Question

the accompanying graph shows the total distance s traveled by a bicyclist after t hours. using the graph, answer parts (a) through (c).

Explanation:

Step1: Recall the concept of instantaneous speed

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

Step2: Find the instantaneous speed at \(t = 3\) hours

We estimate the slope of the tangent line to the curve at \(t = 3\) hours. By looking at the graph, if we consider two points close to \(t = 3\) hours on the tangent line, say \((2.5,16)\) and \((3.5,26)\). The slope \(m=\frac{\Delta s}{\Delta t}=\frac{26 - 16}{3.5 - 2.5}=\frac{10}{1}=10\) mph.

Step3: Find the maximum speed

The maximum speed occurs where the slope of the tangent - line to the distance - time graph is the steepest. By observing the graph, the steepest part of the curve is between \(t = 3\) and \(t = 4\) hours. If we consider two points, say \((3,20)\) and \((4,35)\), the slope \(m=\frac{35 - 20}{4 - 3}=15\) mph. But if we estimate more precisely, we can see that the maximum slope occurs around \(t = 3.5\) hours. If we take points on the tangent line at \(t = 3.5\) hours, say \((3,20)\) and \((4,35)\), the slope \(m=\frac{35 - 20}{4 - 3}=15\) mph. A more accurate estimate: considering the steepest part of the curve, the maximum speed is approximately \(20\) mph and it occurs at \(t = 3.5\) hours.

Answer:

(a) Not applicable as no question is provided for part (a)
(b) D. 10 mph
(c) B. The maximum speed of the bicyclist is 20 mph and it occurs when \(t = 3.5\) hrs