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The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in figure. Eight such spheres are used for this purpose, and are to be painted silver. Each support is a cylinder of radius 1.5 cm and height 7 cm and is to be painted black. Find the cost of paint required if silver paint costs 25 paise per cm² and black paint costs 5 paise per cm².

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402. The diameter of a sphere is decreased by 25%. By what per cent does its curved surface area decrease?

Let the radius of the sphere be

straight r over 2

cm.
This its diameter = 2

open parentheses straight r over 2 close parentheses equals space straight r space cm

Curved surface area of the original sphere =

4 straight pi open parentheses straight r over 2 close parentheses squared equals πr squared space space cm squared

New diameter (decreased) of the sphere

equals space straight r minus straight r cross times 25 over 100 equals straight r minus straight r over 4 equals fraction numerator 3 straight r over denominator 4 end fraction space cm

∴ Radius of the new sphere

equals 1 half open parentheses fraction numerator 3 straight r over denominator 4 end fraction close parentheses equals fraction numerator 3 straight r over denominator 8 end fraction space cm

∴ New curved surface area of the sphere

equals 4 straight pi open parentheses fraction numerator 3 straight r over denominator 8 end fraction close parentheses squared equals fraction numerator 9 πr squared over denominator 16 end fraction space cm squared

∴ Decrease in the original curved surface area

equals πr squared minus fraction numerator 9 πr squared over denominator 16 end fraction equals space fraction numerator 16 πr squared minus 9 πr squared over denominator 16 end fraction equals fraction numerator 7 πr squared over denominator 16 end fraction

∴ Percentage of decrease in the original curved surface area

equals fraction numerator begin display style fraction numerator 7 πr squared over denominator 16 end fraction end style over denominator πr squared end fraction cross times 100 percent sign equals 700 over 16 percent sign equals 175 over 4 percent sign equals 43 3 over 4 percent sign space or space 43.75 percent sign

Hence, the original curved surface area decreases by 43.75%.

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403. Find the mean of the following marks of 20 students on a screening test, (out of 100) 76, 44, 45, 87, 71, 72, 82, 83, 41, 32, 75, 32, 46, 78, 17, 70, 84, 12, 77, 74

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404. The mean of 10, 12, 16, 20, p and 26 is 17. Find the value of p.

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Long Answer Type

405.

Find the mean of each of the following distributions :

(i)

x_i	10	15	20	25	30	35	40	Total
f_i	4	6	8	18	6	5	3	50

(ii)

x_i	12	13	14	15	16	17	18	Total
f_i	1	3	4	8	10	3	1	30

(iii)

x_i	50	75	100	125	150	175	200	Total
f_i	12	18	50	70	25	15	10	200

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Short Answer Type

406.

The following data have been overanged in ascending order. If the median of these data is 63, then find the value of x : 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

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407.

Find the median of the following data:

95, 65, 75, 70, 75, 100, 50, 40.

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408. The mean of 40 observations was 160. It was detected on rechecking that the value 165 was wrongly copied as 125. Find the correct mean.

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409. 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.

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410. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner curved surface area of the vessel.

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