Question

amazon box problem
a small amazon box has a length of 10 cm, width of 10 cm, and height of 10 cm, so it has a volume of 1,000 cm³.
jeff bezos wants a large sized amazon box to have a volume of 2,000 cm³, so he doubles the length, width, and height. is he correct in doing this? explain your reasoning.
what should the length, width, and height be to have a cube thats 2,000 cm³? round to the nearest hundredths if necessary. hint find the scale factor between the small and large box.

Explanation:

Step1: Calculate small box volume

$V_{small} = 10 \times 10 \times 10 = 1000\ \text{cm}^3$

Step2: Calculate volume of doubled box

New dimensions: $20\ \text{cm}, 20\ \text{cm}, 20\ \text{cm}$
$V_{doubled} = 20 \times 20 \times 20 = 8000\ \text{cm}^3$

Step3: Compare to target volume

Target volume is $2000\ \text{cm}^3$; $8000
eq 2000$, so Jeff is incorrect.

Step4: Find scale factor $k$

Volume scales with $k^3$, so $k^3 = \frac{V_{large}}{V_{small}} = \frac{2000}{1000} = 2$
Solve for $k$: $k = \sqrt[3]{2} \approx 1.26$

Answer:

1. Jeff is not correct. When he doubles all dimensions, the volume of the box becomes $8000\ \text{cm}^3$, which is 8 times the original volume, not the desired $2000\ \text{cm}^3$.
2. The scale factor for length, width, and height should be approximately 1.26.