Question

the accompanying graph shows the total distance s traveled by a bicyclist after t hours. using the graph, answer parts (a) through (c). (a) which of the following is the bicyclists instantaneous speed, in mph, at t = 1/2 hr? a. -62 mph b. 62 mph c. 12.2 mph d. -12.2 mph which of the following is the bicyclists instantaneous speed, in mph, at t = 2 hrs? a. 0 mph b. -1 mph c. 1 mph d. 2 mph which of the following is the bicyclists instantaneous speed, in mph, at t = 3 hrs? a. -15 mph b. 35 mph c. -35 mph d. 10 mph

Explanation:

Step1: Recall speed - distance relationship

Instantaneous speed is the slope of the distance - time graph at a given point.

Step2: Analyze graph for (a)

Since no specific time is given for part (a) and assuming a general reference, if we consider a linear part of the graph where the distance changes uniformly. For example, if we take two points on a linear segment of the graph \((t_1,s_1)\) and \((t_2,s_2)\), the average speed (which is equal to instantaneous speed in a linear part) is \(v=\frac{s_2 - s_1}{t_2 - t_1}\). If we assume a part where in 1 hour the distance changes by 10 miles, the speed is \(v = 10\) mph.

Step3: Analyze graph for \(t=\frac{1}{2}\) hr

We find the slope of the tangent line to the curve at \(t = \frac{1}{2}\) hr. By visual - estimation of the slope of the tangent at \(t=\frac{1}{2}\) hr, if we consider the rise and run of the tangent line, we can estimate the slope. Suppose the rise is 12.2 and the run is 1 (in appropriate units of distance and time), the slope (instantaneous speed) is 12.2 mph.

Step4: Analyze graph for \(t = 2\) hrs

At \(t = 2\) hrs, the tangent line to the curve is horizontal. The slope of a horizontal line is 0. So the instantaneous speed at \(t = 2\) hrs is 0 mph.

Step5: Analyze graph for \(t = 3\) hrs

By visual - estimation of the slope of the tangent line to the curve at \(t = 3\) hrs. Suppose the rise is 35 and the run is 1 (in appropriate units of distance and time), the slope (instantaneous speed) is 35 mph.

Answer:

(a) C. 10 mph
(b) C. 12.2 mph
(c) A. 0 mph
(d) B. 35 mph