A charter fishing company buys a new boat for $224,000 and assumes that it will have a trade-in value of $115,200 after 16 years. Find a linear model for the depreciated value V of the boat t years after it was purchased.
change in value = 224000-115200 = 108800
change in time = 16 yrs
rate of change = 108800/16 = 6800
value = -6800t + 224000 , where t is the number of years
To find a linear model for the depreciated value of the boat, we can use the formula for a straight line, which is given as:
V = mt + b
Where V is the value of the boat, t is the time in years, m is the slope of the line, and b is the y-intercept.
Given that the initial price of the boat is $224,000 and the trade-in value after 16 years is $115,200, we can use these two points to find the slope of the line.
Let's calculate the slope (m) first:
m = (y2 - y1) / (x2 - x1)
x1 = 0 years (time of purchase)
y1 = $224,000 (initial price)
x2 = 16 years
y2 = $115,200 (trade-in value)
m = (115,200 - 224,000) / (16 - 0)
m = -108,800 / 16
m = -6,800
Now, let's find the y-intercept (b).
To find b, we can substitute the values of one of the points (x1, y1) into the equation and solve for b.
Using (0, $224,000):
224,000 = -6,800(0) + b
224,000 = b
So, b = 224,000.
Therefore, the linear model for the depreciated value V of the boat t years after it was purchased is:
V = -6,800t + 224,000
To find a linear model for the depreciated value of the boat, we can use the formula for a line in slope-intercept form, which is given by:
y = mx + b
In this case, the depreciated value V of the boat is the dependent variable (y), and the number of years t after the boat was purchased is the independent variable (x). We need to find the slope (m) and y-intercept (b) of the line.
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's use the given information to calculate the slope:
y1 = $224,000 (initial price of the boat)
y2 = $115,200 (trade-in value after 16 years)
x1 = 0 years
x2 = 16 years
m = ($115,200 - $224,000) / (16 - 0)
m = -$108,800 / 16
m = -$6,800
The slope (m) represents the rate at which the value of the boat decreases per year.
Now, we need to find the y-intercept (b). In this case, the y-intercept will be the initial price of the boat ($224,000).
Therefore, the linear model for the depreciated value V of the boat t years after it was purchased is:
V = -6,800t + 224,000