Question

A food company is designing boxes for several products. Each box is a rectangular prism. The company makes a single serving cereal box that contains 1. 2 ounces of cereal. They plan to make an extra large box for school cafeterias. The extra large box will be a dilation of the single serving box using a scale factor of 4. How many ounces of cereal will the extra large box contain? Explain or show your reasoning. The company also wants to offer a family size box of cereal. The family size box will be a dilation of the single serving 1. 2-ounce box. It must contain 18. 75 ounces of cereal. What scale factor does the company need to use for the family size box? Explain or show your reasoning. The food company is now designing soup boxes. The largest box of soup will be a dilation of the smallest box using a scale factor of 2. The smallest box must hold 8 fluid ounces or about 15 cubic inches of soup. Find a set of dimensions for the largest box. Round your answers to the nearest tenth if necessary. Explain or show your reasoning

Answer 1

Answer: For the extra large box, the scale factor is 4. The volume of the single serving box is:

V1 = l x w x h = 1.2 cubic inches

The volume of the extra large box, which is a dilation of the single serving box with a scale factor of 4, is:

V2 = (4l) x (4w) x (4h) = 64(l x w x h) = 64V1

So the extra large box will contain:

1.2 x 64 = 76.8 ounces of cereal.

For the family size box, the volume is 18.75 ounces, which is 15.6 times larger than the single serving box. The scale factor is the cube root of this ratio:

scale factor = (18.75/1.2)^(1/3) = 2.5

So the family size box will be a dilation of the single serving box with a scale factor of 2.5.

For the soup boxes, the largest box is a dilation of the smallest box with a scale factor of 2. The volume of the smallest box is 15 cubic inches, so the volume of the largest box is:

V2 = (2l) x (2w) x (2h) = 8(l x w x h) = 8 x 15 = 120 cubic inches.

We can choose any dimensions that satisfy this volume requirement. For example, we could use:

• Length = 10 inches
• Width = 3 inches
• Height = 4 inches

The volume of this box would be:

V = 10 x 3 x 4 = 120 cubic inches

Therefore, a set of dimensions for the largest soup box is 10 inches by 3 inches by 4 inches, or any other dimensions that satisfy the volume requirement of 120 cubic inches.