Starting at home, person( 1 ) traveled uphill to the store( 1 ) store for TIME_UP minutes at just RATE_UP mph. He( 1 ) then traveled back home along the same path downhill at a speed of K * RATE_UP mph.
What is his( 1 ) average speed for the entire trip from home to the store( 1 ) store and back?
The average speed is not just the average of RATE_UP mph and RATE_DOWN mph.
He( 1 ) traveled for a longer time uphill (since he( 1 ) was going slower), so we can estimate that the average speed is closer to RATE_UP mph than RATE_DOWN mph.
To calculate the average speed, we will make use of the following:
\text{average speed} = \dfrac{\color{KhanUtil.BLUE}{\text{total distance}}}{\color{KhanUtil.ORANGE}{\text{total time}}}
\text{distance uphill} = \text{distance downhill}
What was the total distance traveled?
\color{KhanUtil.BLUE}{\begin{align*}\text{total distance} &= \text{distance uphill} + \text{distance downhill}\\
&= 2 \times \text{distance uphill}\end{align*}}
\begin{align*}\text{distance uphill} &= \text{speed uphill} \times \text{time uphill}\\
&= RATE_UP\text{ mph} \times TIME_UP\text{ minutes}\\
&= DISTANCE\text{ miles}\end{align*}
Substituting to find the total distance:
\color{KhanUtil.BLUE}{\text{total distance} = 2 * DISTANCE\text{ miles}}
What was the total time spent traveling?
\color{KhanUtil.ORANGE}{\text{total time} = \text{time uphill} + \text{time downhill}}
\begin{align*}\text{time downhill} &= \dfrac{\text{distance downhill}}{\text{speed downhill}}\\
&= \dfrac{DISTANCE\text{ miles}}{RATE_DOWN\text{ mph}}\\
&= TIME_DOWN\text{ minutes}\end{align*}
\color{KhanUtil.ORANGE}{\begin{align*}\text{total time} &= TIME_UP\text{ minutes} + TIME_DOWN\text{ minutes}\\
&= TIME_UP + TIME_DOWN\text{ minutes}\end{align*}}
Now that we know both the total distance and total time, we can find the average speed.
\begin{align*}\text{average speed} &= \dfrac{\color{KhanUtil.BLUE}{\text{total distance}}}{\color{KhanUtil.ORANGE}{\text{total time}}}\\
&= \dfrac{\color{KhanUtil.BLUE}{2 * DISTANCE\text{ miles}}}{\color{KhanUtil.ORANGE}{TIME_UP + TIME_DOWN\text{ minutes}}}\\
&= RATE_AVG\text{ mph}\end{align*}
The average speed is RATE_AVG mph, and which is closer to RATE_UP mph than RATE_DOWN mph as we expected.
It takes TIME_INIT minutes for PEOPLE_INIT people to paint WALL_INIT walls.
How many minutes does it take PEOPLE_FINAL people to paint WALL_FINAL walls?
Imagine that each person is assigned one wall, and all PEOPLE_INIT people begin painting at the same time.
Since everyone will finish painting their assigned wall after TIME_INIT minutes, it takes one person TIME_INIT minutes to paint one wall.
If we have PEOPLE_FINAL people and WALL_FINAL walls, we can again assign one wall to each person.
Everyone will take TIME_FINAL minutes to paint their assigned wall.
In other words, it takes TIME_FINAL minutes for PEOPLE_FINAL people to paint WALL_FINAL walls.
PEOPLE_INIT people can paint WALL_INIT walls in TIME_INIT minutes.
How many minutes will it take for PEOPLE_FINAL people to paint WALL_FINAL walls? Round to the nearest minute.
the number of minutes, rounded to the nearest minute
We know the following about the number of walls w
painted by p
people in t
minutes at a constant rate r
.
w = r \cdot t \cdot p
\begin{align*}w &= WALL_INIT\text{ walls}\\
p &= PEOPLE_INIT\text{ people}\\
t &= TIME_INIT\text{ minutes}\end{align*}
Substituting known values and solving for r
:
r = \dfrac{w}{t \cdot p}= \dfrac{WALL_INIT}{TIME_INIT \cdot PEOPLE_INIT} = fractionReduce( WALL_INIT, TIME_INIT * PEOPLE_INIT )\text{ walls painted per minute per person}
We can now calculate the amount of time to paint WALL_FINAL walls with PEOPLE_FINAL people.
t = \dfrac{w}{r \cdot p} = \dfrac{WALL_FINAL}{fractionReduce( WALL_INIT, TIME_INIT * PEOPLE_INIT ) \cdot PEOPLE_FINAL} = \dfrac{WALL_FINAL}{fractionReduce( WALL_INIT * PEOPLE_FINAL, TIME_INIT * PEOPLE_INIT )} = fractionReduce( WALL_FINAL * TIME_INIT * PEOPLE_INIT, WALL_INIT * PEOPLE_FINAL )\text{ minutes}
= mixedFractionFromImproper( WALL_FINAL * TIME_INIT * PEOPLE_INIT, WALL_INIT * PEOPLE_FINAL, true, true )\text{ minutes}
Round to the nearest minute:
t = TIME_FINAL\text{ minutes}