A company paid $110,000 for a new 18-wheeler. When it is 9 years old it will be worth $20,000. Using straight-line depreciation, the value of the trick in dollars, V, is a linear function of its age in years, n. Find the function and the value when the truck is 4 years old.
Write the value of the trick V as a function of its age in years, n.
To find the linear function that represents the value of the truck as it depreciates, we need to determine the depreciation rate per year.
The initial value of the truck is $110,000, and after 9 years, it will be worth $20,000.
The depreciation that occurs over 9 years is $110,000 - $20,000 = $90,000.
The depreciation per year is $90,000 / 9 = $10,000.
So, the linear function that represents the value of the truck can be written as:
V(n) = -$10,000n + $110,000
where V(n) is the value of the truck at age n in years.
To find the value of the truck when it is 4 years old, we substitute n = 4 into the function:
V(4) = -$10,000(4) + $110,000
V(4) = -$40,000 + $110,000
V(4) = $70,000
Therefore, the value of the truck when it is 4 years old is $70,000.
To find the linear function representing the value of the truck as it ages, we can use the formula for the equation of a straight line:
V = mx + b
Where V is the value of the truck, n is the age of the truck in years, m is the slope of the line, and b is the y-intercept (the value of the truck when it is brand new).
Given that the truck is worth $110,000 when it's brand new (n = 0), we have one point on the line: (0, 110000).
We are also given that when the truck is 9 years old (n = 9), it will be worth $20,000. This gives us another point on the line: (9, 20000).
To find the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Using the two points (0, 110000) and (9, 20000), we have:
m = (20000 - 110000) / (9 - 0)
m = -90000 / 9
m = -10000
Now, we can substitute the values of m and the y-intercept (b) into the equation V = mx + b:
V = -10000n + b
To find the value of the y-intercept (b), we can substitute the values of one of the points into the equation. Let's use the point (0, 110000):
110000 = -10000(0) + b
110000 = b
Hence, the value of the truck V as a function of its age n is given by:
V = -10000n + 110000
To find the value of the truck when it is 4 years old (n = 4), we substitute n = 4 into the equation:
V = -10000(4) + 110000
V = -40000 + 110000
V = 70000
Therefore, the value of the truck when it is 4 years old is $70,000.
it loses $90,000 in value over 9 years ... $10,000 per year ... slope
the original value is $110,000 ... at 0 years ... intercept