a cereal maker is currently packaging cereal in cylindrical boxes with a 6" diameter and a height of 10". The cereal maker is considering changing to rectangular boxes. The two boxes need to have the same height to fit on the grocery store shelf and they need to have the same volume.

v = pi * 36 * 10 = 360pi = 1131 in^3

Now you need a rectangular box 10" high with that volume. So, the base needs to have area 113.1 in^2

The rectangle using least material for an area is a square, so the box using least material will have a square base 10.6" square and 10" height

To find the volume of a cylinder, you can use the formula V = πr^2h, where V represents the volume, r represents the radius, and h represents the height.

In this case, the current cylinder has a diameter of 6", so the radius (r) is half of the diameter, which is 3". The height (h) is 10". To find the volume of the current cylinder, you can substitute these values into the formula:

V_cylinder = π(3)^2(10)
= π(9)(10)
= 90π cubic inches

Now, since the rectangular box needs to have the same height, let's denote it as h'.

To find the dimensions of the rectangular box, we need to set up an equation by equating the volume of the rectangular box to the volume of the cylinder, which we already calculated: 90π cubic inches.

The volume of a rectangular box is given by V_box = lwh, where l represents the length and w represents the width. Since the height of the rectangular box is the same as that of the cylinder, we have:

V_cylinder = V_box
90π = lwh'

We also know that the diameter of the cylinder is 6", which means that the width (w) of the rectangular box should be equal to the diameter of the cylinder (since the diameter across the center of a circle is the same as the width of a rectangle with the same height).

Therefore, w = 6".

Now we can rewrite the equation as:

90π = l(6)(h')

To find the length (l), we need to rearrange the equation:

l = (90π) / (6h')

So, by knowing the height (h') of the rectangular box, we can calculate the corresponding length (l) to maintain the same volume.

Please note that the width of the rectangular box is given as 6" based on the diameter of the cylinder.