17. $x$ is directly proportional to the square of $y$ - Edexcel - GCSE Maths - Question 18 - 2021 - Paper 3
Question 18
17. $x$ is directly proportional to the square of $y$.
$y$ is directly proportional to the cube of $z$.
$y = 2$ when $x = 32$.
Find a formula for $x$ in terms of $... show full transcript
Worked Solution & Example Answer:17. $x$ is directly proportional to the square of $y$ - Edexcel - GCSE Maths - Question 18 - 2021 - Paper 3
Step 1
Setting Up the Equations
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Answer
Since x is directly proportional to the square of y, we can express this relationship as:
x=k1y2
where k1 is a constant.
Similarly, since y is directly proportional to the cube of z, this can be expressed as:
y=k2z3
where k2 is another constant.
Step 2
Eliminating Variables
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Answer
Substituting the expression for y into the first equation gives:
x=k1(k2z3)2
This simplifies to:
x=k1k22z6
Let c=k1k22, hence:
x=cz6.
Step 3
Finding the Constant
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Answer
From the problem, we know that y=2 when x=32. First, we need to calculate z using the proportional relationship:
Substituting y into the equation:
2=k2z3,
we can solve for z:
ightarrow z = ext{cube root}igg( rac{2}{k_2}igg).$$
Next, substituting $x = 32$ into the equation:
$$32 = c z^6.$$
Thus, solving for $c$ will give:
$$c = rac{32}{z^6}.$$
Step 4
Combining the Equations
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Answer
By substituting the value of c back into the equation x=cz6, we have:
x = rac{32}{z^6} z^6 = 32.
In conclusion, we find that:
x=32z6.
This gives the required formula for x in terms of z.
