Elwyn D. answered • 03/21/16

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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.L

_{w }= ³√(3/4) L_{f}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}= SA_{f}628cm

^{2}/0.8255 = SA_{f}760.75cm

^{2 }= SA_{f}*I misread the question when I first attempted to answer and thought the 628cm*

^{2}was the surface area of the food container and solved for the surface area of the water container.*SA*

_{w }= [³√(3/4)]² SA_{f }*= [³√(3/4)]²(628cm²)*

*= (0.9086)²(628cm²)*

*= 0.8255 (628cm²)*

*= 518.40 cm²*

Elwyn D.

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 a

^{2}= (kb)^{2 }= k^{2}b^{2}for the areaIf that figure is cubed, the ratio of proportionality is similarly cubed a

^{3}= (kb)^{3}= k^{3}b^{3 }for the volumeIn 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 k

^{2}= (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 (k*

^{2 }= 2^{2}= 4), and*the volume will increase by a factor of (k*

^{3}= 2^{3}= 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 2*

^{2}= 4This explanation was written for simple squares and cubes, but it applies to any two or three dimensional object.

l

_{a}= kl_{b}, w_{a}= kw_{b}, h_{a}= kh_{b},_{ }r_{a }= kr_{b}Areas of Rectangles

l

_{a}w_{a}=k^{2}l_{b}w_{b}Volumes of Rectangular prisms

l

_{a}w_{a}h_{a}= k^{3}l_{b}w_{b}h_{b}Areas of Circles

πr

_{a}^{2 }= πk^{2}r_{b}^{2}Volumes of Spheres

(4/3)πr

_{a}^{3}= (4/3)πk^{3}r_{b}^{3}and rectangular prisms

l

_{a }= kl_{b, }w_{a }= kw_{b}, h_{a }= kh_{b }⇒ l_{a}l_{b}l_{c}=k^{3}l_{b}w_{b}h_{b}
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03/21/16

Elwyn D.

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 k

^{2 }and the ratio of proportionality between their volumes will be k^{3}.This problem gave us k

^{3}= (3/4) to start.The cube root gave us k.

k

^{2}gave us the ratio of their areas.
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03/21/16

Elwyn D.

Aha! I read it wrong. We were given the smaller surface area. Correction posted above.

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03/21/16

AntoriaGirl B.

03/21/16