a paintball is hot at a wheel that has a radius of 40 cm. it hits the very bottom of the wheel next to the road while the car is stopped at the red light. as the light turns green, the wheel rolls, and so does the paint. if it takes 3 seconds for the wheel to make one full revolution, find the equation and graph that model this situation

To model this situation, we can start by understanding the motion of the wheel and the paintball.

Let's assume that the wheel is rolling on a flat surface without slipping. In this case, the linear distance covered by a point on the wheel (such as the bottom of the wheel where the paintball hits) as the wheel rotates is equal to the circumference of the wheel.

The circumference of the wheel can be calculated using the formula: Circumference = 2πr, where r is the radius of the wheel.

In this case, the radius of the wheel is given as 40 cm. Hence, the circumference of the wheel is 2π(40) = 80π cm.

The time taken for the wheel to make one full revolution is given as 3 seconds. So, we can say that in 3 seconds, the bottom of the wheel will cover a distance of 80π cm.

Now let's create the equation and graph to model this situation.

Equation:
Let 'x' represent the time in seconds, and 'y' represent the linear distance covered by the bottom of the wheel in centimeters.

From the above analysis, we know that when x = 3 seconds, y = 80π cm.

So, we can set up the equation as:
y = (80π/3)x

Graph:
To graph this equation, we can plot some points by substituting different values for 'x' and solving for 'y'. Here are a few points:

x = 0: y = (80π/3)(0) = 0
x = 1: y = (80π/3)(1) ≈ 26.67π
x = 2: y = (80π/3)(2) ≈ 53.34π
x = 3: y = (80π/3)(3) = 80π

Now you can plot these points on a graph, where x is on the horizontal axis and y is on the vertical axis. Connect the points with a smooth curve to get the graph.

Note: Remember to convert the angle to radians when calculating values for the equation, as π represents radian measure.

I hope this explanation helps! Let me know if you have any further questions.