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=(30)/(h) \\ w\cdot h=35\longrightarrow w=(35)/(h) \\ l\cdot w=((30)/(h))((35)/(h))=(1050)/(h^2)=42 \\ h^2=(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=(30)/(h)=(30)/(5)=6 \\ w=(35)/(h)=(35)/(5)=7 \end{gathered}`

Answer:

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