Answer :
To determine the change in scale factor for the longer pool Sienna is building, let's start by understanding the initial and final conditions for the pool.
### Initial Scale Factor
1. Sienna initially has a pool that is 22 feet long.
2. According to the problem statement, the initial scale factor is given in the options as [tex]$\frac{1 \text{ in}}{3 \text{ ft}}$[/tex].
### Final Scale Factor
1. Sienna decides to extend the length of the pool to 33 feet.
### Calculation of Final Scale Factor
To find the new scale factor appropriate for the 33 feet long pool, we need to adjust the initial scale factor proportionally to account for the increase in pool length.
- The initial pool length is 22 feet.
- The final pool length is 33 feet.
- The relationship between the final and the initial length can be written as:
[tex]\[ \frac{\text{final length}}{\text{initial length}} = \frac{33 \text{ ft}}{22 \text{ ft}} \][/tex]
Simplify this fraction to find the scaling factor from the initial length to the final length:
[tex]\[ \frac{33}{22} = \frac{3}{2} \][/tex]
### Final Scale Factor Adjustment
- Given the initial scale factor is [tex]$\frac{1 \text{ in}}{3 \text{ ft}}$[/tex], to find the new scale factor, we multiply this by the ratio of the new length to the old length, which is [tex]$\frac{3}{2}$[/tex]:
[tex]\[ \text{New Scale Factor} = \text{Initial Scale Factor} \times \frac{3}{2} \][/tex]
[tex]\[ \text{New Scale Factor} = \left(\frac{1 \text{ in}}{3 \text{ ft}}\right) \times \frac{3}{2} \][/tex]
[tex]\[ \text{New Scale Factor} = \frac{1 \text{ in}}{2 \text{ ft}} \][/tex]
Therefore, the change involves adjusting the initial scale factor by the ratio of the final length to the initial length, expressed as a fraction [tex]$\frac{3}{2}$[/tex]. This process transforms the initial scale factor correctly to accommodate the new dimensions of the pool.
### Answer
The expression that encompasses this change in scale, showing how the scale factor needs to be adjusted, is:
[tex]\[ \boxed{\frac{3 \text{ ft}}{2 \text{ ft}}} \][/tex]
This expresses the factor that the initial scale needs to be multiplied by to achieve the new scale for the 33 feet long pool.
### Initial Scale Factor
1. Sienna initially has a pool that is 22 feet long.
2. According to the problem statement, the initial scale factor is given in the options as [tex]$\frac{1 \text{ in}}{3 \text{ ft}}$[/tex].
### Final Scale Factor
1. Sienna decides to extend the length of the pool to 33 feet.
### Calculation of Final Scale Factor
To find the new scale factor appropriate for the 33 feet long pool, we need to adjust the initial scale factor proportionally to account for the increase in pool length.
- The initial pool length is 22 feet.
- The final pool length is 33 feet.
- The relationship between the final and the initial length can be written as:
[tex]\[ \frac{\text{final length}}{\text{initial length}} = \frac{33 \text{ ft}}{22 \text{ ft}} \][/tex]
Simplify this fraction to find the scaling factor from the initial length to the final length:
[tex]\[ \frac{33}{22} = \frac{3}{2} \][/tex]
### Final Scale Factor Adjustment
- Given the initial scale factor is [tex]$\frac{1 \text{ in}}{3 \text{ ft}}$[/tex], to find the new scale factor, we multiply this by the ratio of the new length to the old length, which is [tex]$\frac{3}{2}$[/tex]:
[tex]\[ \text{New Scale Factor} = \text{Initial Scale Factor} \times \frac{3}{2} \][/tex]
[tex]\[ \text{New Scale Factor} = \left(\frac{1 \text{ in}}{3 \text{ ft}}\right) \times \frac{3}{2} \][/tex]
[tex]\[ \text{New Scale Factor} = \frac{1 \text{ in}}{2 \text{ ft}} \][/tex]
Therefore, the change involves adjusting the initial scale factor by the ratio of the final length to the initial length, expressed as a fraction [tex]$\frac{3}{2}$[/tex]. This process transforms the initial scale factor correctly to accommodate the new dimensions of the pool.
### Answer
The expression that encompasses this change in scale, showing how the scale factor needs to be adjusted, is:
[tex]\[ \boxed{\frac{3 \text{ ft}}{2 \text{ ft}}} \][/tex]
This expresses the factor that the initial scale needs to be multiplied by to achieve the new scale for the 33 feet long pool.
