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Home > Maths I > Relationships > Direct variation

Maths I

Direct variation

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Finding the variation formula (continued)

Example

Study this worked example carefully before doing the questions.

The cost, \pounds{C}, of producing a booklet varies directly as the square root of the number N of pages it contains.

A booklet with 16 pages costs \pounds{1.44} to produce.

Find the cost of producing a booklet with 15 pages.

C \propto \surd{N}

Therefore C = k\surd{N}

so, 1.44 = k \times \surd{16}

(\surd{16} = 4)

1.44 = k \times 4

k = \frac{1.44}{4}

k = 0.36

C = 0.36 \surd{N} is the variation formula

N = 100 for a booklet with 100 pages

Therefore C = 0.36 \times \surd{100}

C = 0.36 \times 10

C = 3.6

The cost is \pounds{3.60}

Now try these questions.

Question

\;P varies directly as the square root of \;Q

P = 10 when Q = 25

Find \;P when Q = 81

Answer

P \propto \surd{Q}

P = k\surd{Q}

10 = k \times \surd{25}

10 = k \times 5

k = 2

Variation formula is P = 2\surd{Q}

P = 2 \times \surd{81}

P = 2 \times 9

P = 18

Question

\;S varies directly as the square of \;T.

S = 20 when T = 2

Find \;S when T = 3

Answer

S \propto T^2

S = kT^2

20 = k \times 2^2

20 = k \times 4

\;k = 5

Variation formula is S = 5T^2

S = 5 \times 3^2

S = 5 \times 9

S = 45

Question

The value, \pounds{V}, of a diamond varies directly as the square of its weight, \;C carats.

A diamond weighing \;9 carats is worth \pounds{567}.

Find the value of a diamond weighing \;20 carats.

Answer

V \propto C^2

V = kC^2

567 = k \times 9^2

567 = k \times 81

81k = 567

k = 7

Variation formula is V = 7C^2

V = 7 \times 20^2

V = 7 \times 400

V = 2800

Value is \pounds{2800}

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