a company packages snack mix in cylindrical tubes.each tube is made from rectangle of cardboard.the bases of the cylinder are plastic.the cardboard comes in.8.5 inches by 11 inches sheets.to hold the greatest amount of mix should the lo side or the shorter side be the height? justify the answer
The longer side should be the height. This will keep the plastic necessary for the base smaller than if the width of cardboard were used.
Sorry, Ms. Sue but your incorrect, the height should be the short side because yield a volume of 81.8 in^3 and if the long side is the height the volume in 63.3 in^3
To hold the greatest amount of mix, the longer side of the cardboard sheet should be the height of the cylinder.
Let's consider the dimensions of the cardboard sheet. The longer side measures 11 inches, and the shorter side measures 8.5 inches.
When the longer side is the height, it allows for a larger circumference of the cylinder. The circumference is given by the formula C = 2πr, where r is the radius of the cylinder. In this case, the longer side of the rectangle will have a greater circumference than the shorter side.
Since the amount of snack mix that can be packed into the cylinder is directly proportional to its circumference, having a larger circumference means a greater amount of snack mix can be held. Therefore, by making the longer side the height of the cylinder, the company can package the greatest amount of snack mix.
In summary, to maximize the amount of snack mix held in the cylindrical tube, the longer side of the cardboard sheet should be the height of the cylinder.
To determine whether the longer side or the shorter side of the cardboard sheet should be the height of the cylinder, we need to consider the volume of the cylinder.
The volume of a cylinder can be calculated using the formula: V = πr^2h, where V represents the volume, r represents the radius of the base, and h represents the height.
Since the bases of the cylinder are plastic and come in rectangular shape, we can assume that the radius of the base (r) will be half the length of one side of the rectangular sheet. Therefore, r = 8.5 inches / 2 = 4.25 inches.
Now, we need to compare the effects of using the longer side and the shorter side as the height (h) of the cylinder.
1. Longer side as the height:
- If we choose the longer side (11 inches) as the height, the volume of the cylinder would be V = π(4.25 inches)^2(11 inches) = 607.4π cubic inches.
2. Shorter side as the height:
- If we choose the shorter side (8.5 inches) as the height, the volume of the cylinder would be V = π(4.25 inches)^2(8.5 inches) = 363.7π cubic inches.
By comparing the two volume calculations, we find that using the longer side as the height results in a greater volume of snack mix that can be held in the cylinder. Specifically, 607.4π cubic inches compared to 363.7π cubic inches.
Therefore, to hold the greatest amount of mix, the longer side should be selected as the height of the cylinder.