To determine the correct sketch of the graph representing the relationship between the number of minutes since the tub started draining and the number of gallons of water, we should consider the situation mathematically.
Initially, the tub has 40 gallons of water. Water drains at a rate of 8 gallons per minute. Therefore, the equation for the amount of water in the bathtub, \( W(t) \), after \( t \) minutes can be expressed as:
\[ W(t) = 40 - 8t \]
Where:
- \( W(t) \) is the water in gallons at time \( t \) minutes,
- \( 40 \) is the initial amount of water,
- \( 8t \) is the amount of water drained after \( t \) minutes.
From this equation, we can calculate how much water is left after different amounts of time:
- At \( t = 0 \): \( W(0) = 40 - 8(0) = 40 \) gallons
- At \( t = 1 \): \( W(1) = 40 - 8(1) = 32 \) gallons
- At \( t = 2 \): \( W(2) = 40 - 8(2) = 24 \) gallons
- At \( t = 3 \): \( W(3) = 40 - 8(3) = 16 \) gallons
- At \( t = 4 \): \( W(4) = 40 - 8(4) = 8 \) gallons
- At \( t = 5 \): \( W(5) = 40 - 8(5) = 0 \) gallons
From the calculations, we see that the water in the bathtub decreases linearly over time. The relationship is a straight line that starts at (0, 40) and ends at (5, 0).
Among the provided options, the correct representation of this relationship is:
A straight line connecting the points (0, 40) and (5, 0).
Thus, the correct choice is the first option, where it describes a straight line between (0, 40) and (5, 0).