Question

1. jada and lin refill their water bottles at a drinking fountain after basketball practice. the graph represents the amount of water in their bottles as they fill it up at the drinking fountain.

a which line represents a proportional relationship between time and the amount of water in the bottle? explain your thinking.
b what does the point (0, 0) mean on the line representing jadas refill?
c what does the point (0, 200) mean on the line representing lins refill?
d which line represents the bottle that was filled at a faster rate? explain your thinking.

Explanation:

Part a

Step1: Recall the property of proportional relationship

A proportional relationship between two variables \(x\) (time) and \(y\) (amount of water) has the form \(y = kx\), and its graph passes through the origin \((0,0)\).

Step2: Identify the line

Looking at the graph, Jada's refill line passes through the origin \((0,0)\). So Jada's line represents a proportional relationship between time and the amount of water in the bottle.

Part b

Step1: Interpret the coordinates

In the context of the problem, the \(x -\) coordinate represents time (in seconds) and the \(y -\) coordinate represents the amount of water (in mL).

Step2: Analyze the point \((0,0)\)

When \(x = 0\) (at the start, time \(t=0\) seconds), \(y = 0\) (the amount of water in Jada's bottle is \(0\) mL). So, at time \(t = 0\) seconds, Jada's bottle has \(0\) mL of water.

Part c

Step1: Interpret the coordinates

The \(x -\) coordinate is \(0\) (start - time \(t = 0\) seconds) and the \(y -\) coordinate is \(200\).

Step2: Analyze the point \((0,200)\)

When \(t=0\) seconds (at the start of refilling), Lin's bottle already has \(200\) mL of water.

Part d

Step1: Recall the formula for the rate of change

The rate of change (slope) of a line \(y=mx + b\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a proportional relationship \(y = kx\), \(k\) is the slope. Let's assume two points \((x_1,y_1)\) and \((x_2,y_2)\) on Jada's line \((y = k_Jx)\). Suppose \((x_1 = 0,y_1=0)\) and \((x_2 = 8,y_2 = 1000)\), then \(k_J=\frac{1000 - 0}{8-0}=125\) mL/s. For Lin's line \(y=k_Lx + 200\), assume \((x_1 = 0,y_1 = 200)\) and \((x_2=6,y_2 = 1000)\). Then \(k_L=\frac{1000 - 200}{6-0}=\frac{800}{6}=\frac{400}{3}\approx133.3\) mL/s.

Step2: Compare the slopes

Another way (without calculation): The steeper the line, the greater the rate of change. Jada's line is steeper. So Jada's line represents the bottle that was filled at a faster rate.

Answer:

a. Jada's line. A proportional relationship graph passes through \((0,0)\), and Jada's line does.
b. At time \(t = 0\) seconds, Jada's bottle has \(0\) mL of water.
c. At time \(t = 0\) seconds, Lin's bottle has \(200\) mL of water.
d. Jada's line. A steeper line indicates a greater rate of change (more water per second).