Variation – Exercise 3.6.3 – Class IX

1. If z varies jointly as x an the square root of y, and if z = 6 when x = 3 and y = 16, find z when x = 7 and y = 4.

Solution:

Zα jointly as x and √y

Z = 6; x = 3 and y = 16

Z = k x√y

6 = k. 3 x √16 = k. 3 x4 = 12 k

6 = 12 k

k = ⁶/₁₂ = ¹/₂

When x = 7 and y = 4 then, Z = k x√y

Z = ¹/₂ x 7x √4  = 7x²/₂ = 7

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2. The power p, watts, of an electrical circuit varies jointly as the resistance R and the square of the current I. for a 240- watt refrigerator that draws a current of 2 amperes, the resistance is 60 ohms. What is the resistance is – watt microwave over that draws a current of 5 amperes?

Solution:

According to the given data Pα RI²

P = KRI²

KRI² = P

K = ^P/_RI² = ²⁴⁰/_60×2² = ²⁴⁰/_60×4 = 1

Thus K = 1

When P = 600W and I = 5amps , then R =?

KRI² = P

K = ^P/_RI² = ⁶⁰⁰/_1x(5)² = ⁶⁰⁰/₂₅ = 24ohms

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3. The work W done in charging a capacitor varies jointly as the charge q and the voltage V. find the equation of joint variation. If a capacitor with a charge of 0.004 coulomb and a voltage of 100 volts performs 0.20 joule of work, find the constant of proportionally.

Solution:

W α Qv

W = kQv

W = kx0.004 x100

0.20 = kx0.004 x100

k = ^0.20/_{0.004 x100} = ^0.2/_0.004 = ^0.2/_0.4 = ¹/₂ = 0/5

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4. The surface area of a cylinder varies jointly as the radius, and the sum of the radius and the height. A cylinder with height 8 cm. and radius 4 cm. has a surface area of 96 πcm². Find the surface area of a cylinder with radius 3 cm and height 10 cm.

Solution:

A α r(r+h)

A = kr(r+h)

96π = 4k(12)

k = ^{96 π}/_4×12

k = 2 π

A = 2 πx3(3+10) = 6 πx13 = 78 πcm²

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5. The number of hours needed to assemble computers varieties directly as the number of computers and inversely as the number of workers. If 4 workers can assemble 12 computers in 9 hours, how many workers are needed to assemble 48 computers in 8 hours?

Solution:

Hrs α number of computer α ¹/_{no. of workers}

hours                no. of computers                no. of works

9                                        12                                  4

8                                        48                                  x

x = ^4x48x9/_12×8  = 18workers

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6. The number of litres of water g in a circular tank varies jointly with the square of the radius r2, and the depth d. if g = 3016 when r = 4 and d =2 , find the number of litres in the tank when r = 3 and d= 1.5.

Solution:

According to the given data g α r²d

Kr²d = g

K = ^g/_R²d  = ³⁰¹⁶/₄²_x2 = ³⁷⁷/₄

When r=3 and d = 15 then g =?

G = Kr²d = ³⁷⁷/₄ x (3)²x1.5 = ^5089.5/₄ = 1272.375litre

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7. If 36 men can build a wall of 140 m long in 21 days, how many men are required to build a similar wall of length 50 m in 18 days?

Solution:

Men                     Length                                  days

36                            140m                                    21

x                               50m                                       18

x= ^36x50x21/_140×18 = 15men

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8. If the total wages of 15 labourers for 6 days is rs. 8,100, find the wages of 21 labourers for 5 days.

Solution:

Labours              days                      rs./wages

15                           6                               8100
21                           5                                    x

x = ^8100x5x21/_6×15 = 9450 Rs.

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9. Tap A can fill a cistern in 8 hours and tap B can empty it in 12 hours. How long will it take to fill the cistern if both of them are opened together?

Solution:

Tap A takes 8 hours to fill the cistern

Part of cistern filled in 1 hour = ¹/₈

Tap B empties the cistern in 12 hours

Part of cistern emptied in 1hr = ¹/₁₂

Part of cistern filled when both the taps open = ¹/₈ + ¹/₁₂ = ^3-2/₂₄ = ¹/₂₄

Time taken to fill the cistern when both taps open = 24hours

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