问题1:求出圆柱体的曲面和总表面积,圆柱体的直径为7厘米,高度为60厘米。
解决方案:
The details given about cylinder are:
Diameter of base of a cylinder = 7 cm
So, radius = (7/2)
Height of cylinder = 60 cm
Curved surface area of a cylinder = 2 * (22/7) * r * h
= 2 * (22/7) * (7/2) * 60
= 1320 cm2
Total surface area of cylinder = 2 * (22/7) * r * (h + r)
= 2 * (22/7) * (7/2) * (60 + (7/2))
= 22 * (60 * 2 + 7)/2
= 22 * (127/2)
= 1397 cm2
问题2:圆柱杆的弯曲表面积为132 cm 2 。如果半径为0.35厘米,请找到长度。
解决方案:
The details given about cylindrical rod are –
Curved surface area of a cylindrical rod = 132 cm2
Length of radius = 0.35 cm
Let length of rod = h
Curved surface area of a cylinder = 2 * (22/7) * r * h
132 = 2 * (22/7) * 0.35 * h
(132 * 7)/(2 * 0.35 * 22) = h
h = 60 cm
问题3:直圆柱的底部面积为616 cm 2 ,高度为2.5 cm。找到圆柱体的曲面区域。
解决方案:
The details given about right circular cylinder are –
Area of the base of a right circular cylinder = 616 cm2
Height of right circular cylinder = 2.5 cm
Let radius of right circular cylinder = r
Area of base of a right circular cylinder = (22/7) * r2
616 = (22/7) * r2
(616 * 7)/22 = r2
196 = r2
r = 14 cm
Curved surface area of cylinder = 2 * (22/7) * r * h
= 2 * (22/7) * 14 * 2.5
= 220 cm2
问题4:圆柱体底部的周长为88厘米,高度为15厘米。找到其弯曲面积和总表面积。
解决方案:
The details given about cylinder are –
Circumference of the base of a cylinder = 88 cm
Height of a cylinder = 15 cm
Let radius of cylinder = r
Circumference of the base of a cylinder = 2 * (22/7) * r
88 = 2 * (22/7) * r
(88 * 7) = (2 * 22) * r
r = 14 cm
Curved surface area of cylinder = 2 * (22/7) * r * h
= 2 * (22/7) * 14 * 15
= 1320 cm2
Total surface area of cylinder = 2 * (22/7) * r * (h + r)
= 2 * (22/7) * 14 * (15 + 14)
= 2 * (22/7) * 14 * 19
= 2552 cm2
问题5:25厘米* 7厘米的矩形条绕着较长的一侧旋转。找到由此产生的固体的总表面积。
解决方案:
The details given about rectangular strip are –
Dimension of rectangular strip = 25 cm * 7 cm
When the strip is rotated about its longer side,
Height of the cylinder becomes = 25 cm
Radius of cylinder = 7 cm
Total surface area of cylinder = 2 * (22/7) * r * (h + r)
= 2 * (22/7) * 7 * (25 + 7)
= 2 * (22/7) * 7 * 32
= 1408 cm2
问题6: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。找到由此产生的圆柱体的总表面积。
解决方案:
The details given about rectangular sheet of paper are –
Dimensions of rectangular sheet = 44 cm * 20 cm
When the sheet of the paper is rolled along its length,
Height of the cylinder = 20 cm
Circumference of base becomes = 44 cm
Let radius of base = r
Circumference of base = 2 * (22/7) * r
44 = 2 * (22/7) * r
(44 * 7)/(2 * 22) = r
r = 7 cm
Total surface area of cylinder = 2 * (22/7) * r * (h + r)
= 2 * (22/7) * 7 * (20 + 7)
= 2 * (22/7) * 7 * 27
= 1188 cm2
问题7:两个圆柱体的半径之比为2:3,高度为5:3。计算曲面面积之比。
解决方案:
The details given cylinders are –
r1 / r2 = 2 : 3
h1 / h2 = 5 : 3
Curved surface area of cylinder 1/Curved surface area of cylinder 2 = (2 * (22/7) * r1 * h1 )/(2 * (22/7) * r2 * h2)
= (2 * (22/7) * 2 * 5)/(2 * (22/7) * 3 * 3)
= 10/9
问题8:直圆柱的曲面表面积与总表面积之比为1:2。证明其高度和半径相等。
解决方案:
Curved surface area of cylinder/Total surface area of a cylinder = 1/2
(2 * (22/7) * r * h)/(2 * (22/7) * r * (h + r) = 1/2
h/(h + r) = 1/2
2 * h = h + r
2h – h = r
h = r
Height = Radius
Hence, Proved.
问题9:圆柱体的曲面面积为1320 cm 2 ,其底部直径为21 cm。找到圆柱体的高度。
解决方案:
The details given about cylinder are –
Curved surface area of cylinder = 1320 cm2
Diameter of base = 21 cm
So, radius = 21/2
Let height of cylinder = h
Curved surface area of cylinder = 2 * (22/7) * r * h
1320 = 2 * (22/7) * (21/2) * h
(1320 * 7 * 2)/(2 * 22 * 21) = h
h = 20 cm
问题10:右圆柱的高度为10.5厘米。如果其两个圆形表面的面积之和是曲面面积的两倍,则为三倍。找到其底面的半径。
解决方案:
The details given about cylinder are –
Height of circular cylinder = 10.5 cm
Let radius of cylinder = r
Area of two bases of cylinder = 2 * (22/7) * r2
Area of curved surface of cylinder = 2 * (22/7) * r * h
3 * (2 * (22/7) * r2) = 2 * (2 * (22/7) * r * h))
6 * r = 4 * h
6 * r = 4 * 10.5
r = (4 * 10.5)/6
r = 7 cm