Answer :
To solve this problem, we need to determine which exponential function best models the relationship between weeks and the remaining weight of the radioactive material. Let's go through the steps to understand which function fits the given data.
1. Understand the Exponential Function:
An exponential decay function generally looks like this: [tex]\( y = a(b)^x \)[/tex], where:
- [tex]\( y \)[/tex] is the amount of material left.
- [tex]\( a \)[/tex] is the initial amount.
- [tex]\( b \)[/tex] is the decay factor (between 0 and 1 for decay).
- [tex]\( x \)[/tex] is the time variable, in weeks in this case.
2. Identify the Starting Amount:
From the table, in week 0, the weight is 100 grams. Therefore, [tex]\( a = 100 \)[/tex].
3. Find the Decay Factor ([tex]\(b\)[/tex]):
To find the decay factor [tex]\( b \)[/tex], we need to estimate how much the quantity decreases each week. We do this by analyzing the ratio of the weights from week to week.
Let's explore the given data. The weights are: 100, 883, 759, 694, 59.1, 51.8, 45.5. Generally, when identifying decay, these numbers should consistently decrease, indicating a consistent decay ratio over the period.
4. Choose the Appropriate Function:
We compare the options and try to see which multiplication factor matches the observed decay rate. The four proposed functions have decay factors of 0.13, 0.87, 1.87, and 8.7.
5. Determine the Closest Option:
By comparing the decay factor derived from calculations to the given options, we find that the exponential function that most closely resembles the rate of change in the data is [tex]\( y = 100(1.87)^x \)[/tex].
Thus, the correct exponential function that models this data is:
[tex]\[ y = 100(1.87)^x \][/tex]
This function best represents how the sample's weight changes over time according to the data collected.
1. Understand the Exponential Function:
An exponential decay function generally looks like this: [tex]\( y = a(b)^x \)[/tex], where:
- [tex]\( y \)[/tex] is the amount of material left.
- [tex]\( a \)[/tex] is the initial amount.
- [tex]\( b \)[/tex] is the decay factor (between 0 and 1 for decay).
- [tex]\( x \)[/tex] is the time variable, in weeks in this case.
2. Identify the Starting Amount:
From the table, in week 0, the weight is 100 grams. Therefore, [tex]\( a = 100 \)[/tex].
3. Find the Decay Factor ([tex]\(b\)[/tex]):
To find the decay factor [tex]\( b \)[/tex], we need to estimate how much the quantity decreases each week. We do this by analyzing the ratio of the weights from week to week.
Let's explore the given data. The weights are: 100, 883, 759, 694, 59.1, 51.8, 45.5. Generally, when identifying decay, these numbers should consistently decrease, indicating a consistent decay ratio over the period.
4. Choose the Appropriate Function:
We compare the options and try to see which multiplication factor matches the observed decay rate. The four proposed functions have decay factors of 0.13, 0.87, 1.87, and 8.7.
5. Determine the Closest Option:
By comparing the decay factor derived from calculations to the given options, we find that the exponential function that most closely resembles the rate of change in the data is [tex]\( y = 100(1.87)^x \)[/tex].
Thus, the correct exponential function that models this data is:
[tex]\[ y = 100(1.87)^x \][/tex]
This function best represents how the sample's weight changes over time according to the data collected.
