Question

grain, precious metals, electricity, oil, beef, orange juice,
and natural gas are traditional examples of commodities.
during uncertain times, commodity prices can fluctuate
rapidly during a single day of trading.
the graph given below shows the price of oil as it varied
with time during a volatile day of trading.
the y - axis represented its price per barrel and the x - axis
represented the number of minutes elapsed since the
opening bell at 9:30 am.
at the opening bell, the price of oil was $56 per
barrel.
for the first 140 minutes of trading, the price of oil
varied with time according to an absolute value
function with its vertex at (80,32).
for the next 150 minutes of trading, the price of oil
varied with time according to a new absolute value
function with its vertex at (230,38).
after 290 minutes, the price of oil varied with time
according to a new absolute value function with a
vertex at (310,62) until trading closed for the day.
at what time did the price of oil trade at $22 per barrel?

Explanation:

Step1: Find the equation for the last - part (after 290 minutes)

The general form of an absolute - value function is \(y=a|x - h|+k\), where \((h,k)\) is the vertex. For the function with vertex \((310,62)\), the equation is \(y=a|x - 310|+62\).
We know that the function is decreasing (since the price is going down after the vertex at \(x = 310\)), so \(a<0\). Let's use the fact that we want to find when \(y = 22\).
Substitute \(y = 22\) into \(y=a|x - 310|+62\), we get \(22=a|x - 310|+62\), then \(a|x - 310|=-40\).
Since the function is a straight - line (absolute - value function) after \(x = 310\) and we assume the slope is constant. Let's find the slope between two known points. But we can also solve the equation \(22=-|x - 310|+62\) (assuming \(a=-1\) for simplicity, because the rate of change of an absolute - value function \(y =-|x - h|+k\) has a slope of \(- 1\) for \(x>h\)).

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Since \(x>310\) (because we are considering the part after the vertex at \(x = 310\)), \(x-310 = 40\)

Step2: Calculate the time

Solve \(x-310 = 40\) for \(x\), we get \(x=350\) minutes.
Since the opening bell is at \(9:30\) am, and \(350\) minutes \(=5\) hours and \(50\) minutes (\(350\div60 = 5\) hours and \(50\) minutes).
\(9\) hours \(30\) minutes \(+5\) hours \(50\) minutes \(=3:20\) pm

Answer:

\(3:20\) pm