Let's go through the problem step-by-step.
For part (a):
Teresa wants to know the range of values that represent a treasure box having an amulet. We know that there is a 58% chance of having an amulet. Since the random number generated will be a whole number between 1 and 100 (inclusive), she can use the range 1 to 58 to represent the amulet. Specifically:
[tex]\[ 1 \leq x \leq 58 \][/tex]
So the range of values that Teresa can use to represent a treasure box having an amulet is:
[tex]\[ \boxed{1 \text{ to } 58} \][/tex]
For part (b):
We are given the random numbers generated for 10 treasure boxes:
[tex]\[ 55, 43, 96, 3, 59, 73, 75, 89, 16, 45 \][/tex]
Using the range determined in part (a) ([tex]\(1 \text{ to } 58\)[/tex]), we need to count how many of these numbers fall within this range.
From the list:
- 55 (falls within the range 1 to 58)
- 43 (falls within the range 1 to 58)
- 96 (does not fall within the range 1 to 58)
- 3 (falls within the range 1 to 58)
- 59 (does not fall within the range 1 to 58)
- 73 (does not fall within the range 1 to 58)
- 75 (does not fall within the range 1 to 58)
- 89 (does not fall within the range 1 to 58)
- 16 (falls within the range 1 to 58)
- 45 (falls within the range 1 to 58)
So, there are 5 numbers (55, 43, 3, 16, 45) that correspond to an amulet.
To calculate the percentage of the 10 treasure boxes that contained an amulet:
[tex]\[ \text{Percentage of amulets} = \left( \frac{\text{Number of amulets}}{\text{Total number of boxes}} \right) \times 100 \][/tex]
Substituting the numbers we have:
[tex]\[ \text{Percentage of amulets} = \left( \frac{5}{10} \right) \times 100 = 50 \% \][/tex]
Therefore, the percentage of the 10 simulated treasure boxes that had an amulet is:
[tex]\[ \boxed{50\%} \][/tex]
