17. y is directly proportional to the square root of t - Edexcel - GCSE Maths - Question 18 - 2022 - Paper 1

Since t is inversely proportional to the cube of x, we can express this relationship as:

$t = \frac{m}{x^3}$

for some constant m. Given that t = 8 when x = 2, substituting these values gives us:

$8 = \frac{m}{(2)^3}$

This simplifies to:

$8 = \frac{m}{8}$

Thus, we find:

$m = 64$

So we can say that:

$t = \frac{64}{x^3}$

Now we have both expressions for y and t. We can substitute the value of t found from the second part into the first equation:

$y = 5 \sqrt{t}$

Replacing t:

$y = 5 \sqrt{\frac{64}{x^3}}$

This simplifies to:

$y = 5 \cdot \frac{8}{x^{3/2}}$

Thus, we can write:

$y = \frac{40}{x^{3/2}}$

This is the final formula for y in terms of x in its simplest form.