Question: The three-toed sloth is an extremely slow animal. On the ground, it travels at a speed of about 6 fe...

Question

The three-toed sloth is an extremely slow animal. On the ground, it travels at a speed of about 6 feet per minute. This is the graph for the distance $(y)$ a sloth will travel in $x$ minutes. Find the constant of variation.

(A) 4 feet per minute
(B) 5 feet per minute
(C) 6 feet per minute
(D) 7 feet per minute

Answer 1

Solution

In this question, we have been given a graph showing the speed of the sloth and we need to find the constant to variation. Constant variation is a direct variation that is a constant ratio of two variable quantities. Constant of variation is the slope of the curve.

Formula used:
Slope = $\dfrac{{\Delta y}}{{\Delta x}}$

Complete step by step solution:
For calculating the slope, we decide any two clearly visible points on the graph.
From the graph, we can see that the line passes through the following points $(0,0)$ and $(1,6)$ .
Slope = $\dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{6 - 0}}{{1 - 0}}$
Slope = $6$
Hence the constant of variation is 6 feet per second.
Since the distance is given in feet and the time is given in seconds, the unit for constant variation will be feet per second.

Hence, the correct answer is option (C), 6 feet per second.

Additional information: Let us consider an equation $y=Kx$ , we can see that y is directly proportional to x, or it can also be said as y is varying constantly with respect to x by a factor of K. Here, K is the constant of variation. We can also represent the above graph in the form of an equation. We have already found that $K=6$ , so the equation for the above graph will be $y=6x$ where y represents the distance in feet and x represents the time in seconds.

Note: For calculating the constant of variation, any two points on the graphs can be used. Taking origin as one of the points is more convenient than taking another point. If the line is not passing from the origin, it is convenient to take Y-intercept as one of the points. If the graph is not line and has some other shape then the slope $dy/dx$ . The slope is also equal to the tangent of the angle made by line with the x-axis.