Question

A jogger runs from her home to a point A, which is 6 km away. For there 6 km, she
begins by running at a constant speed till she reaches a hilly portion 2 km from her
home. Here her speed slows down while she runs up the hill, which is a 1-km run.
Then she speeds up while running down the hill. The last 2 km of the run are again at
constant speed. Draw a graph to show the jogger’s speed as a function of the distance
from her home. Also find the range of this function.

Answer 1

Answer on Question #76669, Math / Calculus

A jogger runs from her home to a point A, which is $6\mathrm{km}$ away. For there $6\mathrm{km}$ , she begins by running at a constant speed till she reaches a hilly portion $2\mathrm{km}$ from her home. Here her speed slows down while she runs up the hill, which is a 1-km run. Then she speeds up while running down the hill. The last $2\mathrm{km}$ of the run are again at constant speed. Draw a graph to show the jogger's speed as a function of the distance from her home. Also find the range of this function.

Solution

Divide the distance into 4 subintervals:

[0,2] - interval, where $V = V_{1} = \text{const}$ .

[2,3] - interval, where $V$ decreases from $V_{1}$ to $V_{\min}$ .

[3,4] - interval, where $V$ increases from $V_{\min}$ to $V_{2}$ .

[4,6] - interval, where $V = V_{2} = \text{const}$ .

How precisely does the velocity depend on the distance on the intervals [2, 3] and [3, 4]?

Is $V_{1}$ equal to $V_{2}$ , greater than $V_{2}$ or less than $V_{2}$ ?

It is undefined.

$V_{1} < V_{2}$

The function $V(s)$ is constant on $(0,2)$ , decreases on $(2,3)$ , increases on $(3,4)$ , and is constant on $(4,6)$ .

Range: $[V_{min}, \max(V_1, V_2)]$

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