Question
The graph shows the relationship between the width `w` in feet of a flower bed and its cost of planting `c` in dollars. Write a quadratic equation in the form y=ax^2 that represents the situation.
Graph
(0,0) <starting point
(1,12)
(2,48)
(3,108)
(4,192)
(5,300)
(1 point) Responses c=12w2c=12w2c is equal to 12 w squared c=16w2c=16w2c is equal to 16 w squared c=4w2c=4w2c is equal to 4 w squared w=12c2
Graph
(0,0) <starting point
(1,12)
(2,48)
(3,108)
(4,192)
(5,300)
(1 point) Responses c=12w2c=12w2c is equal to 12 w squared c=16w2c=16w2c is equal to 16 w squared c=4w2c=4w2c is equal to 4 w squared w=12c2
Answers
GPT-4o mini
To find the quadratic equation that represents the relationship between the width \( w \) of the flower bed and its cost of planting \( c \), we can analyze the points provided in the graph:
- \( (0,0) \)
- \( (1,12) \)
- \( (2,48) \)
- \( (3,108) \)
- \( (4,192) \)
- \( (5,300) \)
We can start by checking if the relationships are quadratic. We will compute the differences between consecutive values of \( c \) to check if they follow a quadratic pattern:
1. First differences of \( c \):
- \( 12 - 0 = 12 \)
- \( 48 - 12 = 36 \)
- \( 108 - 48 = 60 \)
- \( 192 - 108 = 84 \)
- \( 300 - 192 = 108 \)
So the first differences are: \( 12, 36, 60, 84, 108 \).
2. Now, we compute the second differences:
- \( 36 - 12 = 24 \)
- \( 60 - 36 = 24 \)
- \( 84 - 60 = 24 \)
- \( 108 - 84 = 24 \)
The second differences are constant, which confirms that this relationship is indeed quadratic.
Next, we can express the quadratic equation in the form:
\[
c = aw^2 + bw + d
\]
Given the points available, since \( (0, 0) \) indicates that the cost is zero when width is zero, we can deduce that \( d = 0 \).
Let's evaluate the coefficients \( a \) and \( b \) further using the points. Given two points, we can set up the equations:
Using the points \( (1, 12) \) and \( (2, 48) \):
1. From \( (1, 12) \):
\[
c = a(1^2) + b(1) \implies 12 = a + b
\]
2. From \( (2, 48) \):
\[
c = a(2^2) + b(2) \implies 48 = 4a + 2b
\]
Now we have the system of equations:
1. \( a + b = 12 \)
2. \( 4a + 2b = 48 \)
Now we can solve this system. From equation 1, isolate \( b \):
\[
b = 12 - a
\]
Substituting into equation 2:
\[
4a + 2(12 - a) = 48 \\
4a + 24 - 2a = 48 \\
2a + 24 = 48 \\
2a = 24 \\
a = 12
\]
Now substitute \( a \) back into equation 1 to find \( b \):
\[
12 + b = 12 \implies b = 0
\]
Thus, the quadratic equation representing the cost of planting based on the width \( w \) is:
\[
c = 12w^2
\]
So the answer is:
\[
c = 12w^2
\]
- \( (0,0) \)
- \( (1,12) \)
- \( (2,48) \)
- \( (3,108) \)
- \( (4,192) \)
- \( (5,300) \)
We can start by checking if the relationships are quadratic. We will compute the differences between consecutive values of \( c \) to check if they follow a quadratic pattern:
1. First differences of \( c \):
- \( 12 - 0 = 12 \)
- \( 48 - 12 = 36 \)
- \( 108 - 48 = 60 \)
- \( 192 - 108 = 84 \)
- \( 300 - 192 = 108 \)
So the first differences are: \( 12, 36, 60, 84, 108 \).
2. Now, we compute the second differences:
- \( 36 - 12 = 24 \)
- \( 60 - 36 = 24 \)
- \( 84 - 60 = 24 \)
- \( 108 - 84 = 24 \)
The second differences are constant, which confirms that this relationship is indeed quadratic.
Next, we can express the quadratic equation in the form:
\[
c = aw^2 + bw + d
\]
Given the points available, since \( (0, 0) \) indicates that the cost is zero when width is zero, we can deduce that \( d = 0 \).
Let's evaluate the coefficients \( a \) and \( b \) further using the points. Given two points, we can set up the equations:
Using the points \( (1, 12) \) and \( (2, 48) \):
1. From \( (1, 12) \):
\[
c = a(1^2) + b(1) \implies 12 = a + b
\]
2. From \( (2, 48) \):
\[
c = a(2^2) + b(2) \implies 48 = 4a + 2b
\]
Now we have the system of equations:
1. \( a + b = 12 \)
2. \( 4a + 2b = 48 \)
Now we can solve this system. From equation 1, isolate \( b \):
\[
b = 12 - a
\]
Substituting into equation 2:
\[
4a + 2(12 - a) = 48 \\
4a + 24 - 2a = 48 \\
2a + 24 = 48 \\
2a = 24 \\
a = 12
\]
Now substitute \( a \) back into equation 1 to find \( b \):
\[
12 + b = 12 \implies b = 0
\]
Thus, the quadratic equation representing the cost of planting based on the width \( w \) is:
\[
c = 12w^2
\]
So the answer is:
\[
c = 12w^2
\]