Hi, i am happy to help!
So i think the most confusing part about this question is that it asks the effects of changing one attribute of a rectangular prism but it doesn't tell you which attribute. this leads us to think that perhaps it doesn't matter which attribute changes. let's first look at Surface area which i'll denote as SA. now let the sides of the rectangular prism be a, b, and c. to find the surface area, we must find the area of each of the faces of the rectangular prism and add them together:
SA = ab + bc + ac + ab + bc + ac
SA = 2ab + 2bc + 2ac
SA = 2(ab + bc + ac)
looking closely at this formula we'll notice that the formula is symmetric meaning that we can switch any 2 variables and and obtain the same formula. to put it simply, changing a is the same as changing b which is the same as changing c.
without loss of generality, suppose we change a by a factor of k meaning the sides of the rectangular prism that was length a is now length ka. our new surface area then becomes:
SA(new) = 2((ka)b + bc + (ka)(c)).
SA(new) = 2( ka(b + c) + bc))
honestly, im not sure how to relate this to the original surface area, but im sure your teacher will be ok if you just write the new surface area formula.
now moving on to volume: we know the formula for the volume of a prism is:
V = abc
the symmetry in this is even more obvious and like before, let's change side a by a factor of k. our new volume is:
V(new) = (ka)bc
V(new) = k(abc)
now we note that abc is our original volume so we can write a simple relation between the original volume and new volume as:
V(new) = kV(old) and we are done!
let me know if you have any questions!
