Proportion to Powers of x

Worked example

1. 1

   $P\propto n^3$ — “P is directly proportional to the cube of n” means P is always the same multiple of n × n × n.

2. 2

   $P=k{n}^3$ — Swap the ∝ sign for “= k ×”. The number k stays fixed, but we do not know it yet.

3. 3

   $40=k\times 2^3$ — Put in the pair we are given: P = 40 when n = 2.

4. 4

   $40=8k$ — Work out 2³ = 2 × 2 × 2 = 8.

5. 5

   $k=40÷8=5$ — Divide both sides by 8 to get k on its own.

6. 6

   $P=5n^3$ — Write the formula with k filled in. Now it works for any n.

7. 7

   $320=5n^3$ — Now use it. Put in the new value, P = 320.

8. 8

   $n^3=320÷5=64$ — Divide both sides by 5 to get n³ on its own.

9. 9

   $n=\sqrt[3]{64}=4$ — Undo the cube with a cube root. Check: 4 × 4 × 4 = 64.

Worked example

1. 1

   $A=k\sqrt{m}$ — “Proportional to the square root of m” means A = k × √m.

2. 2

   $6=k\sqrt{16}$ — Put in the pair we are given: A = 6 when m = 16.

3. 3

   $6=4k$ — Work out √16 = 4.

4. 4

   $k=6÷4=1.5$ — Divide both sides by 4 to get k.

5. 5

   $A=1.5\sqrt{m}$ — Write the formula with k filled in.

6. 6

   $A=1.5\sqrt{64}$ — Now use it. Put in the new value, m = 64.

7. 7

   $A=1.5\times 8=12$ — √64 = 8, then multiply by k.

Worked example

1. 1

   $y=k(x+3{)}^2$ — “Proportional to the square of (x + 3)”. The whole bracket is squared.

2. 2

   $50=k(2+3{)}^2$ — Put in x = 2 and y = 50. Add inside the bracket first.

3. 3

   $50=25k$ — Work out (2 + 3)² = 5² = 25.

4. 4

   $k=50÷25=2$ — Divide both sides by 25 to get k.

5. 5

   $y=2(x+3{)}^2$ — Write the formula with k filled in.

6. 6

   $y=2(-5+3{)}^2$ — Now put in x = −5. Do the bracket first.

7. 7

   $y=2(-2{)}^2$ — −5 + 3 = −2. Keep the −2 inside its bracket.

8. 8

   $y=2\times 4=8$ — A negative number squared is positive: (−2)² = 4. Then multiply by 2.

Worked example

1. 1

   $y=k\sqrt{x-4}$ — “Proportional to the square root of (x − 4)”. All of x − 4 sits under the root.

2. 2

   $6=k\sqrt{13-4}$ — Put in x = 13 and y = 6. Subtract under the root first.

3. 3

   $6=3k$ — Work out √9 = 3.

4. 4

   $k=6÷3=2$ — Divide both sides by 3 to get k.

5. 5

   $y=2\sqrt{x-4}$ — Write the formula with k filled in.

6. 6

   $y=2\sqrt{29-4}$ — Now put in x = 29.

7. 7

   $y=2\times 5=10$ — Work out √25 = 5, then multiply by 2.

Worked example

1. 1

   $R=k\sqrt{s}$ — R is proportional to √s. R goes up by 30%.

2. 2

   $100\%+30\%=130\%=1.3$ — An increase of 30% means multiply by 1.3.

3. 3

   $\sqrt{s}\times 1.3$ — R is k times √s and k never changes, so √s is multiplied by 1.3 too.

4. 4

   $s\times 1.3^2$ — To undo the square root, square it. So s is multiplied by 1.3².

5. 5

   $1.3^2=1.69$ — Work out 1.3 × 1.3.

6. 6

   $s\text{increases by}69\%$ — 1.69 means 169% of the old amount, which is 69% more.

Worked example

1. 1

   $c=k{d}^2$ — c is proportional to d². Now d is multiplied by 4.

2. 2

   $c=k(4d{)}^2$ — Put 4d in place of d.

3. 3

   $(4d{)}^2=4^2{d}^2=16d^2$ — Square the 4 as well as the d.

4. 4

   $c\times 16$ — So c is multiplied by 16.

Worked example

1. 1

   $y\propto \frac{1}{x},x\propto T^2$ — Two links in a chain: y depends on x, and x depends on T.

2. 2

   $y=\frac{k}{x}$ — Write the first link as an equation.

3. 3

   $x=c{T}^2$ — Write the second link as an equation, with its own constant c.

4. 4

   $y=\frac{k}{c{T}^2}$ — Put cT² in place of x in the first equation.

5. 5

   $y=\frac{K}{T^2}$ — k and c are both fixed numbers, so k ÷ c is one fixed number. Call it K.

6. 6

   $8=\frac{K}{3^2}$ — Find K by putting in y = 8 and T = 3.

7. 7

   $K=8\times 9=72$ — Multiply both sides by 9.

8. 8

   $y=\frac{72}{T^2}$ — Write the combined formula with the number filled in.

9. 9

   $2=\frac{72}{T^2}$ — Now use it. Put in y = 2.

10. 10

    $T^2=72÷2=36$ — T² and 2 swap places, so T² = 72 ÷ 2.

11. 11

    $T=6$ — Take the square root. T is a positive quantity here, so ignore −6.

Worked example

1. 1

   $y=\frac{k}{\sqrt{x}},x=c{w}^2$ — y is inversely proportional to √x, and x is proportional to w².

2. 2

   $y=\frac{k}{\sqrt{c{w}^2}}$ — Put cw² in place of x.

3. 3

   $y=\frac{k}{\sqrt{c}\times w}$ — Take the root of each factor: √(w²) = w, because w is positive.

4. 4

   $y=\frac{K}{w}$ — k ÷ √c is one fixed number. Call it K.

5. 5

   $5=\frac{K}{6}$ — Find K by putting in y = 5 and w = 6.

6. 6

   $K=5\times 6=30$ — Multiply both sides by 6.

7. 7

   $y=\frac{30}{w}$ — The formula for y in terms of w. Check: w = 6 gives 30 ÷ 6 = 5.

Worked example

1. 1

   $M=k\sqrt{h}$ — The question says M is proportional to √h.

2. 2

   $12=k\sqrt{9}$ — The curve passes through (9, 12). Choose a point you can read exactly, so h = 9 and M = 12.

3. 3

   $12=3k$ — Work out √9 = 3.

4. 4

   $k=12÷3=4$ — Divide both sides by 3 to get k.

5. 5

   $M=4\sqrt{h}$ — Write the formula with k filled in.

Worked example

1. 1

   $y=k{x}^3$ — y is proportional to x³. We are told y = 12h when x = h, with h ≠ 0.

2. 2

   $12h=k{h}^3$ — Put in x = h and y = 12h. Treat h like a number.

3. 3

   $k=\frac{12h}{h^3}$ — Divide both sides by h³.

4. 4

   $k=\frac{12}{h^2}$ — Cancel one h from the top and bottom: h ÷ h³ = 1 ÷ h².

5. 5

   $y=\frac{12x^3}{h^2}$ — Put k back into the formula. It is fine for k to contain a letter.

6. 6

   $x=h:\frac{12h^3}{h^2}=12h$ — Check by putting x = h back in. We get the 12h we started with.