Question

the top of a rectangular box has an area of 42cm^2. The sides of the box have areas of 30cm^2 and 35cm^2. what are the dimensions of the box?

Answer 1

We have a rectangular box where we know the area of the faces and we have to find the width w, length l and height h.

The area of the top of the box is equal to the length times the width (l*w) and we also know that it is 42 cm², so we can write:

$l\cdot w=42$

With the same logic, we can write the equations for the other two areas:

$\begin{gathered} l\cdot h=30 \\ w\cdot h=35 \end{gathered}$

NOTE: the area we choose for l or w is indistinct,so we can relate it as we like.

Then, we can solve this system of equations substituting variables as:

$\begin{gathered} l\cdot h=30\longrightarrow l=\frac{30}{h} \\ w\cdot h=35\longrightarrow w=\frac{35}{h} \\ l\cdot w=(\frac{30}{h})(\frac{35}{h})=\frac{1050}{h^2}=42 \\ h^2=\frac{1050}{42} \\ h^2=25 \\ h=\sqrt[]{25} \\ h=5 \end{gathered}$

With the value of h, we can calculate l and w:

$\begin{gathered} l=\frac{30}{h}=\frac{30}{5}=6 \\ w=\frac{35}{h}=\frac{35}{5}=7 \end{gathered}$

Answer:

The dimensions of the box are: length = 6 cm, width = 7 cm and height = 5 cm.