Find the length of AC.

(i)     Write down the size of angle BCA.

(ii)    Calculate the length of AB.

Show that the area of the sandbox is \(31.1{\text{ }}{{\text{m}}^2}\) correct to 3 s.f.

The sandbox is a prism. Its edges are \(40{\text{ cm}}\) high. The sand occupies one third of the volume of the sandbox. Calculate the volume of sand in the sandbox.

\({\text{A}}{{\text{C}}^2} = {9^2} + {4.2^2} - 2 \times 9 \times 4.2 \times \cos {95^ \circ }\)     (M1)(A1)
\({\text{AC}} = 10.3{\text{ m}}\)     (A1)(G2)

Note: (M1) for correct substituted formula and (A1) for correct substitution. If radians used answer is \(6.59\). Award at most (M1)(A1)(A0).

Note: The final A1 is only awarded if the correct units are present; only penalize once for the lack of units or incorrect units.

(i)     \({\text{B}}\hat{\text{C}}{\text{A}} = {25^ \circ }\)     (A1)

(ii)    \(\frac{{{\text{AB}}}}{{\sin {{25}^ \circ }}} = \frac{{10.258 \ldots }}{{\sin {{130}^ \circ }}}\)     (M1)(A1)

\({\text{AB}} = 5.66{\text{ m}}\)     (A1)(ft)(G2)

Note: (M1) for correct substituted formula and (A1) for correct substitution. (A1) for correct answer.

Follow through with angle \({\text{B}}\)\(\hat{\text{C}}\)\({\text{A}}\) and their AC. Allow \({\text{AB}} = 5.68\) if \({\text{AC}} = 10.3\) used. If radians used answer is \(0.938\) (unreasonable answer). Award at most (M1)(A1)(A0)(ft).

OR

Using that ABC is isosceles

\({\text{cos2}}{{\text{5}}^ \circ } = \frac{{\frac{1}{2} \times 10.258 \ldots }}{{{\text{AB}}}}\) (or equivalent)     (A1)(M1)(ft)

\({\text{AB}} = 5.66{\text{ m}}\)     (A1)(ft)(G2)

Note: (A1) for \(\frac{1}{2}\) of their AB seen, (M1) for correct trigonometric ratio and correct substitution, (A1) for correct answer. If \(\frac{1}{2}{\text{AB}}\) seen and correct answer is given award (A1)(G1). Allow \({\text{AB}} = 5.68\) if \({\text{AC}} = 10.3\) used. If radians used answer is \(3.32\). Award (A1)(M1)(A1)(ft). If \(\sin 65\) and radians used answer is \(3.99\). Award (A1)(M1)(A1)(ft).

Note: The final A1 is only awarded in (ii) if the correct units are present; only penalize once for the lack of units or incorrect units.

Area \( = \frac{1}{2} \times 9 \times 4.2 \times \sin {95^ \circ } + \frac{1}{2} \times {(5.6592 \ldots )^2} \times \sin {130^ \circ }\)    (M1)(M1)(ft)(M1)

\( = 31.095 \ldots  = 31.1{\text{ }}{{\text{m}}^2}\) (correct to 3 s.f.)     (A1)(AG)

Note: (M1)(M1) each for correct substitution in the formula of the area of each triangle, (M1) for adding both areas. (A1) for unrounded answer. Follow through with their length of AB but last mark is lost if they do not reach the correct answer.

Volume of sand \( = \frac{1}{3}(31.09 \ldots  \times 0.4)\)     (M1)(M1)

\( = 4.15{\text{ }}{{\text{m}}^3}\)     (A1)(G2)

Note: (M1) for correct formula of volume of prism and for correct substitution, (M1) for multiplying by \(\frac{1}{3}\) and last (A1) for correct answer only.

Note: The final A1 is only awarded if the correct units are present; only penalize once for the lack of units or incorrect units.

It could have been written that the diagram was representing the plan of the sandbox. However, examiner’s comments did not find this lack of information an obstacle for the candidates.

Overall the lengths of AC and AB were well done. Sine rule and cosine rule were in general well used. To find the length of AB many students used correctly right- angled trigonometry. The area of the sandbox was in general well done though some students did not gain the final mark due to premature rounding or for not showing the unrounded answer. The volume of the prism was poorly answered by the majority of the students. Most of the students did not use the correct formula. Very few candidates noticed that the value \(40\) was given in cm. It was good to see very few students losing marks for having their GDC setting in radians.

It could have been written that the diagram was representing the plan of the sandbox. However, examiner’s comments did not find this lack of information an obstacle for the candidates.

Overall the lengths of AC and AB were well done. Sine rule and cosine rule were in general well used. To find the length of AB many students used correctly right- angled trigonometry. The area of the sandbox was in general well done though some students did not gain the final mark due to premature rounding or for not showing the unrounded answer. The volume of the prism was poorly answered by the majority of the students. Most of the students did not use the correct formula. Very few candidates noticed that the value \(40\) was given in cm. It was good to see very few students losing marks for having their GDC setting in radians.

It could have been written that the diagram was representing the plan of the sandbox. However, examiner’s comments did not find this lack of information an obstacle for the candidates.

Overall the lengths of AC and AB were well done. Sine rule and cosine rule were in general well used. To find the length of AB many students used correctly right- angled trigonometry. The area of the sandbox was in general well done though some students did not gain the final mark due to premature rounding or for not showing the unrounded answer. The volume of the prism was poorly answered by the majority of the students. Most of the students did not use the correct formula. Very few candidates noticed that the value \(40\) was given in cm. It was good to see very few students losing marks for having their GDC setting in radians.

It could have been written that the diagram was representing the plan of the sandbox. However, examiner’s comments did not find this lack of information an obstacle for the candidates.

Overall the lengths of AC and AB were well done. Sine rule and cosine rule were in general well used. To find the length of AB many students used correctly right- angled trigonometry. The area of the sandbox was in general well done though some students did not gain the final mark due to premature rounding or for not showing the unrounded answer. The volume of the prism was poorly answered by the majority of the students. Most of the students did not use the correct formula. Very few candidates noticed that the value \(40\) was given in cm. It was good to see very few students losing marks for having their GDC setting in radians.