A tank with some water in it begins to drain. The function v ( t )

Let's solve the problem step by step.

1. Finding the vertical intercept of $v(t) = 40 - 3.5t$

The vertical intercept occurs when $t = 0$. To find this, we substitute $t = 0$ into the equation:

$v(0) = 40 - 3.5(0) = 40$

So, the vertical intercept is $(0, 40)$.

2. What does the $v$-coordinate of your answer to part (a) represent?

The $v$-coordinate of the vertical intercept represents the volume of water in the tank when it starts draining. Therefore, the correct interpretation is:

• The number of gallons of water in the tank when it starts draining.

The other options (weight of the tank when empty, and time taken to drain) are incorrect interpretations in this context.

3. Finding the horizontal intercepts of $v(t) = 40 - 3.5t$

The horizontal intercept occurs when $v(t) = 0$. To find this, we set the equation to 0 and solve for $t$:

$40 - 3.5t = 0$ $3.5t = 40$ $t = \frac{40}{3.5} = 11.43 \, \text{minutes}$

So, the horizontal intercept is $(11.43, 0)$.

4. What does the $t$-coordinate of your answer to part (c) represent?

The $t$-coordinate of the horizontal intercept represents the time it takes for all of the water to drain from the tank. Therefore, the correct interpretation is:

• How many minutes it takes for all of the water to drain from the tank.

The other options (weight of the tank when empty, and initial volume of water) are incorrect interpretations here.

Summary:

• Vertical intercept: $(0, 40)$
• Interpretation of the $v$-coordinate: The number of gallons of water in the tank when it starts draining.
• Horizontal intercept: $(11.43, 0)$
• Interpretation of the $t$-coordinate: How many minutes it takes for all of the water to drain from the tank.

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Would you like more details or have any questions?

Here are 5 related questions to explore this concept further:

1. What happens to the rate of water draining as time increases?
2. How would the vertical intercept change if the initial volume of water were 50 gallons instead of 40?
3. What would the graph of $v(t)$ look like?
4. If the tank drained at a faster rate (say, 5 gallons per minute), how would this affect the horizontal intercept?
5. Can we interpret the slope of the equation as the rate of water flow?

Tip: The vertical intercept of a function often tells us the initial value of the quantity being modeled, while the slope represents the rate of change.