Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\). He decides to design a tray with it.

He removes from each corner the shaded squares of side \(x\,{\text{cm}}\), as shown in the following diagram.

The remainder of the cardboard is folded up to form the tray as shown in the following diagram.

Write down, in terms of \(x\) , the length and the width of the tray.

(i)     State whether \(x\) can have a value of \(5\). Give a reason for your answer.

(ii)    Write down the interval for the possible values of \(x\) .

Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given by

\[V = 4{x^3} - 52{x^2} + 160x.\]

Find \(\frac{{dV}}{{dx}}.\)

Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.

Calculate the maximum volume of the tray.

Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found in part (b)(ii), and \(0 \leqslant V \leqslant 200\) . Clearly label the maximum point.

\(16 - 2x,\,\,10 - 2x\)          (A1)(A1)

(i)     no          (A1)

(when \(x\) is \(5\)) the width of the tray will be zero / there is no short edge to fold / \(10 - 2\,(5) = 0\)          (R1)

Note: Do not award (R0)(A1). Award the (R1) for reasonable explanation.

(ii)    \(0 < x < 5\)          (A1)(A1)

Note: Award (A1) for \(0\) and \(5\) seen, (A1) for correct strict inequalities (accept alternative notation). Award (A1)(A0) for “between \(0\) and \(5\)” or “from \(0\) to \(5\)”.
Do not accept a list of integers.

\(V = (16 - 2x)\,(10 - 2x)\,(x)\)          (M1)

Note: Award (M1) for their correct substitution in volume of cuboid formula.

\( = 160x - 32{x^2} - 20{x^2} + 4{x^3}\) (or equivalent)          (M1)

\( = 160x - 52{x^2} + 4{x^3}\)          (AG)

Note: Award (M1) for showing clearly the expansion and for simplifying the expression, and this must be seen to award second (M1). The (AG) line must be seen for the final (M1) to be awarded.

\(12{x^2} - 104x + 160\)  (or equivalent)          (A1)(A1)(A1)

Note: Award (A1) for \(12{x^2}\), (A1) for \( - 104x\) and (A1) for \( + 160\). If extra terms are seen award at most (A1)(A1)(A0).

\(12{x^2} - 104x + 160 = 0\)          (M1)

Note: Award (M1) for equating their derivative to \(0\).

\(x = 2\)          (A1)(ft)

Note: Award (M1) for a sketch of their derivative in part (d), (A1)(ft) for reading the \(x\)-intercept from their graph.
Award (M0)(A0) for \(x = 2\) with no working seen.
Award at most (M1)(A0) if the answer is a pair of coordinates.
Award at most (M1)(A0) if the answer given is \(x = 2\) and \(x = \frac{{20}}{3}\)
Follow through from their derivative in part (d). Award (A1)(ft) only if answer is positive and less than \(5\).

\(4{(2)^3} - 52{(2)^2} + 160(2)\)          (M1)

Note: Award (M1) for correct substitution of their answer to part (e) into volume formula.

\(144\,({\text{c}}{{\text{m}}^3})\)          (A1)(ft)(G2)

Note: Follow through from part (d).

(A1)(A1)(ft)(A1)(A1)(ft)

Note: Award (A1) for correctly labelled axes and window for \(V\), ie \(0 \leqslant V \leqslant 200\).
Award (A1)(ft) for the correct domain \((0 < x < 5)\). Follow through from part (b)(ii) if a different domain is shown on graph.
Award (A1) for smooth curve with correct shape.
Award (A1)(ft) for their maximum point indicated (coordinates, cross or dot etc.) in approximately correct place.
Follow through from parts (e) and (f) only if the maximum on their graph is different from \((2,\,\,144)\).

Question 5: Differential calculus
In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus
In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus

In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus

In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus

In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus

In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.

Question 5: Differential calculus

In general, many candidates struggled in some parts. Most candidates who could state the dimensions of the box gave a reasonable justification of why \(x\) could not be \(5\). Very few candidates scored the two marks in part (d)(ii). Either their inequalities were not strict or their limits were incorrect or both. Some candidates stated the range of \(x\) as \(1,\,\,2,\,\,3,\,\,4\). The algebra in part (c) caused problems for a number of candidates. It seemed that there was a lack of understanding of what the question required. Some substituted \(x = 2\) in the volume formula. A few candidates wrote the product of the length, width, height, omitting the appropriate brackets. Part (d) was well answered by most candidates. However, its application in the following part was not as good in part (e). In part (e) some candidates left both solutions for \(x\), not appreciating the fact that one was outside the range. Others lost both marks as they did not show that they had used their derivative to part (d) as required by the question. Very few candidates scored full marks for the sketch in part (g). Not following the given instructions about the domain and range let most candidates down in this question.