Solved: Haruka has an eco-friendly bag in the shape of a cuboid with width 12 cm, length... [IB Mathematics Applications & Interpretation (AI)]

Question

SLPaper 2

Haruka has an eco-friendly bag in the shape of a cuboid with width 12 cm, length 36 cm and height of 9 cm. The bag is made from five rectangular pieces of cloth and is open at the top.

Nanako decides to make her own eco-friendly bag in the shape of a cuboid such that the surface area is minimized.

The width of Nanako’s bag is x cm, its length is three times its width and its height is y cm.

The volume of Nanako’s bag is 3888 cm³.

Calculate the area of cloth, in cm², needed to make Haruka’s bag.

[2]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

36× 12 + 2(9×12) + 2(9× 36) (M1)

Note: Award (M1) for correct substitution into surface area of cuboid formula.

= 1300 (cm²) (1296 (cm²)) (A1)(G2)

[2 marks]

Calculate the volume, in cm³, of the bag.

[2]

Verified

Solution

36× 9×12 (M1)

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

= 3890 (cm³) (3888 (cm³)) (A1)(G2)

[2 marks]

Use this value to write down, and simplify, the equation in x and y for the volumeof Nanako’s bag.

[2]

Verified

Solution

3 $x$ × $x$ × $y$ = 3888 (M1)

Note: Award (M1) for correct substitution into volume of cuboid formula and equated to 3888.

$x$ ² $y$ =1296 (A1)(G2)

Note: Award (A1) for correct fully simplified volume of cuboid.

Accept $y = \frac{{1296}}{{{x^2}}}$ .

[2 marks]

Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.

[2]

Verified

Solution

(A =)3x² + 2(xy) + 2(3xy) (M1)

Note: Award (M1) for correct substitution into surface area of cuboid formula.

(A=) 3x²+ 8xy (A1)(G2)

Note: Award (A1) for correct simplified surface area of cuboid formula.

[2 marks]

Use your answers to parts (c) and (d) to show that

$A = 3{x^2} + \frac{{10368}}{x}$ .

[2]

Verified

Solution

$A = 3{x^2} + 8x\left( {\frac{{1296}}{{{x^2}}}} \right)$ (A1)(ft)(M1)

Note: Award (A1)(ft) for correct rearrangement of their part (c) seen (rearrangement may be seen in part(c)), award (M1) for substitution of their part (c) into their part (d) but only if this leads to the given answer, which must be shown.

$A = 3{x^2} + \frac{{10368}}{x}$ (AG)

[2 marks]

Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .

[3]

Verified

Solution

$\left( {\frac{{{\text{d}}A}}{{{\text{d}}x}}} \right) = 6x - \frac{{10368}}{{{x^2}}}$ (A1)(A1)(A1)

Note: Award (A1) for $6x$ , (A1) for −10368, (A1) for ${x^{ - 2}}$ . Award a maximum of (A1)(A1)(A0) if any extra terms seen.

[3 marks]

Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.

[3]

Verified

Solution

$6x - \frac{{10368}}{{{x^2}}} = 0$ (M1)

Note: Award (M1) for equating their ${\frac{{{\text{d}}A}}{{{\text{d}}x}}}$ to zero.

$6{x^3} = 10368$ OR $6{x^3} - 10368 = 0$ OR ${x^3} - 1728 = 0$ (M1)

Note: Award (M1) for correctly rearranging their equation so that fractions are removed.

$x = \sqrt[3]{{1728}}$ (A1)

$x = 12$ (cm) (AG)

Note: The (AG) line must be seen for the final (A1) to be awarded. Substituting $x = 12$ invalidates the method, award a maximum of (M1)(M0)(A0).

[3 marks]

The cloth used to make Nanako’s bag costs 4 Japanese Yen (JPY) per cm².

Find the cost of the cloth used to make Nanako’s bag.

[2]

Verified

Solution

$\left( {3{{\left( {12} \right)}^2} + \frac{{10368}}{{12}}} \right) \times 4$ (M1)

Note: Award (M1) for substituting 12 into the area formula and for multiplying the area formula by 4.

= 5180 (JPY) (5184 (JPY)) (A1)(G2)

[2 marks]

Mathematics Applications & Interpretation (AI) International Baccalaureate (IB) Practice Question: Haruka has an eco-friendly bag in the shape of a cuboid with width 12 cm, length 36 cm and height of 9 cm. The bag is ma...