Question

What is the surface area of two rectangular prisms with a similarity ratio of 2:3?

Answer 1

Similar solutions to your answer, may help

if the ratio of surface areas of prism A and prism B is 3/7, then the ratio of surface areas in terms of the lengths, widths and heights of prims a and b is:

[2(la*wa)+2(wa*ha)+2(ha*la)] / [2(lb*wb)+2(wb*hb)+2(hb*lb)] = 3/7
or,
[(la*wa)+(wa*ha)+(ha*la)] / [(lb*wb)+(wb*hb)+(hb*lb)] = 3/7

since the prisms are similar, we can substitute la=x*lb, wa=x*lb, and ha=x*hb, where "x" is some scaling factor

[(la*wa)+(wa*ha)+(ha*la)] / [(la*wa)*x^2+(wa*ha)*x^2+(ha*la)*x^2] = 3/7

1/x^2 = 3/7, or x = sqrt(7/3)

finally, the ratio of the volumes is:

[la*wa*ha] / [lb*wb*hb]
if we use our substitution,
= [la*wa*ha] / [la*x*wa*x*hb*x]
= 1/x^3
= (3/7)*sqrt(3/7)