Question

A rectangular box with an open top is constructed from cardboard to have a square base of area \[{x^2}\] and height \[h\]. If the volume of this box is 50 cubic units, determine how many square units of cardboard are required to make this box (in terms of \[x\] ).
A) \[5{x^2}\]
B) \[6{x^2}\]
C) \[\dfrac{{200}}{x} + {x^2}\]
D) \[\dfrac{{200}}{x} + 2{x^2}\]

Answer 1

Hint:
Here we will first find the volume of the box and from that we will find the relation between the height of the box and \[{x^2}\]. Then we will find the area of the vertical rectangular area of the box. We will find the total surface area of the box to get the square units of cardboard required to make this box (in terms of \[x\]).

Complete step by step solution:
We know that the height of the rectangular box is \[h\] and the area of the base of the box is \[{x^2}\]. Therefore,
Volume of the box \[ = {x^2}h\]
It is given that the volume of this box is 50 cubic units. Therefore, substituting the value of volume, we get
 \[ \Rightarrow {x^2}h = 50\]
From the above equation we will get the value of height of the box in terms of \[{x^2}\].
So dividing both sides by \[{x^2}\], we get
 \[ \Rightarrow h = \dfrac{{50}}{{{x^2}}}\]…………………. \[\left( 1 \right)\]
Now we will find the area of the vertical rectangular faces of the box. Therefore, we get
Area of the vertical rectangular faces \[ = x \times h\]
Substituting the value of height from equation \[\left( 1 \right)\] in the above equation, we get
 \[ \Rightarrow \] Area of the vertical rectangular faces \[ = x \times \dfrac{{50}}{{{x^2}}}\]
Multiplying the terms, we get
 \[ \Rightarrow \] Area of the vertical rectangular faces \[ = \dfrac{{50}}{x}\]
Now we will find the total surface area of the box by adding the area of the four vertical rectangular faces and one base of the box as the top of the box is open. Therefore, we get
Total surface area of the box \[ = {x^2} + 4\left( {\dfrac{{50}}{x}} \right)\]
 \[ \Rightarrow \] Total surface area of the box \[ = \dfrac{{200}}{x} + {x^2}\]
Hence the square units of cardboard required to make this box (in terms of \[x\]) is \[\dfrac{{200}}{x} + {x^2}\].

So, option C is the correct option.

Note:
Here, the top of the box is open, so we will not take the area of the top face of the box in the total surface area of the box. To solve this question, we need to know some terms, such as area, perimeter and volume. Area is defined as the space occupied by a two dimensional surface, whereas the perimeter is defined as the path that surrounds an object. Volume is defined as the capacity of an object.