Question
the graph shows the number of good deeds performed at a fast food drive thru over the course of a weekend. customers were "paying it forward" by purchasing a food item for the person behind them in line.
time (hours): 0,1,2
number of good deeds: 2,6,18
y=_____
time (hours): 0,1,2
number of good deeds: 2,6,18
y=_____
Answers
Answered by
GPT-4o mini
To express the relationship between time (in hours) and the number of good deeds performed as a function \( y \), we can analyze the data provided:
- At \( t = 0 \) hours, \( y = 2 \) good deeds.
- At \( t = 1 \) hour, \( y = 6 \) good deeds.
- At \( t = 2 \) hours, \( y = 18 \) good deeds.
We can look for a pattern in the way the number of good deeds increases over time. The values appear to increase rather rapidly; let’s calculate the increase in good deeds.
From \( t = 0 \) to \( t = 1 \):
\( 6 - 2 = 4 \) (an increase of 4 good deeds)
From \( t = 1 \) to \( t = 2 \):
\( 18 - 6 = 12 \) (an increase of 12 good deeds)
This suggests that the number of good deeds may be increasing at an accelerating rate.
To form a function, let's explore if there is a polynomial relationship.
The values seem to suggest a quadratic relationship, so you could try fitting a quadratic equation of the form:
\[ y = at^2 + bt + c \]
Using the points (0, 2), (1, 6), and (2, 18) to set up a system of equations:
1. For \( t = 0 \):
\[ c = 2 \]
2. For \( t = 1 \):
\[ a(1)^2 + b(1) + c = 6 \]
\[ a + b + 2 = 6 \]
\[ a + b = 4 \] (Equation 1)
3. For \( t = 2 \):
\[ a(2)^2 + b(2) + c = 18 \]
\[ 4a + 2b + 2 = 18 \]
\[ 4a + 2b = 16 \]
\[ 2a + b = 8 \] (Equation 2)
Now we can solve the system of equations (1 and 2):
From Equation 1:
\[ b = 4 - a \]
Substitute into Equation 2:
\[ 2a + (4 - a) = 8 \]
\[ 2a + 4 - a = 8 \]
\[ a + 4 = 8 \]
\[ a = 4 \]
Substituting \( a \) back into Equation 1:
\[ 4 + b = 4 \]
\[ b = 0 \]
Thus, the quadratic function that fits these points is:
\[ y = 4t^2 + 2 \]
So, the final answer could be described as:
\[ y = 4t^2 + 2 \]
This formula models the number of good deeds \( y \) performed after \( t \) hours.
- At \( t = 0 \) hours, \( y = 2 \) good deeds.
- At \( t = 1 \) hour, \( y = 6 \) good deeds.
- At \( t = 2 \) hours, \( y = 18 \) good deeds.
We can look for a pattern in the way the number of good deeds increases over time. The values appear to increase rather rapidly; let’s calculate the increase in good deeds.
From \( t = 0 \) to \( t = 1 \):
\( 6 - 2 = 4 \) (an increase of 4 good deeds)
From \( t = 1 \) to \( t = 2 \):
\( 18 - 6 = 12 \) (an increase of 12 good deeds)
This suggests that the number of good deeds may be increasing at an accelerating rate.
To form a function, let's explore if there is a polynomial relationship.
The values seem to suggest a quadratic relationship, so you could try fitting a quadratic equation of the form:
\[ y = at^2 + bt + c \]
Using the points (0, 2), (1, 6), and (2, 18) to set up a system of equations:
1. For \( t = 0 \):
\[ c = 2 \]
2. For \( t = 1 \):
\[ a(1)^2 + b(1) + c = 6 \]
\[ a + b + 2 = 6 \]
\[ a + b = 4 \] (Equation 1)
3. For \( t = 2 \):
\[ a(2)^2 + b(2) + c = 18 \]
\[ 4a + 2b + 2 = 18 \]
\[ 4a + 2b = 16 \]
\[ 2a + b = 8 \] (Equation 2)
Now we can solve the system of equations (1 and 2):
From Equation 1:
\[ b = 4 - a \]
Substitute into Equation 2:
\[ 2a + (4 - a) = 8 \]
\[ 2a + 4 - a = 8 \]
\[ a + 4 = 8 \]
\[ a = 4 \]
Substituting \( a \) back into Equation 1:
\[ 4 + b = 4 \]
\[ b = 0 \]
Thus, the quadratic function that fits these points is:
\[ y = 4t^2 + 2 \]
So, the final answer could be described as:
\[ y = 4t^2 + 2 \]
This formula models the number of good deeds \( y \) performed after \( t \) hours.
Answer
this answer is wrong