To solve for the proportion of drinks that cost between [tex]$3.50 and $[/tex]4.00 under a uniform density curve, we can follow these steps:
1. Understand the Problem:
We need to determine the proportion of drinks priced between [tex]$3.50 and $[/tex]4.00. We are given four possible choices for the proportion:
- [tex]\( \frac{1}{30} \)[/tex]
- [tex]\( \frac{1}{6} \)[/tex]
- [tex]\( \frac{1}{3} \)[/tex]
- [tex]\( \frac{1}{2} \)[/tex]
2. Identify the Range:
The range of prices we are considering is from [tex]$3.50 to $[/tex]4.00.
3. Calculate the Interval Length:
The interval length can be calculated by subtracting the lower bound of the interval from the upper bound:
[tex]\[
4.00 - 3.50 = 0.50
\][/tex]
4. Check the Proportion:
Out of the given choices, we need to determine which fraction matches the length of the interval, which in this case is [tex]$0.50$[/tex].
- [tex]\( \frac{1}{30} \approx 0.0333 \)[/tex]
- [tex]\( \frac{1}{6} \approx 0.1667 \)[/tex]
- [tex]\( \frac{1}{3} \approx 0.3333 \)[/tex]
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
By comparing the values, we find that:
[tex]\[
0.5 = 0.50
\][/tex]
Therefore, the correct proportion is [tex]\( \frac{1}{2} \)[/tex].
So, the proportion of drinks costing between [tex]$3.50 and $[/tex]4.00 is [tex]\( \frac{1}{2} \)[/tex]. The correct choice is:
[tex]\[
\frac{1}{2}
\][/tex]
