Question
A television show will be canceled if the estimated number of viewers falls below 2.5 million by Week 10. Use the graph shown to the right to write an exponential function to model the situation. If this pattern continues, will the show be canceled?
0
5
10
0
5
10
Week
Number of Viewers (millions)
y
x
x y graph
Question content area bottom
Part 1
What is the exponential function that models the situation?
A.f left parenthesis x right parenthesis equals 0.9 left parenthesis x right parenthesis Superscript 7
f left parenthesis x right parenthesis equals 0.9 left parenthesis x right parenthesis Superscript 7f(x)=0.9(x)7
B.f left parenthesis x right parenthesis equals 0.9 left parenthesis 7 right parenthesis Superscript x
f left parenthesis x right parenthesis equals 0.9 left parenthesis 7 right parenthesis Superscript xf(x)=0.9(7)x
C.f left parenthesis x right parenthesis equals 7 left parenthesis 0.9 right parenthesis Superscript x
f left parenthesis x right parenthesis equals 7 left parenthesis 0.9 right parenthesis Superscript xf(x)=7(0.9)x
D.f left parenthesis x right parenthesis equals 7 left parenthesis x right parenthesis Superscript 0.9
f left parenthesis x right parenthesis equals 7 left parenthesis x right parenthesis Superscript 0.9f(x)=7(x)0.9
0
5
10
0
5
10
Week
Number of Viewers (millions)
y
x
x y graph
Question content area bottom
Part 1
What is the exponential function that models the situation?
A.f left parenthesis x right parenthesis equals 0.9 left parenthesis x right parenthesis Superscript 7
f left parenthesis x right parenthesis equals 0.9 left parenthesis x right parenthesis Superscript 7f(x)=0.9(x)7
B.f left parenthesis x right parenthesis equals 0.9 left parenthesis 7 right parenthesis Superscript x
f left parenthesis x right parenthesis equals 0.9 left parenthesis 7 right parenthesis Superscript xf(x)=0.9(7)x
C.f left parenthesis x right parenthesis equals 7 left parenthesis 0.9 right parenthesis Superscript x
f left parenthesis x right parenthesis equals 7 left parenthesis 0.9 right parenthesis Superscript xf(x)=7(0.9)x
D.f left parenthesis x right parenthesis equals 7 left parenthesis x right parenthesis Superscript 0.9
f left parenthesis x right parenthesis equals 7 left parenthesis x right parenthesis Superscript 0.9f(x)=7(x)0.9
Answers
GPT-4o mini
To model the situation with an exponential function, we need to identify the initial value (the number of viewers at week 0) and the rate of decrease in viewership per week.
Assuming that the graph implies the following:
1. **Initial value at Week 0**: Let’s assume that the number of viewers at Week 0 is 7 million (you can adjust this if the graph indicates a different initial value).
2. **Rate of decrease**: If the graph shows that the viewers decrease to around 0.9 million each week, that suggests a decay factor of 0.9.
The general form of an exponential decay function can be expressed as:
\[
f(x) = a \cdot b^x
\]
where:
- \( a \) is the initial value (number of viewers at Week 0),
- \( b \) is the decay factor,
- \( x \) is the number of weeks.
Given these assumptions:
- \( a = 7 \) (initial viewers in millions),
- \( b = 0.9 \) (decay factor).
Thus, the exponential function that models this situation is:
\[
f(x) = 7 \cdot (0.9)^x
\]
The correct option from the provided choices is:
**C.** \( f(x) = 7 \cdot (0.9)^x \)
### Part 2: Will the show be canceled?
To determine if the show will be canceled, we need to find out the estimated number of viewers at Week 10:
\[
f(10) = 7 \cdot (0.9)^{10}
\]
Calculating \( (0.9)^{10} \):
\[
(0.9)^{10} \approx 0.3487 \quad (\text{using a calculator})
\]
Now calculating \( f(10) \):
\[
f(10) = 7 \cdot 0.3487 \approx 2.441 \text{ million}
\]
Since \( 2.441 \) million is less than \( 2.5 \) million, **yes**, the show will be canceled if the trend continues as modeled by this function.
Assuming that the graph implies the following:
1. **Initial value at Week 0**: Let’s assume that the number of viewers at Week 0 is 7 million (you can adjust this if the graph indicates a different initial value).
2. **Rate of decrease**: If the graph shows that the viewers decrease to around 0.9 million each week, that suggests a decay factor of 0.9.
The general form of an exponential decay function can be expressed as:
\[
f(x) = a \cdot b^x
\]
where:
- \( a \) is the initial value (number of viewers at Week 0),
- \( b \) is the decay factor,
- \( x \) is the number of weeks.
Given these assumptions:
- \( a = 7 \) (initial viewers in millions),
- \( b = 0.9 \) (decay factor).
Thus, the exponential function that models this situation is:
\[
f(x) = 7 \cdot (0.9)^x
\]
The correct option from the provided choices is:
**C.** \( f(x) = 7 \cdot (0.9)^x \)
### Part 2: Will the show be canceled?
To determine if the show will be canceled, we need to find out the estimated number of viewers at Week 10:
\[
f(10) = 7 \cdot (0.9)^{10}
\]
Calculating \( (0.9)^{10} \):
\[
(0.9)^{10} \approx 0.3487 \quad (\text{using a calculator})
\]
Now calculating \( f(10) \):
\[
f(10) = 7 \cdot 0.3487 \approx 2.441 \text{ million}
\]
Since \( 2.441 \) million is less than \( 2.5 \) million, **yes**, the show will be canceled if the trend continues as modeled by this function.