The fact that the two boxes are similar in shape means that their lengths will vary proportionately.
Volumes will vary by the factor of proportionality cubed. So, if the volume of the water box is 3/4 the volume of the food box, the ratio of proportionality between the lengths of of the sides will mean that the water box is 3√(3/4) times the length of the food box.
Lw = ³√(3/4) Lf
The ratio of the surface areas will be the square of the ratio of the lengths,
so the surface area of the water box is 1/[³√(3/4)]² of the food box
SAw/[³√(3/4)]2 = SAf
628cm2/0.8255 = SAf
760.75cm2 = SAf
I misread the question when I first attempted to answer and thought the 628cm2 was the surface area of the food container and solved for the surface area of the water container.
SAw = [³√(3/4)]² SAf
= [³√(3/4)]²(628cm²)
= (0.9086)²(628cm²)
= 0.8255 (628cm²)
= 518.40 cm²
Elwyn D.
tutor
To keep it clear, lets discuss simple square and cubes first.
The ratio of proportionality between two similar objects is the ratio between their linear dimensions: a = kb
If that figure is squared, the ratio of proportionality is similarly squared a2 = (kb)2 = k2b2 for the area
If that figure is cubed, the ratio of proportionality is similarly cubed a3 = (kb)3 = k3b3 for the volume
In this problem, we started with a ratio in volume of 3/4, this is the cube of ratio of proportionality.
So, k = ³√(3/4) = 0.9086 which is the linear ratio.
The area ratio will be k2 = (0.9086)2 = 0.8255
Further example
If I double the edges of the cube (k=2),
the surface area will increase by a factor of 4 (k2 = 22 = 4), and
the volume will increase by a factor of (k3= 23 = 8)
So, if someone had told me the volume of the food container was 8 times that of the water container, I would have concluded that the linear ratio of proportionality was 2, and that the ratio of surfaces areas was 22 = 4
This explanation was written for simple squares and cubes, but it applies to any two or three dimensional object.
la = klb, wa = kwb, ha = khb, ra = krb
Areas of Rectangles
lawa =k2lbwb
Volumes of Rectangular prisms
lawaha = k3lbwbhb
Areas of Circles
πra2 = πk2rb2
Volumes of Spheres
(4/3)πra3 = (4/3)πk3rb3
and rectangular prisms
la = klb, wa = kwb, ha = khb ⇒ lalblc =k3lbwbhb
Report
03/21/16
Elwyn D.
tutor
I just provided an extensive response to your question and it has not posted. Forgive me but I do not have the energy to do it again. I hope it does eventually post.
If the linear ratio of proportionality between two similar objects is k, then the ratio of proportionality between their areas will be k2 and the ratio of proportionality between their volumes will be k3.
This problem gave us k3 = (3/4) to start.
The cube root gave us k.
k2 gave us the ratio of their areas.
Report
03/21/16
Elwyn D.
tutor
Aha! I read it wrong. We were given the smaller surface area. Correction posted above.
Report
03/21/16
AntoriaGirl B.
03/21/16