Pule has 3 identical cuboids, each has length / cm and width w cm. The distance around the base of this shape is given by an equation 61 + 2w = 72, while the area of the base of the shape is given by an equation 3lw = 243.

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a) Pule has 3 identical cuboids, each with length $l$ cm and width $w$ cm. They are placed together to form a larger shape. The distance around the base of this shape (perimeter) is given by $6l + 2w = 72$. The area of the base of this shape is given by $3lw = 243$.

b) Step 1: Form an equation in $w$ (Part b(i)). The perimeter of the combined base is $2(3l + w)$. $2(3l + w) = 72$ Divide by 2: $3l + w = 36$ Express $3l$ in terms of $w$: $3l = 36 - w$ The area of the combined base is $(3l)w$. $(3l)w = 243$ Substitute the expression for $3l$ into the area equation: $(36 - w)w = 243$ $36w - w^2 = 243$ Rearrange the terms to form a quadratic equation in standard form: $w^2 - 36w + 243 = 0$

Step 2: Solve the equation for $w$ (Part b(ii)). The equation is $w^2 - 36w + 243 = 0$. We need to find two numbers that multiply to $243$ and add up to $-36$. These numbers are $-9$ and $-27$. $w^2 - 27w - 9w + 243 = 0$ Factor by grouping: $w(w - 27) - 9(w - 27) = 0$ $(w - 27)(w - 9) = 0$ Set each factor to zero to find the possible values for $w$: $w - 27 = 0 \implies w = 27$ $w - 9 = 0 \implies w = 9$ The possible values for $w$ are $\boxed{27 cm or 9 cm}$.

Step 3: Find the length of the cuboid (Part b(iii)). We use the relation $3l = 36 - w$. Case 1: If $w = 9$ cm $3l = 36 - 9$ $3l = 27$ $l = \frac{27}{3} = 9 \text{ cm}$ In this case, the dimensions of each cuboid's base are $l=9$ cm and $w=9$ cm. The combined shape's base would be $3l \times w = 3(9) \times 9 = 27 \times 9$ cm. This is consistent with the diagram showing a shape longer than it is wide.

Case 2: If $w = 27$ cm $3l = 36 - 27$ $3l = 9$ $l = \frac{9}{3} = 3 \text{ cm}$ In this case, the dimensions of each cuboid's base are $l=3$ cm and $w=27$ cm. The combined shape's base would be $3l \times w = 3(3) \times 27 = 9 \times 27$ cm. This would mean the combined shape is wider than it is long, which contradicts the visual representation in the diagram.

Therefore, we choose the dimensions that are consistent with the diagram. The length of the cuboid is $l=9$ cm. The length of the cuboid is $\boxed{9 cm}$.

Step 4: Calculate the volume of each cuboid (Part b(iv)). The height of the cuboid is given as $h = 4$ cm. Using the dimensions $l=9$ cm and $w=9$ cm. The volume of one cuboid is given by the formula $V = l \times w \times h$. $V = 9 cm \times 9 cm \times 4 \text{ cm}$ $V = 81 cm^2 \times 4 \text{ cm}$ $V = 324 cm^3$ The volume of each cuboid is $\boxed{324 cm^3}$.

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