Straightened 26861

A Mazda van with a box body with internal dimensions of 1.6 m x 3.1 m x 1.7 m (w x d x h) with a payload of 1.7 tons delivers crates with pastries. Pastry box 520: dimensions 600mm x 400mm x 200mm (l x w x h), load capacity 15 kg.
Calculate:

a) How many crates can be straightened into one layer at most to fully use the box body's loading area?
b) How many layers can the crates be placed on top of each other to make the most of the height of the box body?
c) One crate, even with pastries, weighs 14 Kg. What is the weight of the load at the maximum use of the box body,
d) Compare the calculated weight of the load with the vehicle's payload and state whether or not the vehicle can transport the load. Calculate the maximum number of crates a vehicle can carry to make the most of the utility load capacity of the vehicle.

Correct 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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