QuestionA motorist traveled from Town A to B, averaging $54 \mathrm{~km/h}$ . If the first $\frac{1}{3}$ was at $45 \mathrm{~km/h}$ and he traveled 480 km after, find the speed for the last $\frac{2}{3}$ .

Studdy Solution

STEP 1

Assumptions1. The motorist traveled $\frac{1}{3}$ of the distance at an average speed of $45 \mathrm{~km/h}$ . . The motorist traveled the remaining distance of480 km to reach Town B.
3. The average speed for the entire journey was $54 \mathrm{~km/h}$ .
4. We need to find the average speed for the last $\frac{}{3}$ of the distance.

STEP 2

First, we need to find the total distance of the journey. We know that the remaining distance after the motorist traveled $\frac{1}{}$ of the distance is480 km, so the total distance is three times this remaining distance.
$Total\, distance = \times Remaining\, distance$

STEP 3

Now, plug in the given value for the remaining distance to calculate the total distance.
$Total\, distance =3 \times480 \mathrm{km}$

STEP 4

Calculate the total distance.
$Total\, distance =3 \times480 \mathrm{km} =1440 \mathrm{km}$

STEP 5

Now that we have the total distance, we can find the distance the motorist traveled at an average speed of $45 \mathrm{~km/h}$ . This is $\frac{1}{3}$ of the total distance.
$First\, part\, distance = \frac{1}{3} \times Total\, distance$

STEP 6

Plug in the value for the total distance to calculate the first part distance.
$First\, part\, distance = \frac{1}{3} \times1440 \mathrm{km}$

STEP 7

Calculate the first part distance.
$First\, part\, distance = \frac{1}{3} \times1440 \mathrm{km} =480 \mathrm{km}$

STEP 8

Now, we can find the time it took for the motorist to travel the first part of the distance. The time is the distance divided by the speed.
$First\, part\, time = \frac{First\, part\, distance}{First\, part\, speed}$

STEP 9

Plug in the values for the first part distance and the first part speed to calculate the first part time.
$First\, part\, time = \frac{480 \mathrm{km}}{45 \mathrm{~km/h}}$

STEP 10

Calculate the first part time.
$First\, part\, time = \frac{480 \mathrm{km}}{45 \mathrm{~km/h}} =10.67 \mathrm{hours}$

STEP 11

Now, we can find the total time for the journey. The total time is the total distance divided by the average speed.
$Total\, time = \frac{Total\, distance}{Average\, speed}$

STEP 12

Plug in the values for the total distance and the average speed to calculate the total time.
$Total\, time = \frac{1440 \mathrm{km}}{54 \mathrm{~km/h}}$

STEP 13

Calculate the total time.
$Total\, time = \frac{1440 \mathrm{km}}{54 \mathrm{~km/h}} =26.67 \mathrm{hours}$

STEP 14

Now, we can find the time it took for the motorist to travel the last $\frac{2}{3}$ of the distance. This is the total time minus the first part time.
$Last\, part\, time = Total\, time - First\, part\, time$

STEP 15

Plug in the values for the total time and the first part time to calculate the last part time.
$Last\, part\, time =26.67 \mathrm{hours} -10.67 \mathrm{hours}$

STEP 16

Calculate the last part time.
$Last\, part\, time =26.67 \mathrm{hours} -10.67 \mathrm{hours} =16 \mathrm{hours}$

STEP 17

Finally, we can find the average speed for the last $\frac{2}{3}$ of the distance. The average speed is the distance divided by the time.
$Last\, part\, speed = \frac{Last\, part\, distance}{Last\, part\, time}$

STEP 18

Plug in the values for the last part distance and the last part time to calculate the last part speed.
$Last\, part\, speed = \frac{480 \mathrm{km}}{16 \mathrm{hours}}$

STEP 19

Calculate the last part speed.
$Last\, part\, speed = \frac{480 \mathrm{km}}{16 \mathrm{hours}} =30 \mathrm{km/h}$ The average speed for the last $\frac{}{3}$ of the distance was $30 \mathrm{km/h}$ .

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