A corner shop's storage crates are stacked in a square arrangement of side \(n\) crates. The total number of tins on display, \(T\), is given by \(T = n^{2}...
Question 1 Report
A corner shop's storage crates are stacked in a square arrangement of side \(n\) crates. The total number of tins on display, \(T\), is given by \(T = n^{2} + 3n\). The shopkeeper's calculator shows a total of 154 tins today.
- Show that \(n\) satisfies \(n^{2} + 3n - 154 = 0\). (1)
- Use your calculator to solve this equation for \(n\), giving the positive solution. (3)
- State why the negative solution is not valid in this context. (1)
Answer Details
Setting an algebraic formula equal to a known value and rearranging into the form "\(=0\)" turns a word problem into a quadratic equation that can be solved with the quadratic formula.
(a) Setting the formula for \(T\) equal to the given total, 154:
\[n^{2}+3n=154\]Subtracting 154 from both sides puts the equation into the standard form required for solving:
\[n^{2}+3n-154=0\] [1 mark](b) Comparing with \(an^{2}+bn+c=0\) gives \(a=1\), \(b=3\), \(c=-154\). Substituting into the quadratic formula:
\[n=\dfrac{-3\pm\sqrt{3^{2}-4(1)(-154)}}{2(1)}=\dfrac{-3\pm\sqrt{9+616}}{2}=\dfrac{-3\pm\sqrt{625}}{2}=\dfrac{-3\pm25}{2}\]Taking the positive solution:
\[n=\dfrac{-3+25}{2}=\dfrac{22}{2}=11\] [3 marks](c) The other root is \(n=\dfrac{-3-25}{2}=-14\). Since \(n\) represents the number of crates along one side of a physical square arrangement, it must be a positive whole number; a negative side length has no meaning in this context, so \(n=-14\) is rejected. [1 mark]
A quadratic equation from a real-world context almost always produces two mathematically valid roots, but only the one consistent with the physical situation (here, a positive count of crates) is kept as the final answer.
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