QuestionGraph the distance from home after $x$ hours for a trip to a park 20 miles away at $40 \mathrm{mph}$ , 2 hours at the park, then return at $20 \mathrm{mph}$ .

Studdy Solution

STEP 1

Assumptions1. The person drives at a constant speed of40 mph to the park. . The park is20 miles away from home.
3. The person stays at the park for hours.
4. The person drives back home at a constant speed of20 mph.
5. The person does not stop or change speed during the journey, except at the park.

STEP 2

First, we need to calculate the time it takes to reach the park. We can do this by dividing the distance to the park by the speed of the person.
$Time_{to\_park} = \frac{Distance_{to\_park}}{Speed_{to\_park}}$

STEP 3

Now, plug in the given values for the distance to the park and the speed to the park to calculate the time to the park.
$Time_{to\_park} = \frac{20 \, miles}{40 \, mph}$

STEP 4

Calculate the time it takes to reach the park.
$Time_{to\_park} = \frac{20 \, miles}{40 \, mph} =0. \, hours$

STEP 5

Next, we need to calculate the time it takes to return home. We can do this by dividing the distance to home (which is the same as the distance to the park) by the speed of the person.
$Time_{to\_home} = \frac{Distance_{to\_home}}{Speed_{to\_home}}$

STEP 6

Now, plug in the given values for the distance to home and the speed to home to calculate the time to home.
$Time_{to\_home} = \frac{20 \, miles}{20 \, mph}$

STEP 7

Calculate the time it takes to return home.
$Time_{to\_home} = \frac{20 \, miles}{20 \, mph} =1 \, hour$

STEP 8

Now, we can sketch the graph. On the x-axis, we have time in hours and on the y-axis, we have distance in miles.1. From0 to0.5 hours, the person is driving to the park at40 mph. The distance from home increases linearly from0 to20 miles.
2. From0.5 to2.5 hours, the person is at the park. The distance from home remains constant at20 miles.
3. From2.5 to3.5 hours, the person is driving home at20 mph. The distance from home decreases linearly from20 to0 miles.

STEP 9

The graph is a piecewise function, defined as follows $Distance(x) =\begin{cases}40x & \text{if } \leq x <.5 \\ 20 & \text{if }.5 \leq x <2.5 \\ 40 -20(x -2.5) & \text{if }2.5 \leq x \leq3.5\end{cases}$

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