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.

Step2: Find the linear - equation formula

The general form of a linear equation is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept. The amount of dog food decreases by $1.5$ cups per day. So the equation is $y=-1.5x + 38$.

Step3: Graph the equation (a)

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

Step4: Find the y - intercept (b)

For the equation $y=-1.5x + 38$, the y - intercept $b = 38$. In the context of the problem, it means that when the bag is just opened ($x = 0$), there are 38 cups of dog food in the bag.

Step5: Find the slope (c)

The slope $m=-1.5$. In the context of the problem, it means that the amount of dog food in the bag decreases by 1.5 cups per day.

Step6: Find the x - intercept (d)

Set $y = 0$ in the equation $y=-1.5x + 38$. Then $0=-1.5x+38$, so $1.5x = 38$ and $x=\frac{38}{1.5}=\frac{76}{3}\approx25.33$. In the context of the problem, it means that after approximately 25.33 days, all the dog food in the bag will be used up.

For question 2:

Let two points on the line be $(0,2)$ and $(2,5)$.

Step1: Calculate the slope

The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 2}{2-0}=\frac{3}{2}$.

Step2: Find the y - intercept

The y - intercept $b$ is the y - value when $x = 0$. From the point $(0,2)$, $b = 2$.

Step3: Write the equation of the line

The equation of the line in slope - intercept form $y=mx + b$ is $y=\frac{3}{2}x+2$.

Answer:

1.
a. Graph with points $(0,38)$ and $(\frac{76}{3},0)$ connected by a line.
b. The y - intercept is 38. It means there are 38 cups of dog food when the bag is first opened.
c. The slope is - 1.5. It means the amount of dog food decreases by 1.5 cups per day.
d. The x - intercept is $\frac{76}{3}\approx25.33$. It means the dog food will be used up in approximately 25.33 days.

1. $y=\frac{3}{2}x + 2$