The license fee for a car varies directly with the value of the car. The fee for a car that is valued at $19,500 is $292.50. a) What is the constant of proportionality? b) Write the equation that shows the license fee as a function of the value of the car. c) If you were to draw the graph of this function, what would be the slope of the line? d) If you were to draw the graph of this function, what would be the y-intercept? e) At this rate, what is the license fee for a car that is valued at $25,400?
a) To find the constant of proportionality, we can set up a proportion using the given information. Let's denote the constant of proportionality as k. We can set up the following proportion:
fee/value = k
We know that the fee for a car valued at $19,500 is $292.50, so the proportion becomes:
292.50/19,500 = k
Simplifying, we find:
k ≈ 0.015 (rounded to three decimal places)
Therefore, the constant of proportionality is approximately 0.015.
b) Now that we have the constant of proportionality, we can write the equation that shows the license fee as a function of the value of the car. Let's denote the license fee as F and the value of the car as V. The equation can be written as:
F = kV
Substituting the value of k from part a:
F = 0.015V
c) If we were to draw the graph of this function, the slope of the line would be equal to the constant of proportionality, which is 0.015. The slope represents the rate of change, in this case, the increase in license fee for each unit increase in the value of the car.
d) To find the y-intercept, we need to substitute the given information into the equation and solve for F when V = 0 (since the y-intercept occurs when the value of the car is zero).
Using the given information that the fee for a car valued at $19,500 is $292.50:
292.50 = 0.015(19,500)
Simplifying, we find that the y-intercept is approximately $292.50.
Therefore, the equation of the line relating the license fee to the value of the car is:
F = 0.015V + 292.50
e) Finally, to find the license fee for a car valued at $25,400, we can substitute this value into the equation and solve for F:
F = 0.015(25,400) + 292.50
Calculating, the license fee for a car valued at $25,400 is approximately $387.00.