Question 995862: In 2002, Bill bought a truck for $30,000. The value of the truck has been decreasing by $2600 every three years. In the same year, Bill also bought a painting from a starving artist at a farmer's market for $50. The artist then became famous and the value of the painting has been increasing at a rate of $800 per year since. Let x be the number of years after 2002.
a)Write the value of the truck as a linear function of x
b)Write the value of the painting as a linear function of x
c)In what year will the painting be worth as much as the truck? What is the value of each item in that year?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! truck = 30,000
depreciation = 2600 every 3 years.
painting = 50.
appreciation = 800 every year.
x = number of years since 2002.
x = 0 = 2002
x = 1 = 2003
etc.
y1 = value of the truck in year x.
value of the truck in year x is given by the formula:
y1 = -2600/3 * x + 30000.
y2 = value of the painting in year x.
value of the painting in year x is given by the formula:
y2 = 800 * x + 50.
your break even point is when y1 = y2.
y1 = y2 is the same as -2600/3 * x + 30000 = 800 * x + 50 because:
y1 can be replaced with its equivalent value of -2600/3 * x + 30000.
y2 can be replaced with its equivalent value of 800 * x + 50.
start with:
-2600/3 * x + 30000 = 800 * x + 50
add 2600/3 * x to both sides of the equation and subtract 50 from both sides of the equation to get:
30000 - 50 = 800 * x + 2600/3 * x
combine like terms on the left side of the equation and multiply 800 by 3/3 on the right side of the equation to get:
29950 = 2400/3 * x + 2600/3 * x
combine like terms on the right side of the equation to get:
29950 = 5000/3 * x
divide both sides of the equation by (5000/3) to get:
29950 / (5000/3) = x
this is the same as 29950 * 3 / 5000 = x
solve for x to get:
x = 17.97
your break even point hould be when x = 17.97.
that would be somewhere between x = 17 and x = 18.
when x = 17, the year is 2002 + 17 = 2019.
when x = 18, the year is 2002 + 18 = 2020.
the breakeven point is in the year 2019.
both equations can be graphed and their intersection is the break even point.
the graph looks like this:
you can see that, at the 3 year point, the truck has depreciated by 2600 because 30000 - 2600 = 27400.
you can also see on the graph that, at the 3 year point, the painting has appreciated by 2400 because 50 + 2400 = 2450.
2400 is the result of 3 years appreciation at 800 per year because 800 * 3 = 2400.
the value of the painting and the truck at the breakeven point is 14426 as shown on the graph.
this point is found by calculating the value of the truck and the value of the painting when x = 17.97.
800 * 17.97 + 50 = 14426.
-2600/3 * 17.97 + 30000 = 14426.
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