Question

Consider ramps whose length and height are in the ratio 11 to 4. Let X be an unspecified number of feet, standing for the length of the ramp, and let Y be the corresponding height of the ramp in feet. Think of X and Y as varying together to make ramps of different lengths in heights, all in ratio 11 to 4. A.) sketch a graph in a coordinate plane to show the relationship between the lengths in heights of these ramps. B.) reason about quantities, and our definition of multiplication, and use math drawings to derive and explain equations of the form y=c•x and x=y•k where C & Kare suitable constants of proportionality ( find c and k). C.) explain how to interpret the constant of proportionality C & K from part B in terms of your graph in part A in several different ways.

Answer 1

B. Recall that the general equation of a line is of the form y=mx+b where b is the value of y at which the graph crosses the y axis and m is the slope of the line. Based on the graph, we can tell that the lines crosses the y-axis at the value y=0. So b=0. We must determine m. We are told that the length and height are in the ration 11 to 4, this means that whenever we increment the value of X by 11, the value of Y increments by 4. So, having our equation, whenever we replace x = 11, we must have y =4. then, we have the equation

4 = 11m. Dividing both sides by 11 we get m = 4/11. So the equation we were looking for is y = 4/11 x. From this equation, we can multiply by 11 and then divide by 4 on both sides, so we get 11/4 y = x.

C. Consider an approximation of the graph as follows:

You can pick any point that is on the line. Take for example the one I've drawn. By having m = 4/11 it means that whenever we draw a horizontal line that is parallel to the x-axis that start at the point we chose, and whose lenght is 11 (red line in the drawing), if we then draw a vertical line (blue line)that is parallel to the y-axis, as soon as this vertical line touches the original line, we will form a triangle whose height necessarily has length equal to 4.