JOINT VARIATION WORD PROBLEMS

Problem 1 :

z varies directly with the sum of squares of x and y. z = 5 when x = 3 and y = 4. Find the value of z when x = 2 and y = 4.

Solution :

Since z varies directly with the sum of squares of x and y,

z ∝ x² + x²

z = k(x² + y²) ----(1)

Substitute z = 5, x = 3 and y = 4 to find the value k.

5 = k(3² + 4²)

5 = k(9 + 16)

5 = 25k

Divide both sides by 25.

1/5 = k

Substitute k = 1/5 in (1).

z = (1/5)(x² + y²)

Substitute x = 2, y = 4 and evaluate z.

z = (1/5)((2² + 4²)

z = (1/5)((4 + 16)

z = (1/5)((20)

z = 4

Problem 2 :

M varies directly with the square of d and inversely with the square root of x. M = 24 when d = 4 and x = 9. Find the value of M when d = 5 and x = 4.

Solution :

Since m varies directly with the square of d and inversely with the square root of x

M ∝ d²√x

M = kd²√x ----(1)

Substitute M = 24, d = 4 and x = 9 to find the value k.

24 = k4²√9

24 = k(16)(3)

24 = 48k

Divide both sides by 48.

1/2 = k

Substitute k = 1/2 in (1).

M = (1/2)(d²√x)

Substitute d = 5, x = 4 and evaluate M.

M = (1/2)(5²√4)

M = (1/2)((25)(2)

M = 25

Problem 3 :

Square of T varies directly with the cube of a and inversely with the square of d. T = 2 when a = 2 and d = 4. Find the value of square of T when a = 4 and d = 2

Solution :

Since square of T varies directly with the cube of a and inversely with the square of d

T² ∝ a³d²

T² = ka³d² ----(1)

Substitute T = 2, a = 2 and d = 4 to find the value k.

2² = k2³4²

4 = k(4)(16)

4 = 64k

Divide both sides by 64.

1/16 = k

Substitute k = 1/16 in (1).

T² = (1/16)a³d²

Substitute a = 4, d = 2 and evaluate T². 

T² = (1/16)(4³)(2²)

T² = (1/16)(64)(4)

T² = 16

Problem 4 :

The area of a rectangle varies directly with its length and square of its width. When the length is 5 cm and width is 4 cm, the area is 160 cm². Find the area of the rectangle when the length is 7 cm and the width is 3 cm.

Solution :

Let A represent the area of the rectangle, l represent the length and w represent width.

Since the area of the rectangle varies directly with its length and square of its width,

A ∝ lw²

A = klw² ----(1)

Substitute A = 160, l = 5 and d = 4 to find the value k.

160 = k(5)(4²)

160 = k(5)(16)

160 = 80k

Divide both sides by 80.

2 = k

Substitute k = 2 in (1).

A = 2lw²

Substitute l = 7, w = 3 and evaluate A. 

A = 2(7)(3²)

A = 2(7)(9)

Area of the rectangle = 126 cm²

Problem 5 :

The volume of a cylinder varies jointly as the square of radius and two times of its height. A cylinder with radius 4 cm and height 8 cm has a volume 128π cm³. Find the volume of a cylinder with radius 3 cm and height 10 cm.

Solution :

Let V represent volume of the cylinder, r represent radius and h represent height.

Since the volume of a cylinder varies jointly as the radius and the sum of the radius and the height.

V ∝ r²(2h)

V = kr²(2h) ----(1)

Substitute V = 128π, r = 4 and h = 8 to find the value of k.

128π = k(4²)(2 ⋅ 8)

128π = k(16)(16)

128π = 256k

Divide both sides by 256.

π/2 = k

Substitute k = π/2 in (1).

V = (π/2)r²(2h)

V = πr²h

Substitute r = 3, h = 10 and evaluate V.

V = π(3²)(10)

V = π(9)(10)

Volume of the cylinder = 90π cm³

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