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Question 476549: Oil is pumped from an oil field at a constant rate each year, so that its oil reserves have been decreasing linearly with time. Geologists estimate that the field reserves were 400,000 barrels in 1980 and 320,000 barrels in 1990.
(a) Write an equation describing the amount of oil left in the field at any time.
(b) If the trend continues, when will the oil well dry out?
(c) Interpret the slope.
thank you very much, appreciate if you could guide me. I don't seems to understand what I am looking for.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the reserves were 400,000 in 1980.
the reserves were 320,000 in 1990.
that means the reserves lost 80,000 barrels between 1980 and 1990.
that's 80,000 in 10 years which is an average straight line loss of 8,000 barrels per year.
your trend is a straight line.
the equation for a straight line is y = mx + b where m is the slope and b is the y intercept.
y is the number of barrels of oil reserves.
x is the number of years.
m is equal to the slope.
the slope is equal to the change in the value of y divided by the change in the value of x.
the change in the value of y is 320,000 minus 400,000 = -80,000
the change in the value of x is 1990 minus 1980 = 10
the slope is equal to -80,000/10 = -8,000.
the equation of y = mx + b becomes:
y = -8000x + b
the value of b is the value of y when x is equal to 0.
since you want 1980 to be the starting point of your equation, you subtract 1980 from all x values to get:
x = 0 means 1980
x = 10 means 1990
x = 20 means 2000
etc.
since the value of b is the value of y when x is equal to 0, then the value of b will be 400,000 because the reserves contain 400,000 barrels in 1980 which you just made equivalent to year 0.
your equation therefore becomes:
y = -8,000x + 400,000
when will the oil reserves run out?
they will run out when y = 0.
replace y with 0 in your equation and solve for x.
your equation of:
y = -8,000x + 400,000 becomes:
0 = -8,000x + 400,000
add 8000x to both sides of this equation to get:
8,000x = 400,000
divide both sides of this equation by 8000 to get:
x = 400,000/8,000 = 400/8 = 50 years
at the present rate of consumption, the oil will run out in 50 years.
a graph of the equation of :
y = -8,000x + 400,000 is shown below:
the graph shows the number of barrels in thousands.
400 is equal to 400,000
320 is equal to 320,000
etc.
the horizontal lines are at 400 and 320.
400 is the value when x = 0.
320 is the value when x = 10
that's equivalent to 400,000 when x = 0 and 320,000 when x = 10
the graph goes to 0 when x = 50.
that's when the oil runs out at the present rate of consumption.
to find the value of x when y = 320, just drop a vertical line from the intersection of the line with the equation of y = 320 (the horizontal line) and the line of the equation y = -8x + 400 (the line that slants down from the left to the right).
remember that the graph is in thousands.
the original equation is y = -8,000x + 400,000
divide the right side of the equation by 1,000 and you get the result in thousands.
the equation becomes y = -8x + 400
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