Ratio, proportion and rates of changeGCSE Higher

Direct proportion to a cube

Form and use .

  • Start with the correct power of .
  • Use the known pair to find .
  • Substitute the new value, or rearrange to find the positive value of .

Example 1

is directly proportional to the cube of . When , . Find when .

Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Simplify the numerical terms and collect like terms.

Example 2

is directly proportional to the cube of . When , . Find the positive value of when .

Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.Evaluate the discriminant and simplify the numerator and denominator. Keep both signs.

Your turn

Question 1

is directly proportional to the cube of . When , . Find when .

Check answer
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Simplify the numerical terms and collect like terms.
Question 2

is directly proportional to the cube of . When , . Find the positive value of when .

Check answer
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.Evaluate the discriminant and simplify the numerator and denominator. Keep both signs.
Question 3

is directly proportional to the cube of . When , . Find when .

Check answer
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Simplify the numerical terms and collect like terms.
Question 4

is directly proportional to the cube of . When , . Find the positive value of when .

Check answer
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Apply this to both sides; then write the subject on the left.Simplify the numerical terms and collect like terms.Take the positive cube root.Evaluate the discriminant and simplify the numerator and denominator. Keep both signs.
Question 5

is directly proportional to the cube of . When , . Find when .

Check answer
Write the proportionality as an equation with constant .Substitute the known pair.Simplify the numerical terms and collect like terms.Use the same constant for the new values.Simplify the numerical terms and collect like terms.
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