Questions: Hassan is signing up for a gym membership with a one-time fee to join and then a monthly fee to remain a member. Let C represent the total cost of the gym membership over t months. A graph of C is shown below. Write an equation for C then state the y-intercept of the graph and determine its interpretation in the context of the problem.

Hassan is signing up for a gym membership with a one-time fee to join and then a monthly fee to remain a member. Let C represent the total cost of the gym membership over t months. A graph of C is shown below. Write an equation for C then state the y-intercept of the graph and determine its interpretation in the context of the problem.

Transcript text: Hassan is signing up for a gym membership with a one-time fee to join and then a monthly fee to remain a member. Let $C$ represent the total cost of the gym membership over $t$ months. A graph of $C$ is shown below. Write an equation for $C$ then state the $y$-intercept of the graph and determine its interpretation in the context of the problem.

Solution

Solution Steps

Step 1: Find two points on the line.

Two points that are easy to read from the graph are (0, 200) and (4, 400).

Step 2: Calculate the slope.

The slope 'm' can be calculated using the formula: m = (y2 - y1) / (x2 - x1)

Plugging in the values from the two points: m = (400 - 200) / (4 - 0) m = 200 / 4 m = 50

Step 3: Determine the y-intercept.

The y-intercept is the point where the line crosses the y-axis. This occurs when x = 0. From the graph, we can see the y-intercept is 200. It is also given by the point (0, 200).

Step 4: Write the equation in slope-intercept form.

The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. In this problem, the total cost 'C' is the dependent variable (y), and the number of months 't' is the independent variable (x). Substituting the values calculated:

C = 50t + 200

Step 5: Interpret the y-intercept.

The y-intercept represents the total cost when the number of months (t) is zero. This means the y-intercept of 200 is the initial one-time joining fee for the gym membership.