h is inversely proportional to p p is directly proportional to \sqrt{t} Given that h = 10 and t = 144 when p = 6 find a formula for h in terms of t - Edexcel - GCSE Maths - Question 21 - 2019 - Paper 1

Given that p is directly proportional to \sqrt{t}, we can express this as:

$p = k_1 \sqrt{t}$

for some constant k_1.

From the information provided, when h = 10 and t = 144, we also have p = 6. Thus, we can find k:

1. Substitute (p = 6) into the second equation: $6 = k_1 \sqrt{144} \implies 6 = k_1 \cdot 12 \implies k_1 = 0.5$.

2. Plugging k_1 into the equation for p gives: $p = 0.5 \sqrt{t}$.

Now we can substitute this into the first equation.

Substituting ( p = 0.5 \sqrt{t} ) into the equation for h:

$h = \frac{k}{0.5 \sqrt{t}} \implies h = \frac{2k}{\sqrt{t}}$

Now we need to determine the constant k using the known values of h and p:

1. From h = 10 when p = 6, we have: $10 = \frac{k}{6} \implies k = 60$.

Therefore, substituting k back into the equation gives:

$h = \frac{120}{\sqrt{t}}$.