1. Carl's constant rate of speed is the slope of the straight line graph.
This straight line passes through: (0,0), (5,1), (10,2) etc
We can use the slope formula with any two points to find the slope of this line.
The slope formula is [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex].
Let [tex](x_1,y_1)=(0,0)[/tex] and [tex](x_2,y_2)=(5,1)[/tex]
Then [tex]m=\frac{1-0}{5-0}[/tex], [tex]\implies m=\frac{1}{5}[/tex].
Carl's speed is [tex]\frac{1}{5}[/tex] miles per minute.
But we must leave our answer in miles per hour
Hence Carl's speed is [tex]\frac{1}{5}\times 60=12[/tex] miles per hour
After 2 hours, Carl will travel [tex]12\times 2=24[/tex] miles.
2. The given line has equation [tex]-4x+y=10[/tex]
We write this in slope-intercept form by solving for y.
[tex]\implies y=4x+10[/tex]
This is in the form [tex]y=mx+c[/tex], where [tex]m=4[/tex] is the slope.
When x=3, [tex]y=4(3)+10[/tex]
[tex]\implies y=12+10=22[/tex]
When x=3, y=22
3. The given straight line graph that models the situation passes through:
(0,0) and (20,30).
The slope of this line is [tex]\frac{Rise}{Run}=\frac{300}{20}=15[/tex]
Therefore the rate is $ 15 per ticket.
If the theater sells 150 tickets, the earnings will be: [tex]150\times 15=2,250[/tex] dollars.
