Van crate calculation

A Mazda van with a box body has internal dimensions of 1.6 m × 3.1 m × 1.7 m (width × depth × height) and a payload of 1.7 tonnes. It delivers crates of pastries. Each pastry crate (type 520) has dimensions 600 mm × 400 mm × 200 mm (l × w × h) and a maximum load capacity of 15 kg.
Calculate:
a) How many crates can be placed in a single layer on the loading floor, making full use of the available area?
b) How many layers of crates can be stacked to make the most of the van's interior height?
c) One fully loaded crate weighs 14 kg. What is the total load weight when the van is fully loaded?
d) Compare the calculated load weight with the vehicle's payload and state whether the vehicle can carry the full load. Calculate the maximum number of crates the vehicle can carry while staying within its payload capacity.

Final Answer:

x₁ =  20
y₁ =  8
m =  2240 kg
t =  121

Step-by-step explanation:

$$\begin{gathered} a=1.6\text{m} \\ b=3.1\text{m} \\ c=1.7\text{m} \\ x=600\text{mm}\to \text{m}=600:1000\text{m}=0.6\text{m} \\ y=400\text{mm}\to \text{m}=400:1000\text{m}=0.4\text{m} \\ z=200\text{mm}\to \text{m}=200:1000\text{m}=0.2\text{m} \\ {d}_1=a/x=1.6/0.6=\frac{8}{3}=2\frac{2}{3}\doteq 2.6667 \\ {d}_2=b/x=3.1/0.6=\frac{31}{6}=5\frac{1}{6}\doteq 5.1667 \\ {d}_3=a/y=1.6/0.4=4 \\ {d}_4=b/y=3.1/0.4=\frac{31}{4}=7\frac{3}{4}=7.75 \\ {n}_1=\lfloor d_1\rfloor \cdot \lfloor d_4\rfloor =\lfloor 2.6667\rfloor \cdot \lfloor 7.75\rfloor =14 \\ {n}_2=\lfloor d_2\rfloor \cdot \lfloor d_3\rfloor =\lfloor 5.1667\rfloor \cdot \lfloor 4\rfloor =20 \\ {x}_1=n_2=20 \end{gathered}$$

$y_1=\lfloor c/z\rfloor =\lfloor 1.7/0.2\rfloor =8$

$$\begin{gathered} m_1=14\text{kg} \\ m=m_1\cdot x_1\cdot y_1=14\cdot 20\cdot 8=2240\text{kg} \end{gathered}$$

$$\begin{gathered} m_2=1.7\text{t}\to \text{kg}=1.7\cdot 1000\text{kg}=1700\text{kg} \\ {m}_2>m \\ {t}_1=x_1\cdot y_1=20\cdot 8=160 \\ t=\lfloor m_2/m_1\rfloor =\lfloor 1700/14\rfloor =121 \end{gathered}$$

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