To determine the correct sketch of the graph that represents the situation described, we need to analyze the scenario where a bathtub starts with 40 gallons of water and drains at a constant rate of 8 gallons per minute.
- Initial Condition: At time \(t = 0\) minutes, the amount of water is 40 gallons.
- Rate of Draining: The water drains at a constant rate of 8 gallons per minute.
Using this information, we can derive a linear equation for the amount of water in the tub over time. The equation can be represented as:
\[ \text{Gallons of water} = 40 - 8t \]
where \(t\) is the time in minutes.
Key points to consider:
- At \(t = 0\): Water = 40 gallons (this corresponds to the point (0, 40)).
- At \(t = 5\): Water = \(40 - 8 \times 5 = 0\) gallons (this corresponds to the point (5, 0)).
- At \(t = 8\): Water = \(40 - 8 \times 8 = -16\) gallons, which isn't physically meaningful since the water can't go below 0 gallons.
This indicates that the correct graph should be a straight line connecting the points (0, 40) and (5, 0), and it should not extend to 8 minutes since the water would have fully drained by that time.
Now, analyzing the given options:
- Option (a): Connects points (0, 40) and (5, 0) with a straight line, which matches our expectations.
- Option (b): Connects points (0, 40) and (8, 0) which is incorrect because the tub would not be at 0 gallons when t = 8.
- Option (c): A concave downward curve, which does not match the linear nature of the draining water.
- Option (d): A concave upward curve, which is also incorrect.
Thus, the correct response is:
a. A coordinate plane shows the x-axis representing Number of Minutes, ranging from 0 to 7 in increments of 1. The y-axis represents the Number of Gallons of Water in Tub, ranging from 0 to 50 in increments of 5. A straight line connects the following two plotted points: (0, 40) and (5, 0).