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(1 pt) Suppose that in the year 2000, the cost of a compact car averaged $ 12600 and that in 2003, the cost averaged $ 14100. Assuming that the relationship between year t and cost c is linear, develop a formula for predicting the average cost of a compact car in the future measuring the variable t as years since 2000. Be sure to use the lower case variables t and c. .
What do you predict the average cost of a compact car will be in the year 2008?
Well, if we assume that the relationship is linear, we can calculate the slope of the line using the formula:
slope, m = (change in cost)/(change in years)
So, using the given information:
m = (14100 - 12600)/(2003 - 2000) = 1500/3 = 500
Now that we have the slope, we can use the equation of a straight line to predict the average cost in the year 2008 (8 years since 2000):
c = mt + b
where c is the cost, m is the slope, t is the number of years since 2000, and b is the y-intercept.
Since we don't know the y-intercept, we can substitute the values we have into the equation and solve for b:
$14100 = 500(3) + b
$14100 = 1500 + b
b = $14100 - $1500
b = $12600
Now that we have the y-intercept, we can substitute it back into the equation and solve for the cost in 2008 (t = 8):
c = 500(8) + 12600
c = 4000 + 12600
c = $16,600
So, according to my calculations, I predict that the average cost of a compact car in 2008 will be $16,600. But hey, prices are known to fluctuate, so take it with a grain of clownish humor!
To develop a formula for predicting the average cost of a compact car in the future, we can use the concept of a linear relationship between the variables t (years since 2000) and c (cost).
First, we need to calculate the slope of the line. The slope represents how much the cost changes per year. We can use the formula:
slope (m) = (change in cost) / (change in years)
Given that the cost in 2000 was $12600 and in 2003 was $14100, the change in cost is $14100 - $12600 = $1500. The change in years is 2003 - 2000 = 3. Therefore, the slope is:
m = (change in cost) / (change in years) = $1500 / 3 = $500 per year
Next, we need to determine the y-intercept of the line. The y-intercept represents the cost at year t = 0 (year 2000). We can use the formula:
y-intercept (b) = c - (m * t)
Using the cost in 2000 ($12600), the slope (m = $500), and substituting t = 0, we can solve for the y-intercept:
b = $12600 - ($500 * 0) = $12600
Now we can write the equation for predicting the average cost of a compact car in the future:
c = m * t + b
c = $500 * t + $12600
To predict the average cost of a compact car in the year 2008 (t = 2008 - 2000 = 8), we can substitute t = 8 into the equation:
c = $500 * 8 + $12600
c = $4000 + $12600
c = $16600
Therefore, the predicted average cost of a compact car in the year 2008 is $16,600.
To develop a formula for predicting the average cost of a compact car in the future, we need to determine the equation of the linear relationship between the year t and the cost c.
First, let's define t as the number of years since 2000. In 2000, t would be 0, and in 2003, t would be 3.
From the information given, we have two data points: (0, $12600) and (3, $14100). These points represent the years and their corresponding average costs of compact cars.
To find the equation of the line, we need to calculate the slope (m) and the y-intercept (b) using the formula:
m = (c2 - c1) / (t2 - t1)
Here, c1 = $12600, c2 = $14100, t1 = 0, t2 = 3.
So, plugging in these values:
m = ($14100 - $12600) / (3 - 0)
= $1500 / 3
= $500
The slope of the line is $500.
Now, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where y is the cost (c), x is the year (t), and (x1, y1) is any point on the line. We can use either of the given data points.
Let's use the point (0, $12600):
c - $12600 = $500(t - 0)
c - $12600 = $500t
c = $500t + $12600
Now, we have our formula for predicting the average cost of a compact car in the future based on the number of years since 2000.
To predict the average cost in the year 2008, we substitute t = 2008 - 2000 = 8 into the formula:
c = $500 * 8 + $12600
c = $4000 + $12600
c = $16600
Therefore, we predict that the average cost of a compact car in the year 2008 will be $16600.