Question

Thomas buys a cardboard sheet that is 8 by 12 inches. Let x be the side length of each cutout. Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function. What is the size of the cutout he needs to make so that he can fit the most marbles in the box? If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be? What would be the dimensions of this box? Using complete sentences, explain the connection between the cutout and the volume of the box. Design an equation that would work for any cardboard sheet length, q, and width, p

Answer 1

Answer:

Thomas buys a cardboard sheet that is 8 by 12 inches.

Let x be the side length of each cutout. So, from two sides, the cut out is 2x.

Part A:

The volume of the food dish is given by :

$V =(12-2x)(8-2x)(x)$

= $96x -40x^{2}+4x^{3}$

To find the zeros, we equate the above equation to 0

$96x -40x^{2}+4x^{3}=0$ or

$4x^{3} -40x^{2}+96x=0$

Taking out 4 common, we get

$x^{3} -10x^{2}+24x=0$

Solving this we get;

x=0, x=4, x=6

Part B:

Now we will differentiate the equation to get the value of x, and to obtain the maximum capacity. We will further equate that to zero. So, we get the equation as ;

$d/dx =12x^{2}-80x+96=0$

Solving this we get x = 1.57 and x = 5.1

Hence, x = 1.57 will give the maximum capacity.

Part C:

Thomas wants a volume of 12 cubic inches, we will equate to 12.

$12= 4x^{3}-40x^{2}+96x$

=> $4x^{3}-40x^{2}+96x-12=0$

We get,

x = 0.1321

x = 6.217 (neglect biggest value)

x = 3.6502

Part D:

The connection between the cutout and the volume of the box can be described as: When we increase the cutout dimensions of the box, the volume increases up to the measurement of 1.57 inches. More than this the volume starts decreasing.

Part E:

The equation that would work for any cardboard sheet length, q, and width, p is ; $V =(q -2x)(p -2x)(x)$

Answer 2

Answer: Your equation would be y = x(12 - 2x)(8 - 2x).

The volume of a rectangular prism is V = LWH. The initial length and width are given as 12 and 8, however that makes the height 0.

If we cut out the corners, they would be the length of x. When we cut out the corners, we are subtract the length x from both ends of the length and width.

So now our measurements are:
Length: 12 - 2x
Side: 8 - 2x
Height: x

Multiply them altogether and you have your equation. Then, you can use your equation to find any measurements that you need.