A student is investigating a relationship between two quantities. The table below shows recorded values of \(x\) and \(y\), where \(y\) is directly proporti...
Question 1 Report
A student is investigating a relationship between two quantities. The table below shows recorded values of \(x\) and \(y\), where \(y\) is directly proportional to \(x^2\).
| x | 5 | 8 |
|---|---|---|
| y | 75 | ? |
- Find the constant of proportionality. (1)
- Complete the table by finding the missing value of \(y\). (2)
- Find the positive value of \(x\) for which \(y = 48\). (1)
Answer Details
"Directly proportional to \(x^2\)" means \( y = kx^2 \) for a fixed constant \(k\), which must first be found from the known pair of values before it can be used elsewhere in the table or to solve for \(x\).
(a) Substituting \( x = 5 \), \( y = 75 \) gives \( 75 = k \times 5^2 = 25k \), so \( k = 75 \div 25 = 3 \) [1 mark].
(b) The completed table below uses \( y = 3x^2 \) for \( x = 8 \): \( y = 3 \times 8^2 = 3 \times 64 = 192 \) [2 marks].
| x | 5 | 8 |
|---|---|---|
| y | 75 | 192 |
(c) Setting \( y = 48 \) in the formula gives \( 48 = 3x^2 \), so \( x^2 = 16 \), and taking the positive square root (as requested) gives \( x = 4 \) [1 mark].
Squaring \(x\) means the formula \( y = 3x^2 \) always has two solutions for \(x\) given a value of \(y\), \( x = 4 \) or \( x = -4 \); asking specifically for the positive value avoids any ambiguity.
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