Question

crackers a box of crackers has a volume of 48 cubic inches. what is the volume of a similar box that is smaller by a scale factor of \\(\frac{2}{3}\\)?

Explanation:

Step1: Recall volume scale factor rule

For similar solids, the ratio of volumes is the cube of the scale factor. The scale factor here is $\frac{2}{3}$, so the volume scale factor is $(\frac{2}{3})^3$.

Step2: Calculate volume scale factor

$(\frac{2}{3})^3=\frac{2^3}{3^3}=\frac{8}{27}$.

Step3: Find the new volume

Multiply the original volume (48 cubic inches) by the volume scale factor. So new volume $= 48\times\frac{8}{27}$. Simplify: $48\div27=\frac{16}{9}$, then $\frac{16}{9}\times8=\frac{128}{9}\approx14.22$ (or keep as fraction $\frac{128}{9}$). Wait, wait, no: Wait, 48 times 8 is 384, 384 divided by 27: 384÷3=128, 27÷3=9. So $\frac{128}{9}$ cubic inches or approximately 14.22 cubic inches. Wait, let's check again. Original volume V1 = 48, scale factor k = 2/3. Volume of similar solid V2 = V1 k³. So V2 = 48 (2/3)³ = 48 8/27 = (48/27)8 = (16/9)*8 = 128/9 ≈14.22. Yes.

Answer:

The volume of the smaller box is $\frac{128}{9}$ cubic inches (or approximately 14.22 cubic inches).