unknown volume formula for a box?

[SilentSymphony]

A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the 10 in. side. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with a lid. What is the formula for the volume of the box?

 

[Deleted member 4993]

Hint:
First draw a picture of the situation

Let the squares be L * L

Then the rectangles - such that a lid of the box is made would be made upon folding - would be of size L * (L + 15-2L) = L * (15 - L)

 

[soroban]

Hello, SilentSymphony!

Did you make a sketch?

A piece of cardboard measures 10 in. by 15 in.
Two equal squares are removed from the 10 in. side.
Two equal rectangles are removed from the other corners
so that the tabs can be folded to form a rectangular box with a lid.
What is the formula for the volume of the box?

Let \(\displaystyle x\) = side of the squares.
Let \(\displaystyle y\) = width of the box.

      : - - - - - -15 - - - - - - :
    - *---*---------*-------------* -
    : |///|         |///://///////| x
    : * - * - - - - * - * - - - - * -
    : |   :         :   :         | :
    : |   :         :   :         | :
   10 |   :         :   :         | 10-2x
    : |   :         :   :         | :
    : |   :         :   :         | :
    : |   :         :   :         | :
    : * - * - - - - * - * - - - - * -
    : |///|         :///://///////| x
    - *---*---------*---*---------* -
      : x :    y    : x :    y    :

The box looks like this:

          *-------------*
           \             \
            \             \y
             \             \
              * - - - - - - *
            / |           / | x
          /   * - - - - /   *
        /    /        /   /
      *-------------*   / y
    x |             | / 
      *-------------*
          10-2x

The length is \(\displaystyle 10-2x\), the width is \(\displaystyle y\), the height is \(\displaystyle x\).

The volume is: \(\displaystyle \:V \;=\;(10\,-\,2x)(y)(x)\;\) [1]

From the first diagram, we see that: \(\displaystyle \:2x\,+\,2y\:=\:15\;\;\Rightarrow\;\;y \:=\:\frac{15\,-\,2x}{2}\)

Substitute into [1]: \(\displaystyle \:V \;=\;(10\,-\,2x)\left(\frac{15\,-\,2x}{2}\right)x\)

Therefore, the volume is: \(\displaystyle \L\:V \;=\;x(5\,-\,x)(15\,-\,2x)\)