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=ax2
that represents the situation.
graph shows a line passing through (1,12) (2,48) (3,108) (4,192) (5,300)
(1 point)
Responses
w=12c2
w is equal to 12 c squared
c=12w2
c is equal to 12 w squared
c=4w2
c is equal to 4 w squared
c=16w2
that represents the situation.
graph shows a line passing through (1,12) (2,48) (3,108) (4,192) (5,300)
(1 point)
Responses
w=12c2
w is equal to 12 c squared
c=12w2
c is equal to 12 w squared
c=4w2
c is equal to 4 w squared
c=16w2
Answers
GPT-4o mini
To find a quadratic equation that represents the relationship between the width \( w \) of the flower bed and its cost \( c \), we'll look at the points provided:
- (1, 12)
- (2, 48)
- (3, 108)
- (4, 192)
- (5, 300)
We can analyze the given data to find a pattern.
If we assume that \( c \) can be expressed as a quadratic function \( c = a w^2 \), we can use the points where \( w \) takes on the values of 1 through 5 and see how \( c \) behaves.
Calculating the ratios of \( c \) to \( w^2 \):
- For \( w = 1 \): \( c = 12 \) and \( w^2 = 1^2 = 1 \) → \( \frac{c}{w^2} = \frac{12}{1} = 12 \)
- For \( w = 2 \): \( c = 48 \) and \( w^2 = 2^2 = 4 \) → \( \frac{c}{w^2} = \frac{48}{4} = 12 \)
- For \( w = 3 \): \( c = 108 \) and \( w^2 = 3^2 = 9 \) → \( \frac{c}{w^2} = \frac{108}{9} = 12 \)
- For \( w = 4 \): \( c = 192 \) and \( w^2 = 4^2 = 16 \) → \( \frac{c}{w^2} = \frac{192}{16} = 12 \)
- For \( w = 5 \): \( c = 300 \) and \( w^2 = 5^2 = 25 \) → \( \frac{c}{w^2} = \frac{300}{25} = 12 \)
The ratio \( \frac{c}{w^2} \) is consistently equal to 12.
Thus, the equation that represents the relationship is:
\[ c = 12w^2 \]
In terms of the responses given, you have:
**c = 12w^2**
This can be interpreted as the correct answer, and it implies that the cost \( c \) is equal to 12 times the square of the width \( w \).
Therefore, the correct response is: **c = 12w^2**.
- (1, 12)
- (2, 48)
- (3, 108)
- (4, 192)
- (5, 300)
We can analyze the given data to find a pattern.
If we assume that \( c \) can be expressed as a quadratic function \( c = a w^2 \), we can use the points where \( w \) takes on the values of 1 through 5 and see how \( c \) behaves.
Calculating the ratios of \( c \) to \( w^2 \):
- For \( w = 1 \): \( c = 12 \) and \( w^2 = 1^2 = 1 \) → \( \frac{c}{w^2} = \frac{12}{1} = 12 \)
- For \( w = 2 \): \( c = 48 \) and \( w^2 = 2^2 = 4 \) → \( \frac{c}{w^2} = \frac{48}{4} = 12 \)
- For \( w = 3 \): \( c = 108 \) and \( w^2 = 3^2 = 9 \) → \( \frac{c}{w^2} = \frac{108}{9} = 12 \)
- For \( w = 4 \): \( c = 192 \) and \( w^2 = 4^2 = 16 \) → \( \frac{c}{w^2} = \frac{192}{16} = 12 \)
- For \( w = 5 \): \( c = 300 \) and \( w^2 = 5^2 = 25 \) → \( \frac{c}{w^2} = \frac{300}{25} = 12 \)
The ratio \( \frac{c}{w^2} \) is consistently equal to 12.
Thus, the equation that represents the relationship is:
\[ c = 12w^2 \]
In terms of the responses given, you have:
**c = 12w^2**
This can be interpreted as the correct answer, and it implies that the cost \( c \) is equal to 12 times the square of the width \( w \).
Therefore, the correct response is: **c = 12w^2**.