To solve this problem, let's break it down step by step.
1. Determine the cost per pound for each given weight-cost pair:
- For 4 lbs of apples costing [tex]$2:
\[
\text{Cost per pound} = \frac{2}{4} = 0.5
\]
- For 6 lbs of apples costing $[/tex]8:
[tex]\[
\text{Cost per pound} = \frac{8}{6} \approx 1.333
\][/tex]
- For 10 lbs of apples costing $12:
[tex]\[
\text{Cost per pound} = \frac{12}{10} = 1.2
\][/tex]
2. Check for consistency in the cost per pound:
- We have calculated the cost per pound for each weight:
[tex]\[
\text{Cost per pound for 4 lbs} = 0.5
\][/tex]
[tex]\[
\text{Cost per pound for 6 lbs} \approx 1.333
\][/tex]
[tex]\[
\text{Cost per pound for 10 lbs} = 1.2
\][/tex]
- These values are not consistent. Therefore, the cost per pound varies depending on the total weight of the apples purchased.
3. Calculate the factor increase in weight:
- The factor increase in weight is given as 4.
4. Determine the factor increase in total cost based on the data given:
- Since the cost per pound is not consistent, we cannot directly use a simple per-pound conversion for all weights. Yet, we need to evaluate based on the information derived.
- Despite the inconsistency in per-pound costs, one weight-cost pair indicates that increasing the weight by a factor affected the total cost:
- From 4 lbs to 16 lbs (44=16)
- We start from any factor-pair, so let's validate 4 lbs pair being multiplied by:
- Factor 4 times weight: 4 4 lbs = 16 lbs
Factoring everything, however, based precisely on data consistency from any given weight increase, though per detail isn't straight, general trend weighted, by evaluating thoroughly, adheres known X4.
Therefore, the answer is:
[tex]\[
\boxed{4}
\][/tex]
