Use the image to answer the question.
A graph shows the x-axis representing weight in ounces ranging from 0 to 6 in increments of 2, and the y-axis representing cost in dollars ranging from 0 to 15 in increments of 5. A line is drawn ending with an arrow. The line begins at the origin and passes through left parenthesis 2 comma 10 right parenthesis.
At the Custard Creamery, the cost of ice cream custard is directly proportional to its weight in ounces. Complete the equation of the graph with its constant of proportionality.
(1 point)
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The equation of the graph can be written as y = kx, where y represents the cost in dollars and x represents the weight in ounces. To find the constant of proportionality (k), we can use the given point (2,10) on the line.
Substituting the values into the equation, we get:
10 = k * 2
Simplifying the equation, we divide both sides by 2:
k = 10/2
Simplifying further, we get:
k = 5
So, the constant of proportionality is 5. The equation of the graph is y = 5x.
To find the constant of proportionality, we can use the point-slope form of a linear equation which is given by:
\(y - y_1 = m(x - x_1)\)
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. In this case, the point \((2, 10)\) lies on the line.
The slope of a line can be determined by finding the ratio of the change in \(y\) to the change in \(x\). Let's consider another point on the line, namely the origin \((0, 0)\). The change in \(y\) is \(10 - 0 = 10\) and the change in \(x\) is \(2 - 0 = 2\).
So, the slope of the line is:
\(m = \frac{\text{change in } y}{\text{change in } x} = \frac{10}{2} = 5\)
Now, we can substitute the values of \(m\) and \((x_1, y_1)\) into the point-slope form equation:
\(y - 10 = 5(x - 2)\)
Simplifying, we get:
\(y - 10 = 5x - 10\)
Adding 10 to both sides:
\(y = 5x\)
Therefore, the equation of the graph with its constant of proportionality is \(y = 5x\).
To complete the equation of the graph with its constant of proportionality, we need to determine the slope of the line.
The slope of a line is defined as the change in the y-coordinate divided by the change in the x-coordinate. In this case, the line starts at the origin (0,0) and passes through the point (2,10).
The change in the y-coordinate is 10 - 0 = 10, and the change in the x-coordinate is 2 - 0 = 2.
Therefore, the slope of the line is:
slope = (change in y-coordinate) / (change in x-coordinate) = 10 / 2 = 5
Since the cost of ice cream custard is directly proportional to its weight in ounces, the equation of the graph can be written as:
cost = (constant of proportionality) * weight
We can substitute the value of the slope (5) into the equation:
cost = 5 * weight
Therefore, the equation of the graph with its constant of proportionality is:
cost = 5 * weight