Video Transcript
John and Tom worked for 140 minutes
to fill two holes of the same depth with the same type of sand. The depth of the holes is 60
feet. The given graph shows the depth of
the two holes at 20-minute intervals. Which of the following statements
correctly compares the average rates at which the depth of the two holes
changed?
There are four possible options,
which we will create some space to write down.
Option A, in the interval from zero
to 20 minutes, the rate of change in the depth of Tom’s hole is of greater magnitude
than John’s. Whereas in the interval from 120 to
140 minutes, it is the other way round. Option B, in every 20-minute
interval, the magnitude of the rate of change in the depth of Tom’s hole is greater
than John’s. Option C, in every 20-minute
interval, the magnitude of the rate of change in the depth of John’s hole is greater
than Tom’s. And option D, in the interval from
zero to 20 minutes, the rate of change in the depth of John’s hole is of greater
magnitude than Tom’s. Whereas in the interval from 120 to
140 minutes, it is the other way round.
The rate of change is the slope or
gradient between two points. The steeper the slope, then the
greater the rate of change. Based on our four options, there
are two key time frames, from zero to 20 minutes and 120 to 140 minutes. If we firstly consider John’s hole,
then the orange lines represent the rates of change. From zero to 20 minutes, the depth
of John’s hole decreased from 60 feet to around 53 feet. This was a decrease of seven
feet. From 120 to 140 minutes, John’s
hole decreased in depth from 35 feet to 30 feet. This is a decrease of five
feet.
The rate of change in depth of
Tom’s hole is shown by the pink lines. From zero to 20 minutes, the depth
of Tom’s hole decreased from 60 feet to 41 feet. This is a decrease of 19 feet. From 120 minutes to 140 minutes,
Tom’s hole decreased from 26 feet to 25 feet. This means that it only decreased
by one foot in this 20-minute period. The decrease in feet was greater
for Tom from zero to 20 minutes, but greater for John from 120 to 140 minutes. We can, therefore, conclude that
Tom had the greater rate of change from zero to 20 minutes. And John had the greater rate of
change from 120 to 140 minutes.
This means that we can rule out
options B and C. As option B said that the rate of
change was always greater for Tom. Whereas option C said the rate of
change was always greater for John. We need the option where Tom’s rate
of change was greater for the first period but John’s rate of change was greater for
the final period. The correct answer is option A. In the interval from zero to 20
minutes, the rate of change in the depth of Tom’s hole is of greater magnitude than
John’s. Whereas in the interval 120 to 140
minutes, it is the other way round, John’s is greater than Tom’s.
We could have answered this without
doing any calculations and just looking at the lines on the graph. From zero to 20 minutes, the pink
line is steeper. Therefore, there is a greater rate
of change for Tom. From 120 to 140 minutes, the orange
line is steeper. Therefore, John has a greater rate
of change.
