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

We know that y = 15 when t = 9. Plugging these values into the equation gives:

$15 = k \sqrt{9} \Rightarrow 15 = k \cdot 3 \Rightarrow k = 5$

Since t is inversely proportional to the cube of x, we can write:

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

where m is another constant of proportionality.

We know that t = 8 when x = 2. Plugging these values into the equation gives:

$8 = \frac{m}{2^3} \Rightarrow 8 = \frac{m}{8} \Rightarrow m = 64$

Now we substitute the expression for t back into the equation for y:

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

Now simplifying:

$y = 5 \cdot \frac{8}{x^{3/2}} = \frac{40}{x^{3/2}}$

Hence, the formula for y in terms of x is:

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