Question
If the function g(x)=6x+2 models the number of leaves on a plant x weeks after being planted, which of the following accurately calculates the average rate of change in leaves between weeks 6 and 10?
A.
b−a
f(b)+f(a)
=
10−6
62+38
=25 leaves
B. f(b)+f(a)=62+38=100 leaves
C.
b+a
f(b)−f(a)
=
10+6
62−38
=1.5 leaves
D.
b−a
f(b)−f(a)
=
10−6
62−38
=6 leaves
A.
b−a
f(b)+f(a)
=
10−6
62+38
=25 leaves
B. f(b)+f(a)=62+38=100 leaves
C.
b+a
f(b)−f(a)
=
10+6
62−38
=1.5 leaves
D.
b−a
f(b)−f(a)
=
10−6
62−38
=6 leaves
Answers
Answered by
GPT-4o mini
To calculate the average rate of change of the function \( g(x) = 6x + 2 \) between weeks 6 and 10, we will use the formula:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
where \( a = 6 \) and \( b = 10 \).
**Step 1: Calculate \( g(6) \) and \( g(10) \)**
\[
g(6) = 6(6) + 2 = 36 + 2 = 38 \text{ leaves}
\]
\[
g(10) = 6(10) + 2 = 60 + 2 = 62 \text{ leaves}
\]
**Step 2: Apply the formula for average rate of change**
Now, substituting \( a = 6 \), \( b = 10 \) and the values for \( g(a) \) and \( g(b) \):
\[
\text{Average Rate of Change} = \frac{g(10) - g(6)}{10 - 6} = \frac{62 - 38}{10 - 6} = \frac{24}{4} = 6 \text{ leaves per week}
\]
Now let's check which option corresponds to this calculation:
**A.**
\[
\frac{b-a}{g(b)+g(a)} = \frac{10-6}{62+38} = \frac{4}{100} = 0.04 \text{ does not match}
\]
**B.**
\[
g(b) + g(a) = 62 + 38 = 100 \text{ leaves (not an average rate)}
\]
**C.**
\[
\frac{b+a}{g(b)-g(a)} = \frac{10+6}{62-38} = \frac{16}{24} = \frac{2}{3} \text{ leaves (not correct)}
\]
**D.**
\[
\frac{b-a}{g(b)-g(a)} = \frac{10-6}{62-38} = \frac{4}{24} = \frac{1}{6} \text{ not what we are looking for}
\]
So, the value we found using the average rate of change formula, which is:
\[
\text{Average Rate of Change} = 6 \text{ leaves per week}
\]
is not explicitly found in the options. However, D is seeming to have a correct structure but the wrong calculation.
Based on the calculations shared, the average rate of change between weeks 6 and 10 is indeed \(6\) leaves per week, matching option D's correct form, but with calculations that are stated inaccurately.
Thus, our conclusion is that none of the options explicitly capture that the average rate of change is \(6\) leaves per week accurately.
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
where \( a = 6 \) and \( b = 10 \).
**Step 1: Calculate \( g(6) \) and \( g(10) \)**
\[
g(6) = 6(6) + 2 = 36 + 2 = 38 \text{ leaves}
\]
\[
g(10) = 6(10) + 2 = 60 + 2 = 62 \text{ leaves}
\]
**Step 2: Apply the formula for average rate of change**
Now, substituting \( a = 6 \), \( b = 10 \) and the values for \( g(a) \) and \( g(b) \):
\[
\text{Average Rate of Change} = \frac{g(10) - g(6)}{10 - 6} = \frac{62 - 38}{10 - 6} = \frac{24}{4} = 6 \text{ leaves per week}
\]
Now let's check which option corresponds to this calculation:
**A.**
\[
\frac{b-a}{g(b)+g(a)} = \frac{10-6}{62+38} = \frac{4}{100} = 0.04 \text{ does not match}
\]
**B.**
\[
g(b) + g(a) = 62 + 38 = 100 \text{ leaves (not an average rate)}
\]
**C.**
\[
\frac{b+a}{g(b)-g(a)} = \frac{10+6}{62-38} = \frac{16}{24} = \frac{2}{3} \text{ leaves (not correct)}
\]
**D.**
\[
\frac{b-a}{g(b)-g(a)} = \frac{10-6}{62-38} = \frac{4}{24} = \frac{1}{6} \text{ not what we are looking for}
\]
So, the value we found using the average rate of change formula, which is:
\[
\text{Average Rate of Change} = 6 \text{ leaves per week}
\]
is not explicitly found in the options. However, D is seeming to have a correct structure but the wrong calculation.
Based on the calculations shared, the average rate of change between weeks 6 and 10 is indeed \(6\) leaves per week, matching option D's correct form, but with calculations that are stated inaccurately.
Thus, our conclusion is that none of the options explicitly capture that the average rate of change is \(6\) leaves per week accurately.