Question

If two solids are similar and the ratio of their areas are 36:49 ft2, what is their scale factor?

Answer 1

\(\bf 36:49\qquad \textit{scale factor}\implies \cfrac{36}{49}\)

Answer 2

shouldnt i be taking the square root or something

Answer 3

The "scale factor" is the coefficient by which each "side" was scaled. That is, if we were describing a rectangle, we would multiply both it's width and it's length by the same number.

Now, using a rectangle as an example, let's see what would happen to the area of the rectangle if we did this.

A = l * w

A is the area, l is length, w is width.
Now, multiply length by "a," and width by "a" also:

A = a*l*a*w = (a^2)*l*w

So the scaling factor "a" shows up as a square in the area. (It can be shown that it's true in general).

All this to say that you are correct. You should take the square root.

Answer 4

could you please show me an example with a sphere

Answer 5

Sure, with surface areas on a sphere:

\[A_{sphere} = 4 \times \pi \times r^2\]
We multiply the radius by a scaling factor "a" to get a new area.
\[A_{scaled} = 4 \times \pi \times (ra)^2\]
Taking a ratio, we see:
\[\frac{A_{scaled}}{A_{sphere}}=\frac{4 \times \pi \times a^2r^2}{4 \times \pi \times r^2} = a^2\]

Answer 6

hmmm

Answer 7

ohhh... ahemm yes, you should.... .so \(\bf \cfrac{area1}{area2}=\left(\cfrac{side1}{side2}\right)^2\implies \cfrac{area1}{area2}=\cfrac{side1^2}{side2^2}
\\ \quad \\
ratio\implies \cfrac{side1}{side2}\qquad or\qquad \cfrac{\sqrt{area1}}{\sqrt{area2}}\)

Answer 8

\(\bf 36:49\qquad ratio1:ratio2\qquad \cfrac{36}{49}=\left(\cfrac{ratio1}{ratio2}\right)^2
\\ \quad \\
\sqrt{\cfrac{36}{49}}=\cfrac{ratio1}{ratio2}\)

Answer 9

recall \(\bf \sqrt{\cfrac{36}{49}}\implies \cfrac{\sqrt{36}}{\sqrt{49}}\)

Answer 10

im still confused.
so for problem: similar sphere volumes with ratio 27:64 in^3,
how do i work out the scale factor from there.

Answer 11

Volumes go as the cube of the radius.

\[V_{sphere} = \frac{4}{3} \times \pi \times r^3\]

So, how would a number multiplying the radius effect the volume?

Answer 12

well, the idea is that they're similar
\( \bf \cfrac{volume1}{volume2}=\left(\cfrac{ratio1}{ratio2}\right)^{\color{red}{ 3}}\implies{\Large \sqrt[{\color{red}{ 3}}]{\cfrac{volume1}{volume2}}}=\cfrac{ratio1}{ratio2}\)

Answer 13

keep in mind that an Area has 2 units "encapsulated" into it
and the Volume has 3 units "encapsulated" into it

the ratio is a flat single unit, so to equate it about, for 2 "encapsulated units" raise by 2
when 3 "encapsulated units" raise by 3