问题1：一个木制书架的外部尺寸如下：高度= 110厘米，深度= 25厘米，宽度= 85厘米(见图13.31)。木板的厚度到处都是5厘米。外表面要打磨，内表面要上漆。如果抛光速率为20帕斯卡每cm 2且涂漆速率为10帕斯卡每cm 2 ，请找到对书架表面进行抛光和喷漆所需的总费用。
解决方案：
Given :External dimensions of book self,
Length, (l) = 85cm
Breadth, (b) = 25 cm
Height, (h) = 110 cm
Formula to calculate external surface area of shelf while leaving out the front face of the shelf = lh+2(lb+bh)
= [85×110+2(85×25+25×110)]
= (9350+9750)
= 19100 cm2
Therefore, the external surface area of shelf is 19100 cm2
Area of front face = [85×110-75×100+2(75×5)]
= 1850+750
= 2600 cm2
Thus, the area is 2600 cm2
Area to be polished surface = (19100+2600) cm2
= 21700 cm2
Given cost of polishing 1 cm2 area = Rs 0.20
Therefore, the cost of polishing 21700 cm2 area Rs. (21700×0.20)
= Rs 4340
Dimensions of the row of the bookshelf
Length, (l) = 75 cm
Breadth, (b) = 20 cm
Height, (h) = 30 cm
Area to be painted in 1 row= 2(l+h)b+lh
= [2(75+30)× 20+75×30]
= (4200+2250)
= 6450 cm2
Thus, the area is 6450 cm2
Area to be painted in 3 rows = (3×6450)cm2
= 19350 cm2.
Given cost of painting 1 cm2 area = Rs. 0.10
Therefore, the cost of painting 19350 cm2 area = Rs (19350 x 0.1) = Rs 1935
Total expense required for polishing and painting = Rs. (4340+1935) = Rs. 6275
Therefore, the cost for polishing and painting the surface of the bookshelf is Rs. 6275.
问题2：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2所示。 13.32。目的是使用八个这样的球体，并将其涂成银色。每个支架都是一个半径为1.5cm，高度为7cm的圆柱体，并被涂成黑色。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。
解决方案：
Given: Diameter of wooden sphere = 21 cm
Radius of wooden sphere, r = (21/2) cm = 10.5 cm
Formula for surface area of wooden sphere = 4πr2
= 4×(22/7)×(10.5)2
= 1386 cm3
Thus, the surface area is 1386 cm3
The radius of the circular end of cylindrical support (r) = 1.5 cm
Height of cylindrical support (h) = 7 cm
Formula for curved surface area = 2πrh
= 2×(22/7)×1.5×7
= 66
Thus, the CSA is 66 cm2
The area of the circular end of cylindrical support = πr2
= (22/7)×(1.5)2
= 7.07 cm2
Area of the circular end is 7.07 cm2
Area to be painted silver = [8 ×(1386-7.07)]
= 8×1378.93
= 11031.44 cm2
Area to be painted is 11031.44 cm2
Given cost for painting 1cm2 with silver colour = Rs 0.25
Therefore, the cost for painting 11031.44 cm2 with silver colour = Rs(11031.44×0.25)
=Rs 2757.86
Area to be painted black = (8×66) cm2
= 528 cm2
Given cost for painting 1cm2 with black colour = Rs 0.05
Therefore, the cost for painting with black colour = Rs (528×0.05) = Rs 26.40
Therefore, the total painting cost is:
= Rs (2757.86 + 26.40)
= Rs 2784.26
Hence, the total painting cost will be Rs. 2784.26
问题3：球体的直径减小了25％。它的弯曲表面积减少了百分之几?
解决方案：
Let us assume the diameter of the sphere be “d”.
Radius of sphere, (r1) = d/2
New radius of sphere, (r2) = (d/2)×(1-25/100)
= 3d/8
The curved surface area of sphere, (CSA)1 = 4πr12 = 4π×(d/2)2
= πd2 … (i)
The curved surface area of sphere, (CSA)2 = 4πr22 = 4π×(3d/8)2
= (9/16)πd2 … (ii)
From equation (i) and (ii), we get
Decrease in surface area of sphere will be = (CSA)1 – (CSA)2
= πd2 – (9/16)πd2
= (7/16)πd2
The percentage decrease in surface area of sphere = ((CSA)1 – (CSA)2) / (CSA)1) × 100
= (7πd2/16πd2)×100
= 700/16
= 43.75%
Therefore, the decrease in the percentage of the surface area of sphere is 43.75%