为2.5厘米的拉多犬?
解决方案:
Given that,
Volume of laddoo having radius 5 cm (V1) = 4/3×22/7×(5)3 (Using Volume of Sphere formula)
= 11000/21 cm3
Also, Volume of laddoo having radius 2.5 cm (V2) = 4/3πr3
= 4/3×22/7×(2.5)3 = 1375/21 cm3
Hence, Number of laddoos of radius 2.5 cm that can be made are = V1/V2 = 11000/1375 = 8
问题7.将一个直径3厘米的铅球形球熔化并重铸成三个球形球。如果两个球的直径分别为3 / 2cm和2 cm,请找到第三个球的直径。
解决方案:
Given that,
Volume of lead ball with radius 3/2 cm = 4/3πr3 = 4/3×π×(3/2)3
Lets,
Diameter of first ball (d1) = 3/2cm,
Radius of first ball (r1) = 3/4 cm,
Diameter of second ball (d2) = 2 cm,
Radius of second ball (r2) = 2/2 cm = 1 cm,
Diameter of third ball (d3) = d,
Radius of third ball (r3) = d/2 cm,
As we know that,
Volume of lead ball = 4/3πr13 + 4/3πr23 + 4/3πr33
Volume of lead ball = 4/3π(3/4)3 + 4/3π(2/2)3 + 4/3π(d/2)3
4/3π(3/2)3 = 4/3π[(3/4)3 + (2/2)3 + (d/2)3]
27/8 = 27/64 + 1 + d3/8
d3 = (125 x 8) / 64
d = 10 / 4
Hence, d = 2.5 cm
问题8.将半径为5厘米的球体浸入装在圆柱体中的水中,水位上升5/3厘米。找到圆柱体的半径。
解决方案:
Given that,
Radius of sphere = 5cm,
Height of water rised = 5/3cm,
Let us assume that radius of Cylinder is r cm,
As we know that Volume of Sphere = 4/3πr3
= 4/3 × π × (5)3
As we know that, Volume of water rised in cylinder = πr2h
Therefore,
Volume of water rises in cylinder = Volume of sphere
πr2h = 4/3πr3
r2 × 5/3 = 4/3 × π × (5)^3
r2 × 5/3 = 4/3 × 22/7 × 125
r2 = 20 × 5
r = √100
r = 10 cm
Hence the radius of cylinder is 10 cm.
问题9.如果球体的半径加倍,则第一个球体与第二个球体的体积比是多少?
解决方案:
Let us assume that v1 and v2 be the volumes of the first and second sphere respectively,
Radius of the first sphere = r,
Radius of the second sphere = 2r
therefore (Volume of first sphere) / (Volume of second sphere)
= 4/3πr3 / 4/3π(2r)3 = 1 / 8
Hence the ratio is 1 : 8
问题10.圆锥和半球具有相等的底和相等的体积。找到他们的身高比例。
解决方案:
Given that,
Volume of Cone = Volume of Hemisphere
1/3πr2h = 2/3πr3
r2h = 2r3
h = 2r
h/r = 1/1 × 2 = 2
Hence, Ratio of their heights is 2 : 1
问题11:半球形碗形式的容器装满了水。它的内容物在一个右圆柱中倒空。碗和圆柱的内部半径分别为3.5厘米和7厘米。求出水在缸中上升的高度。
解决方案:
Given that,
Volume of water in the hemispherical bowl = Volume of water in the cylinder
Let h be the height to which water rises in the cylinder
Inner radii of the bowl = r1 = 3.5 cm
Inner radii of the bowl = r2 = 7 cm
2/3π(r13 )= π(r22)h
h = 2r13 / 3r22
h = 2(3.5)3 / 3(72)
h = 7 / 12 cm
Hence the height to which the water will rise in the cylinder is 7/12 cm.
问题12.高度为直径三分之二的圆柱体的体积与半径为4 cm的球体的体积相同。计算圆柱体的半径。
解决方案:
Given that,
Height of the cylinder = 2/3 diameter
We know that
Diameter = 2(radius)
h = 2/3 × 2r = 4/3r
Volume of Cylinder = Volume of Sphere
πr2h = 4/3πr3
π × r2 × (4/3r) = 4/3π(4)3
(r)3 = (4)3
r = 4 cm
Hence, the radius of the base of the Cylinder is 4 cm.
问题13:半球形碗形式的容器装满了水。将内装物倒入圆筒中。碗和圆柱的内部半径分别为6厘米和4厘米。找出缸中水的高度。
解决方案:
Given that,
Volume of water in Hemispherical bowl = Volume of Cylinder
2/3πr13 = πr22h
h = 2x(6)3 / 3x(4)2
h = 9 cm
Hence the height of water in the cylinder is 9 cm.
问题14.半径为16厘米的圆柱桶盛有30厘米深的水。将球形铁球放入浴缸,使水位上升9厘米。球的半径是多少?
解决方案:
Given that,
Radius of the cylinder = 16 cm,
Let’s r be the radius of the iron ball
Then,
Volume of iron ball = Volume of water raised in the hub
4/3 x π x r3 = π x (r)2 x h
4/3 x r3 = (16)2 x 9
r^3 = 1728 = (12)^3
Hence radius of ball is 12 cm.
问题15.半径为12厘米的圆柱体中盛有20厘米深的水。将球形铁球放入圆柱体中,从而使水位上升6.75厘米。找到球的半径。 (使用= 227)。
解决方案:
Given that,
Radius of the cylinder = r1 = 12cm,
Raised in raised = r2 = 6.75 cm,
Volume of water raised = Volume of the sphere
π x (r1)2 x h = 4/3 x π x (r2)3
12 x 12 x 6.75 = 4/3 x (r2)3
= (r2)3 = (12 x 12 x 6.75 x 3) / 4
= r2 = 9 cm
Hence radius of Sphere is 9 cm.
问题16.铜球的直径为18厘米。球体熔化并被拉成具有均匀圆形横截面的长丝。如果电线的长度为108 m,请找到其直径。
解决方案:
Given that,
Diameter of a copper sphere = 18 cm,
Radius of the sphere = 9 cm,
Length of the wire = 108 m = 10800 cm,
Volume of cylinder = Volume of sphere
π x (r1)^2 x h = 4/3 x π x (r2)^3
= (r1)^2 x 10800 = 4/3 x 9 x 9 x 9
= (r1)^2 = 0.009
= r1 = 0.3 cm
Hence Diameter is 0.6 cm.