Question

check your understanding

1. imani buys dog food in a 12 - pound bag, amounting to 38 cups of food. each day she feeds her dog 1 1/2 cups of dog food.

a. graph the relationship between the number of cups of dog food remaining in the bag, and the days since opening a new bag of dog food.
b. what is the y - intercept of this line? what does this mean in the context of this problem?
c. what is the slope of this line? what does this mean in the context of this problem?
d. what is the x - intercept of this line? what does this mean in the context of this problem?

1. write the equation of the line shown.

Explanation:

Step1: Define variables

Let $x$ be the number of days since opening the bag and $y$ be the number of cups of dog - food remaining. The initial amount of dog food is 38 cups and 1.5 cups are used per day. So the linear relationship is $y = 38-1.5x$.

Step2: Find points for graphing (a)

When $x = 0$, $y=38$. When $y = 0$, we solve $0 = 38 - 1.5x$ for $x$.
$1.5x=38$, so $x=\frac{38}{1.5}=\frac{76}{3}\approx25.33$. Plot the points $(0,38)$ and $(\frac{76}{3},0)$ and draw a line through them.

Step3: Find y - intercept (b)

The equation of the line is $y=-1.5x + 38$. In the slope - intercept form $y = mx + b$, where $b$ is the y - intercept. Here, $b = 38$. It means that when the bag is just opened ($x = 0$), there are 38 cups of dog food in the bag.

Step4: Find slope (c)

The equation is $y=-1.5x + 38$, and the slope $m=-1.5$. It means that the number of cups of dog food in the bag decreases by 1.5 cups per day.

Step5: Find x - intercept (d)

Set $y = 0$ in $y=-1.5x + 38$. We get $1.5x=38$, so $x=\frac{38}{1.5}=\frac{76}{3}\approx25.33$. It means that after approximately 25.33 days, all the dog food in the bag will be used up.

Step6: Find equation of line in 2 (using slope - intercept form)

The line passes through the points $(0,2)$ and $(2,5)$. The slope $m=\frac{5 - 2}{2-0}=\frac{3}{2}=1.5$. The y - intercept $b = 2$. So the equation of the line is $y = 1.5x+2$.

Answer:

a. Plot points $(0,38)$ and $(\frac{76}{3},0)$ and draw a line.
b. y - intercept is 38. It means there are 38 cups of dog food when the bag is just opened.
c. Slope is - 1.5. It means the amount of dog food decreases by 1.5 cups per day.
d. x - intercept is $\frac{76}{3}\approx25.33$. It means the dog food will be used up in about 25.33 days.

1. The equation of the line is $y = 1.5x+2$.