Question

the equation and graph show the distance traveled by a convertible and a limousine in miles, y, as a function of time in hours, x. convertible: y = 35x compare the speeds of the two vehicles. the rate of change of the limousine is? the rate of change of the convertible. the convertible is? than the limousine.

Explanation:

Step1: Find rate of convertible

The equation for the convertible is \( y = 35x \). In the slope - intercept form \( y=mx + b \) (where \( m \) is the slope, representing the rate of change), the slope \( m = 35 \). So the rate of change (speed) of the convertible is 35 miles per hour.

Step2: Find rate of limousine

To find the rate of change (speed) of the limousine, we use the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). From the graph, when \( x = 2 \) hours, \( y=60 \) miles (we can also use other points, for example, when \( x = 1 \), \( y = 30 \); when \( x=3 \), \( y = 90 \)). Using the points \( (0,0) \) and \( (2,60) \), the slope \( m=\frac{60 - 0}{2 - 0}=\frac{60}{2}=30 \) miles per hour. So the rate of change (speed) of the limousine is 30 miles per hour.

Step3: Compare the rates

Since \( 30<35 \), the rate of change of the limousine is less than the rate of change of the convertible. And the convertible is faster than the limousine.

Answer:

The rate of change of the limousine is less than the rate of change of the convertible. The convertible is faster than the limousine.