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Inverse proportion to the square root

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Inverse proportion to the square root worksheet
Inverse proportion to the square root worksheet

Inverse proportion to the square root occurs when one quantity decreases as the square root of another quantity increases. This concept is often found in areas like physics, such as the relationship between pressure and volume in gases or certain diffusion processes. Understanding this relationship helps in solving problems where the change in one value slows down as another value increases. Jump to the questions

Practise now

Question

y is inversely proportional to √p

y is when p is

Find the equation for y and p

Step 1 - describe the relationship using the correct symbol

y ∝ 1 / √p

Step 2 - replace the symbol with the constant of proportionality, k

y = / √p

Step 3 - substitute the values from the question into the equation

= /

Step 4 - rearrange the equation to isolate k

× = k

Step 5 - determine the value of k

k =

Step 6 - write the final equation linking y and √p

y = / √p

Find y when p =

Step 1 - restate the equation

y = / √p

Step 2 - substitute the value you know

y = /

Step 3 - determine the final value of y (to 3 decimal places)

y =

Find p when y =

Step 1 - restate the equation

y = / √p

Step 2 - substitute the value you know

= / √p

Step 3 - rearrange the equation to isolate √p

/ = √p

Step 4 - determine the value of √p

√p =

Step 5 - calculate p

p =

Topic guide

What this worksheet practises

This worksheet provides practice on solving inverse proportion problems where one variable is inversely proportional to the square root of another variable (e.g., y ∝ 1/√x). As one value gets larger, the other gets smaller, but at a slowing rate due to the square root.

Key method

The structured process of finding the constant 'k' remains exactly the same as all other proportion topics.

  • Write out the relationship algebraically: y = k / √x.
  • Substitute the complete pair of known values into this equation.
  • Square root the x value first, and then solve the equation to find 'k'. Usually, you will multiply the 'y' value by the square rooted 'x' value.
  • Rewrite your full formula, inserting your newly calculated 'k'.
  • Use this completed formula to find the missing value requested in the question.

Worked example

y is inversely proportional to the square root of x. When x = 36, y = 3. Find y when x = 9.

Step 1: Write the base equation.

y = k / √x

Step 2: Substitute the known pair (x=36, y=3) to find 'k'.

3 = k / √36

3 = k / 6

k = 3 × 6 = 18.

Step 3: Write the completed formula.

y = 18 / √x

Step 4: Use the formula to answer the question (Find y when x = 9).

y = 18 / √9

y = 18 / 3

y = 6.

Common mistakes to avoid

A frequent mistake occurs when finding 'x' rather than 'y'. If your formula is y = 18 / √x, and the question asks you to find 'x' when y = 2, you set up 2 = 18 / √x. Rearranging gives √x = 9. Many students stop here. But the question asked for 'x', not '√x'. To remove a square root, you must square both sides. So x = 9² = 81.

How to check your answer

Always verify the inverse relationship. In the example, x went down from 36 to 9. Therefore, because the relationship is inverse, y must go up. It went up from 3 to 6. This logical check confirms you haven't accidentally set up a direct proportion equation.