Answer :
Certainly! Let's find the exponential model for the function based on the given data:
### Step-by-Step Solution:
1. Understand the Problem:
You are given a table showing the weight over several weeks, and it seems like the weight is decreasing in a pattern that could be modeled with an exponential function. The goal is to find which of the given options best represents this model.
2. Identify the Pattern:
From the data:
- Week 0: Weight = 100
- Week 1: Weight = 88.3
- Week 2: Weight = 75.9
- Week 3: Weight = 69.4
- Week 4: Weight = 59.1
- Week 5: Weight = 51.8
- Week 6: Weight = 45.5
3. Calculate Ratios Between Consecutive Weeks:
To find if there's an exponential relationship, calculate the ratio between each week's weight and the previous week's weight:
- Ratio from Week 0 to 1: [tex]\( \frac{88.3}{100} \)[/tex]
- Ratio from Week 1 to 2: [tex]\( \frac{75.9}{88.3} \)[/tex]
- Ratio from Week 2 to 3: [tex]\( \frac{69.4}{75.9} \)[/tex]
- Ratio from Week 3 to 4: [tex]\( \frac{59.1}{69.4} \)[/tex]
- Ratio from Week 4 to 5: [tex]\( \frac{51.8}{59.1} \)[/tex]
- Ratio from Week 5 to 6: [tex]\( \frac{45.5}{51.8} \)[/tex]
4. Find the Average Ratio:
- Sum these ratios and divide by the number of calculated ratios to find the average ratio. This average ratio represents the base of the exponential decay function.
5. Match the Model:
- The general form of an exponential model given by the data is [tex]\( y = 100 \cdot (b)^x \)[/tex], where [tex]\( b \)[/tex] is the average ratio.
- From the ratios calculated and averaged, the average ratio is approximately 0.877.
6. Choose the Closest Option:
- The option closest to this calculated base of approximately 0.877 is [tex]\( y = 100 \cdot (0.88)^x \)[/tex].
Therefore, the exponential model that best fits the given data is:
[tex]\[ y = 100 \cdot (0.88)^x \][/tex]
### Step-by-Step Solution:
1. Understand the Problem:
You are given a table showing the weight over several weeks, and it seems like the weight is decreasing in a pattern that could be modeled with an exponential function. The goal is to find which of the given options best represents this model.
2. Identify the Pattern:
From the data:
- Week 0: Weight = 100
- Week 1: Weight = 88.3
- Week 2: Weight = 75.9
- Week 3: Weight = 69.4
- Week 4: Weight = 59.1
- Week 5: Weight = 51.8
- Week 6: Weight = 45.5
3. Calculate Ratios Between Consecutive Weeks:
To find if there's an exponential relationship, calculate the ratio between each week's weight and the previous week's weight:
- Ratio from Week 0 to 1: [tex]\( \frac{88.3}{100} \)[/tex]
- Ratio from Week 1 to 2: [tex]\( \frac{75.9}{88.3} \)[/tex]
- Ratio from Week 2 to 3: [tex]\( \frac{69.4}{75.9} \)[/tex]
- Ratio from Week 3 to 4: [tex]\( \frac{59.1}{69.4} \)[/tex]
- Ratio from Week 4 to 5: [tex]\( \frac{51.8}{59.1} \)[/tex]
- Ratio from Week 5 to 6: [tex]\( \frac{45.5}{51.8} \)[/tex]
4. Find the Average Ratio:
- Sum these ratios and divide by the number of calculated ratios to find the average ratio. This average ratio represents the base of the exponential decay function.
5. Match the Model:
- The general form of an exponential model given by the data is [tex]\( y = 100 \cdot (b)^x \)[/tex], where [tex]\( b \)[/tex] is the average ratio.
- From the ratios calculated and averaged, the average ratio is approximately 0.877.
6. Choose the Closest Option:
- The option closest to this calculated base of approximately 0.877 is [tex]\( y = 100 \cdot (0.88)^x \)[/tex].
Therefore, the exponential model that best fits the given data is:
[tex]\[ y = 100 \cdot (0.88)^x \][/tex]
