Danny is sitting on a rectangular box. The area of the front face of the box is half the area of the top face, and the area of the top face is 1.5 times the area of the side face. If the volume of the box is 24, what is the area of the side face of the box?
You grossly underestimated the time this question took you. You actually solved it in 10 minutes and 33 seconds.
Applying the relationship of the surface areas of the three sides, you can get the values of w and h in terms of l:
Side face = w x h
Top face = w x l = 1.5 x w x h which implies l = 1.5 h --> h = 2/3 l
Front face = l x h = 1.5/2 x w x h which implies l = 1.5/2 w --> w =4/3 l
Thus, the sides of the box: l, h and w, have the relationship of 1, 2/3 and 4/3 respectively such that the sides of the box are factors of 24. Expand the ratio times 3 to get rid of the fraction: l:h:w have the ratio of 3:2:4. It just so happens that 3×2×4 = 24, so these values also satisfy the question' requirements.
Hence, the area of side face is w x h = 4 x 2 = 8.
Wow, so I understand all the concepts here but I have to say, to be able to understand everything fully, and to do all the calculations required in this question would take a LOT of org and concentration skills not to make an easy error.
Anyone have any special insight on this on how to break it down in a more easier manner?
Lets say, front face is made of l ,b
top face is made of l,w
and side face is made of b,w
front face of the box is half the area of the top face, these faces will share one common side, lb =lw/2 => b=w/2
area of the top face is 1.5 times the area of the side face, lw = 1.5 bw => l=3/2w
We want area of side face, A=bw
We know Volume =lbw or V = A*l
Thus all we need to know is l.
Now, Volume =24
l^3 = 24*9/2^3
l = 3
Also, volume = Area of side face * l
=>A = Volume/l = 24/3 = 8
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