Question

If 675 cm² of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Note: Write only the numerical value of the volume, the units are already written. Write the exact answer not the decimal approximation (for example write 4/5 not 0.8 ). The largest possible volume of the box is ______
∑cm³

Answer 1

The largest possible volume of the box is
`\( (5062.5)/(4) \)` cm³ or 1265.625 cm³ (exact value, not decimal).

Let's denote the side length of the square base as x and the height of the box as h. Since the box has an open top, the volume (V) is given by the product of the base area and the height:
`\( V = x^2h \).`

The total surface area of the box, including the open top, is the sum of the base area and the four vertical sides. The surface area (A) is given by:
`\( A = x^2 + 4xh \).`

The available material to make the box is the surface area, which is given as 675 cm²:
`\( x^2 + 4xh = 675 \).`

To maximize the volume, we need to express h in terms of x . Solving the surface area equation for h :

`\[ h = (675 - x^2)/(4x) \]`

Now, substitute this expression for h into the volume equation:

`\[ V = x^2 \left((675 - x^2)/(4x)\right) \]`

Simplify the expression for V :

`\[ V = (675x - x^3)/(4) \]`

To find the maximum volume, take the derivative of V with respect to x and set it equal to zero:

`\[ 675 - 3x^2 = 0 \]`

Solving for x:

`\[ x^2 = (675)/(3) \]`

`\[ x^2 = 225 \]`

`\[ x = 15 \]`

Now, substitute x = 15 back into the expression for h :

`\[ h = (675 - 15^2)/(4 * 15)`=
`(225)/(4) \]`

Now, find the volume V with x = 15 and
`\( h = (225)/(4) \):`

`\[ V = (675 * 15 - 15^3)/(4)`=
`(675 * 15 - 3375)/(4)`=
`(5062.5)/(4) \]`

Therefore, the largest possible volume of the box is
`\( (5062.5)/(4)`=
`1265.625 \)`cm³. The exact answer is
`\( (5062.5)/(4) \)` cm³.