Question

explain what the slope and y - intercept mean in each situation.
part a
a graph represents the perimeter, y, in units, for an equilateral triangle with side length x units. the slope of the line is 3 and the y - intercept is 0.
enter your answer(s) here
part b
the amount of money, y, in a cash box after x tickets are purchased for carnival games. the slope of the line is \\( \frac { 1 } { 2 } \\) and the y - intercept is 8.

Explanation:

Step1: Analyze the perimeter - side length relationship

The formula for the perimeter \(y\) of an equilateral triangle with side length \(x\) is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y -\)intercept. For an equilateral triangle, \(y=3x+0\).
The slope \(m = 3\) means that for every 1 - unit increase in the side length \(x\) of the equilateral triangle, the perimeter \(y\) increases by 3 units. The \(y -\)intercept \(b = 0\) means that when the side length \(x = 0\) (a degenerate triangle, which is just a point), the perimeter \(y=0\).

Step2: Analyze the money - ticket relationship

The equation of the line is \(y=\frac{1}{4}x + 8\).
The slope \(m=\frac{1}{4}\) means that for every 1 - ticket purchased (\(x\) increases by 1), the amount of money \(y\) in the cash box increases by \(\frac{1}{4}\) of a unit (assuming the unit is currency, say dollars). The \(y -\)intercept \(b = 8\) means that when no tickets (\(x = 0\)) are purchased, there is $8 in the cash box.

Answer:

• Part A: The slope \(3\) means the perimeter increases by \(3\) units per 1 - unit increase in side length. The \(y -\)intercept \(0\) means perimeter is \(0\) when side length is \(0\).
• Part B: The slope \(\frac{1}{4}\) means money in the cash box increases by \(\frac{1}{4}\) per ticket. The \(y -\)intercept \(8\) means there is $8 when no tickets are sold.