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Maths

Changing the subject of a formula - Higher

Sometimes we will need to rearrange a formula to find the value of a subject.

Changing the subject of a formula

We may know the area of a circle and need to find the radius. To do this, we rearrange the formula to make the radius the subject.

The area of a circle (A) is πr2. So:

A = πr2

We will now rearrange the formula to make 'r' the subject.

A = πr2

Start by dividing both sides by π.

a  over pi - r to the power of 2

Then take the square root of both sides.

square root of A over pi = r or square root of A over pi

Question

The formula for the volume (V) of a sphere is:

V = 4/3 π r3

Rearrange the formula to make 'r' the subject.

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Answer

V = 4/3 π r 3

Multiply both sides by 3.

3V = 4πr3

Divide by 4π.

= 3V over 4 pi = r to the power of 3

Take the cube root of both sides.

cube root 3V over 4pi = r

Or r = cube root 3V over 4pi

Question

A circular pond is surrounded by a square lawn.

The area (A) of the lawn is:

A = 4x2 - πx2

Rearrange the formula to make 'x' the subject.

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Answer

A = 4x2 - πx2

Factorise so the term with 'x's only appears once.

A = x2 (4 – π)

Divide both sides by 4 – π.

A ÷ 4 – π = x2

Square root both sides.

x equals the square root of A divided by 4 minus pi

Question

This triangle and rectangle have equal perimeters.

Diagram of a triangle and rectangle

This means that 9x + 2 = 6x + 2y.

Rearrange the formula to make 'x' the subject.

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Answer
  1. 9x + 2 = 6x + 2y
  2. Gather the x terms on the left side of the equation and the remaining terms on the right.
  3. 9x – 6x = 2y – 2
  4. Simplify it.
  5. 3x = 2y – 2
  6. And then divide by 3.
  7. x equals 2y - 2 divided by 2

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