Answer :
To determine the value of \( d \) for the table to represent exponential decay, we need to examine the pattern of the data and apply the properties of exponential decay.
Exponential decay can be expressed typically as:
[tex]\[ y = a \cdot b^x \][/tex]
where:
- \( a \) is the initial value,
- \( b \) (with \( 0 < b < 1 \)) is the decay factor,
- \( x \) is the domain value.
Given the initial data points:
- For \( x = 0 \), \( y = 32 \),
- For \( x = 1 \), \( y = 24 \).
We can test values for \( a \) and \( b \) using these two points to establish the exponential decay relationship.
For \( x = 0 \):
[tex]\[ y = a \cdot b^0 \][/tex]
[tex]\[ 32 = a \][/tex]
This indicates that \( a = 32 \).
Next, using the data point \( x = 1 \) where \( y = 24 \):
[tex]\[ 24 = 32 \cdot b^1 \][/tex]
[tex]\[ 24 = 32 \cdot b \][/tex]
[tex]\[ b = \frac{24}{32} \][/tex]
[tex]\[ b = 0.75 \][/tex]
Hence, the exponential decay equation becomes:
[tex]\[ y = 32 \cdot (0.75)^x \][/tex]
Now, to find \( d \) when \( x = 2 \):
[tex]\[ d = 32 \cdot (0.75)^2 \][/tex]
Calculating the value:
[tex]\[ d = 32 \cdot 0.5625 \][/tex]
[tex]\[ d = 18 \][/tex]
Thus, the value of \( d \) that maintains the exponential decay pattern as outlined would be:
[tex]\[ 18 \][/tex]
So the value of \( d \) in the table must be:
[tex]\[ d = 18 \][/tex]
Exponential decay can be expressed typically as:
[tex]\[ y = a \cdot b^x \][/tex]
where:
- \( a \) is the initial value,
- \( b \) (with \( 0 < b < 1 \)) is the decay factor,
- \( x \) is the domain value.
Given the initial data points:
- For \( x = 0 \), \( y = 32 \),
- For \( x = 1 \), \( y = 24 \).
We can test values for \( a \) and \( b \) using these two points to establish the exponential decay relationship.
For \( x = 0 \):
[tex]\[ y = a \cdot b^0 \][/tex]
[tex]\[ 32 = a \][/tex]
This indicates that \( a = 32 \).
Next, using the data point \( x = 1 \) where \( y = 24 \):
[tex]\[ 24 = 32 \cdot b^1 \][/tex]
[tex]\[ 24 = 32 \cdot b \][/tex]
[tex]\[ b = \frac{24}{32} \][/tex]
[tex]\[ b = 0.75 \][/tex]
Hence, the exponential decay equation becomes:
[tex]\[ y = 32 \cdot (0.75)^x \][/tex]
Now, to find \( d \) when \( x = 2 \):
[tex]\[ d = 32 \cdot (0.75)^2 \][/tex]
Calculating the value:
[tex]\[ d = 32 \cdot 0.5625 \][/tex]
[tex]\[ d = 18 \][/tex]
Thus, the value of \( d \) that maintains the exponential decay pattern as outlined would be:
[tex]\[ 18 \][/tex]
So the value of \( d \) in the table must be:
[tex]\[ d = 18 \][/tex]
