Answer :
To solve this problem, you need to calculate the number of lily pads in a pond starting with 22 lily pads and growing at a rate of 17% per week after 7 weeks. The compound growth can be calculated using the formula:
[tex]\[ y = a \cdot b^x \][/tex]
where:
- [tex]\( y \)[/tex] is the final number of lily pads,
- [tex]\( a \)[/tex] is the initial number of lily pads,
- [tex]\( b \)[/tex] is the growth factor, and
- [tex]\( x \)[/tex] is the number of time periods (weeks).
### Step-by-Step Solution
1. Identify the Given Values:
- Initial number of lily pads, [tex]\( a \)[/tex]: 22
- Growth rate per week: 17%, which can be written as a decimal 0.17
- Number of weeks, [tex]\( x \)[/tex]: 7
2. Calculate the Growth Factor, [tex]\( b \)[/tex]:
- Since the growth rate is 17%, the growth factor [tex]\( b \)[/tex] is [tex]\( 1 + 0.17 = 1.17 \)[/tex]
3. Substitute the Values into the Formula:
- Using the formula [tex]\( y = 22 \cdot 1.17^7 \)[/tex]
4. Calculate the Final Number of Lily Pads:
- First, calculate [tex]\( 1.17^7 \)[/tex].
- Then multiply the initial number of lily pads by this result:
[tex]\[ y = 22 \cdot 1.17^7 \approx 22 \cdot 3.00124 \approx 66.027 \][/tex]
5. Round the Result to the Nearest Whole Number:
[tex]\[ y \approx 66 \][/tex]
Therefore, the number of lily pads after 7 weeks will be approximately 66 when rounded to the nearest whole number.
### Final Answer:
66 lily pads
[tex]\[ y = a \cdot b^x \][/tex]
where:
- [tex]\( y \)[/tex] is the final number of lily pads,
- [tex]\( a \)[/tex] is the initial number of lily pads,
- [tex]\( b \)[/tex] is the growth factor, and
- [tex]\( x \)[/tex] is the number of time periods (weeks).
### Step-by-Step Solution
1. Identify the Given Values:
- Initial number of lily pads, [tex]\( a \)[/tex]: 22
- Growth rate per week: 17%, which can be written as a decimal 0.17
- Number of weeks, [tex]\( x \)[/tex]: 7
2. Calculate the Growth Factor, [tex]\( b \)[/tex]:
- Since the growth rate is 17%, the growth factor [tex]\( b \)[/tex] is [tex]\( 1 + 0.17 = 1.17 \)[/tex]
3. Substitute the Values into the Formula:
- Using the formula [tex]\( y = 22 \cdot 1.17^7 \)[/tex]
4. Calculate the Final Number of Lily Pads:
- First, calculate [tex]\( 1.17^7 \)[/tex].
- Then multiply the initial number of lily pads by this result:
[tex]\[ y = 22 \cdot 1.17^7 \approx 22 \cdot 3.00124 \approx 66.027 \][/tex]
5. Round the Result to the Nearest Whole Number:
[tex]\[ y \approx 66 \][/tex]
Therefore, the number of lily pads after 7 weeks will be approximately 66 when rounded to the nearest whole number.
### Final Answer:
66 lily pads
