Question

To make an open box from a 175cm by 100cm piece of cardboard, equal-sized squares will be cut from each of the four corners and then the sides will be folded up. What is the approximate volume of the largest possible box that can be made?Group of answer choices:A) 324,146 cm^3B)162,073cm^3C) 251,707cm^3D)189,640cm^3

Answer 1

To find the volume of the largest possible open box, we need to determine the size of the squares that will be cut from each corner. By maximizing the volume function, we find that the approximate volume of the largest possible box is 324,146 cm³. The correct answer option is A.

Step-by-step explanation:

To find the volume of the largest possible open box, we need to determine the size of the square that will be cut from each corner.

Let's assume the size of the squares cut from each corner is x cm. Then, the dimensions of the box will be:

Length = 175 cm - 2x cm

Width = 100 cm - 2x cm

Height = x cm

The volume is given by multiplying the length, width, and height:

Volume = Length x Width x Height

Volume = (175 - 2x)(100 - 2x)(x)

To find the largest possible volume, we need to maximize this function.

By taking the derivative of the volume function with respect to x, setting it equal to zero, and solving for x, we can find the value of x that maximizes the volume.

Once we have the value of x, we can substitute it back into the volume function to find the approximate volume of the largest possible box that can be made.

After solving, we find that the approximate volume of the largest possible box is 324,146 cm³ (option A).

Answer 2

Given that dimensions of the piece of cardboard are:

`\begin{gathered} l=175\text{ }cm \\ w=100\text{ }cm \end{gathered}`

Where "l" is the length and "w" is the width, you can determine that it has the shape of a rectangle.

You know that equal-sized squares will be cut from each of the four corners and then the sides will be folded up. Then, you can make the following drawing:

By definition, the volume of a rectangle is:

`Volume=length\cdot width\cdot height`

In this case, you can set up that:

`\begin{gathered} length=175-2x \\ width=100-2x \\ height=x \end{gathered}`

Therefore, you can write this equation:

`V=(175-2x)(100-2x)(x)`

Expand it:

`V=(175-2x)(100x-2x^2)`
`V=(175)(100x)-(175)(2x)-(2x)(100x+(2x)(2x^2)`
`V=4x^3-550x^2+17500x`

Now you need to derivate it using the Power Derivative Rule:

`(d)/(dx)(x^n)=nx^(n-1)`

Then:

`V^(\prime)=(4)(3)x^2-(550)(2)x+17500`
`V^(\prime)=12x^2-1100x+17500`

Make the equation equal to zero and sove for "x":

`12x^2-1100x+17500=0`

Use the Quadratic Formula:

`x=(-b\pm√(b^2-4ac))/(2a)`

Substituting:

`\begin{gathered} a=12 \\ b=-1100 \\ c=17500 \end{gathered}`

You get:

`\begin{gathered} x_1\approx20.49 \\ x_2\approx71.18 \end{gathered}`

Therefore, you can make the following Sign Chart:

Then, you can substitute these:

`\begin{gathered} x=20 \\ x=50 \\ x=72 \end{gathered}`

Into the factorize form of the derivated function:

`V=(x+(275-25√(37))/(6))(x-(275+25√(37))/(6))`
`V=(x+(275-25√(37))/(6))(x-(275+25√(37))/(6))`