问题1:找出半径为球体的表面积:
(i)10.5厘米(ii)5.6厘米(iii)14厘米
解决方案:
(i) Radius (r) = 10.5 cm
Surface area = 4ᴨr2
= 4 * (22/7) * (10.5)2 cm2
= 4 *(22/7) * (21/2) * (21/2) cm2
= 1386 cm2
(ii) Radius (r) = 5.6 cm
Surface area = 4ᴨr2
= 4 * (22/7) * (5.6)2 cm2
= 4 *(22/7) * (56/10) * (56/10) cm2
= 39424/100 = 394.24 cm2
(iii) Radius (r) = 14 cm
Surface area = 4ᴨr2
= 4 * (22/7) * (14)2 cm2
= 4 *(22/7) * (14) * (14) cm2
= 2464 cm2
问题2:找出直径为球体的表面积:
(i)14厘米(ii)21厘米(iii)3.5厘米
解决方案:
(i) Diameter of a sphere = 14 cm
Radius (r) = 14/2 cm = 7 cm
Surface area = 4ᴨr2
= 4 * (22/7) * (7)2 cm2
= 4 *(22/7) * (7) * (7) cm2
= 616 cm2
(ii) Diameter of a sphere = 21 cm
Radius (r) = 21/2 cm
Surface area = 4ᴨr2
= 4 *(22/7) * (21/2) * (21/2) cm2 = 1386 cm2
(iii) Diameter of a sphere = 3.5 cm
Radius (r) = 3.5/2 = 7/4 cm
Surface area = 4ᴨr2
= 4 *(22/7) * (7/4) * (7/4) cm2
= 77/2 = 38.5 cm2
问题3:求出半径分别为10 cm的半球和实心半球的总表面积。 (π= 3.14)
解决方案:
1.Radius of hemisphere = 10 cm
Total surface area of hemisphere = 2ᴨr2
= 2 * 3.14 * 10 * 10 cm2 = 628 cm 2
2. Total surface area of solid hemisphere = 3ᴨr2
= 3 * 3.14 * 10 * 10 cm2 = 942 cm2
问题4:球的表面积为5544平方厘米,求出直径。
解决方案:
Let r be the radius of a sphere, then surface area = 4ᴨr2
So, 5544 = 4 * (22/7) * r2
r2 = (5544 * 7)/(4 * 22) = 63 * 7 cm2
= 441 = (21)2
So, r = 21 cm
Now diameter = 2 * r = 2 * 21 = 42 cm
问题5:黄铜制的半球形碗的内径为10.5厘米。找到以每100 cm 2卢比4的比率在内部镀锡的成本。
解决方案:
Inner diameter of a hemispherical bowl = 10.5 cm
Radius (r) = 10.5/2 = 5.25 cm = 525/100 = 21/4 cm
Surface area of inner part of bowl = 2ᴨr2 = 2 *(22/7) * (21/4) * (21/4) cm2 = 693/4 cm2
Rate of tinplanting = Rs 4 per 100 cm2
Total cost = (693 * 4) / (4 * 100) = Rs 693/100 = Rs 6.93
问题6:建筑物的圆顶是半球形。半径为63 dm。以Rs的比率查找绘画的成本。每平方米2个。
解决方案:
Radius of dome (hemispherical) = 63 dm
Area of curved surface = 2ᴨr2 = 2 * (22/7) * 63 * 63 dm2 = 24948 dm2
Rate of painting = Rs 2 per sq. meter
Total cost = (24948 * 2) / 100 = Rs 249.48 * 2 = Rs 498.96
问题7:假设地球是半径6370公里的球体,如果地球表面的四分之三被水覆盖,则土地面积为多少平方公里?
解决方案:
Radius of earth (sphere) = 6370 km
Water on earth = 3/4 % of total area
Required area = (1/4) * (4ᴨr2) = ᴨr2
= (22/7) * (6370)2 km2 = 22 * 910 * 6310 km2 = 127527400 km2
问题8:将一个高度和半径相同的圆柱体放在半球的顶部。如果形状的长度为7厘米,则找到形状的曲面区域。
解决方案:
Total height of the so formed shape = 7cm
Radius = height of cylinder = 7/2 cm
Curved surface area = 2ᴨrh + 2ᴨr2 = 2ᴨr(h+r) cm2
= 2 * (22/7) * (7/2) * (7/2 + 7/2) cm2 = 22 * 7 = 154 cm2
问题9:月球的直径大约是地球直径的四分之一。求出它们的表面积之比。
解决方案:
Diameter of moon = 1/4 of diameter of earth
Let radius of earth be r km
Then the radius of moon = (r / 4) km
Now, surface area of earth = 4ᴨr2
Surface area of moon = 4ᴨ(r/4)2
= 4ᴨ * (1/16) r2 = (1/4) * ᴨr2
Ratio between surface area of moon and earth = (1/4) * ᴨr2 : 4ᴨr2 = (1/4) : 4 = 1/16
问题10:需要绘制建筑物的半球形圆顶。如果圆顶底部的周长为17.6 m,则考虑油漆成本,因为油漆成本为₹5每100 cm 2 。 [NCERT]
解决方案:
Circumference (c) of the base of dome (r) = 17.6 cm
Radius = c/2ᴨ = (17.6 * 7) / (2 * 22) = 2.8 m
Surface area = 2ᴨr2 = 2 * (22/7) * (2.8)2 m2 = 49.28 m2
Rate of painting the surface = Rs 5 per 100 cm2
Total cost = (49.28 * 5 * 10000) / 100 = Rs 24640
问题11:一个木制玩具是圆锥形的,悬在半球上。圆锥体的底部直径为16厘米,高度为15厘米。以每100 cm 2 ₹7的价格查找玩具的涂漆费用。
解决方案:
Diameter of toy = 16 cm
Radius (r) = 16/2 = 8 cm
Height of conical part (h) =15 cm
Slant height (l) = sqrt. (r2 + h2)
= sqrt. (82 + 152) = sqrt (64 + 225) = sqrt (289) = 17 cm
Total surface area of the toy = ᴨrl + 2ᴨr2
= (22/7) *8 * 17 + 2 * (22/7) * 8 * 8 cm2 = (176/7) * 33 cm2 = (5808/7) cm2
Rate of painting the surface of the toy = Rs 7 per cm2
Total cost = (5808/7) * (7/100) = Rs (5808/100) = Rs 58.08
问题12:储罐由一个圆柱体组成,圆柱体的两端均与一个半球相连。如果圆柱体的外径为1.4 m,圆柱体的长度为8 m,则以每m 2 ₹10的比率找到在外部进行喷涂的成本。
解决方案:
Diameter of tank = 1.4 m
Radius (r) = 1.4/2 = 0.7 m
Height of cylindrical portion = 8m
Outer surface area of tank = 2ᴨrh + 2ᴨr2 = 2ᴨr (h + r)
= 2 * (22/7) * 0.7 * (8 + 0.7) m2 = (44/10) * 8.7 m2
Rate of painting = Rs 10 per m2
Total cost = (44 * 8.7 *10) /10 = Rs 382.80
问题13:房屋的前复合墙由直径21厘米的木球装饰,并放置在小支架上,如图所示。为此使用了八个这样的球体,并将其涂成银色。每个支架是一个半径为1.5厘米,高度为7厘米的圆柱体,应涂成黑色。如果银色涂料的成本为每厘米2 25帕,黑色涂料的成本为每厘米2 5帕,请找出所需的涂料成本。 [NCERT]
解决方案:
Diameter of each sphere = 21 cm
Radius (R) = 21/2 cm
Radius of each cylinder(r) = 1.5 cm and height (h) = 7 cm
Now surface area of one sphere = 4ᴨR2
= 4 * (22/7) * (21/2) * (21/2) cm2 = 1386 cm2
Surface area of one cylinder = 2ᴨrh
= 2 * (22/7) * 1.5 * 7 cm2 = 66 cm2
Surface area of 8 spheres = 8 * 1386 cm2 = 11088 cm2
Surface area of 8 cylinders tops = 8ᴨr2 = 8 * (22/7) * 1.5 * 1.5 cm2 = 56.57 cm2
Surface area of 8 cylinders = 8 * 66 cm2
Surface area of spheres excluding base area = 11088 – 56.57 = 11031.43 cm2
Rate of silver painting the spheres = Rs 25 per cm2
Total cost = Rs (11031.43 *25)/100 = Rs 2757.86
Rate of black painting the spheres = Rs 5 per cm2
Total cost = Rs (528 * 5)/100 = Rs 26.40
Total cost of painting = Rs 2757.86 + Rs 26.40 = Rs 2784.26