Question

3 a square cake measures 6 units by 6 units by 4 units. the cake falls into a vat of frosting and comes out frosted on all six faces. the cake is then cut up into cubes, each measuring 1 unit on an edge. how many of these cubes will have exactly one face frosted?

Explanation:

Step1: Identify the formula for one - face - frosted cubes

For a rectangular prism of dimensions $l\times w\times h$ cut into unit - cubes, the number of one - face - frosted cubes is given by $2[(l - 2)(w - 2)+(l - 2)(h - 2)+(w - 2)(h - 2)]$. Here, $l = 6$, $w = 6$, and $h = 4$.

Step2: Calculate $(l - 2)(w - 2)$

$l-2=6 - 2=4$, $w - 2=6 - 2=4$, so $(l - 2)(w - 2)=4\times4 = 16$.

Step3: Calculate $(l - 2)(h - 2)$

$l-2=6 - 2=4$, $h - 2=4 - 2=2$, so $(l - 2)(h - 2)=4\times2 = 8$.

Step4: Calculate $(w - 2)(h - 2)$

$w-2=6 - 2=4$, $h - 2=4 - 2=2$, so $(w - 2)(h - 2)=4\times2 = 8$.

Step5: Calculate the total number of one - face - frosted cubes

$2[(l - 2)(w - 2)+(l - 2)(h - 2)+(w - 2)(h - 2)]=2(16 + 8+8)=2\times32 = 64$.

Answer:

64