Video: Pack 1 β€’ Paper 1 β€’ Question 13

Pack 1 β€’ Paper 1 β€’ Question 13

02:20

Video Transcript

The table shows a set of values for π‘₯ and 𝑦. The variable 𝑦 is inversely proportional to the cube of π‘₯. Find an equation for 𝑦 in terms of π‘₯.

If two items are inversely proportional to one another, we can say 𝑦𝛼 one over π‘₯. This symbol 𝛼 is just our way of saying is proportional to. So 𝑦 is proportional to one over π‘₯. We can then form this equation: 𝑦 is equal to π‘˜ over π‘₯. In this case though, 𝑦 is inversely proportional to the cube of π‘₯. So we write 𝑦 is proportional to one over π‘₯ cubed. Our equation, therefore, becomes 𝑦 is equal to π‘˜ over π‘₯ cubed.

You might have noticed that we have a table of values linking π‘₯ and 𝑦. At this stage, we can use any one of these pairs of numbers to help us calculate the value of the constant π‘˜. Since this is a noncalculator paper, let’s choose the simplest numbers possible. When π‘₯ is two, 𝑦 is equal to five over four. Substituting these values into our equation gives us five over four is equal to π‘˜ over two cubed. To work out the value of π‘˜, we multiply both sides of the equation by two cubed or eight. π‘˜ is therefore equal to 40 over four, which is equal to 10. We mustn’t forget to substitute our new value of π‘˜ back into the original equation. 𝑦 is therefore equal to 10 over π‘₯ cubed.

Find the value of 𝑦 when π‘₯ equals 10. We now have an equation linking π‘₯ and 𝑦. We’re asked to find the value of 𝑦 when π‘₯ equals 10. So it’s a simple matter of substituting π‘₯ equals 10 into that equation to find 𝑦. 𝑦 is therefore equal to 10 over 10 cubed. 10 cubed is 1000, giving us a value of 𝑦 equals one over 100 or 0.01. Remember as long as our answer is a fraction in its simplest form, we don’t need to work out as a decimal, unless specifically told to. 𝑦 is equal to one over 100.

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