Find scale factor that compares new surface are to original box and scale factor used to reduce original packaging

Question

A company that makes cookies and crackers wants to reduce there packaging by 25% l. The products are sold in a box shaped like a rectangular prism measuring 25cm by 20cm by 12cm.

A) what scale factor compares the surface area of the new box to the surface are of the original packaging?

B) what scale factor would be used to reduce the measurements of the original packaging?

Answer 1

The term scale factor implies that the schematic for the current packaging is going to be scaled down while keeping all aspect ratios constant like a vector graphic. The formula for the surface area of a rectangular prism is 

 

A = 2A_B + 2A_sl + 2A_sw

 

A_B = area of base

A_sl = area of side attached to base length

A_sw = area of side attached to base width

 

The correct area scaling factor to achieve a 25% reduction in packaging is 3/4 since this is 1 less 25%. Applying this scaling factor to the area formula:

 

3/4A = 3/4(2A_B + 2A_sl + 2A_sw) = 2(3/4A_B) + 2(3/4A_sl) + 2(3/4A_sw)

 

Since A_B, A_sl, and A_sw are each the product of two measurements of the box, to produce the appropriate area scaling factor of 3/4 each measurement must be scaled by sqrt(3/4). To check:

 

A_B-new = (sqrt(3/4)·l)(sqrt(3/4)·w) = 3/4(l·w) = 3/4(A_B-old)