A rain barrel is partially filled at noon and then water flows into it at a constant rate for the...

1. TRANSLATE the problem information

The problem gives us:

• \(\mathrm{V(t)}\) = volume of water (in gallons) in the barrel
• \(\mathrm{t}\) = time in hours after noon
• We need the 'initial volume at noon'

Key translation: Since t measures hours after noon, then noon itself is when \(\mathrm{t = 0}\).

So 'initial volume at noon' translates to: Find \(\mathrm{V(0)}\)

2. INFER where to find \(\mathrm{V(0)}\) on the graph

To find \(\mathrm{V(0)}\), we need to determine what V equals when t equals 0.

On a graph:

• The horizontal axis shows t values
• The vertical axis shows V values
• When \(\mathrm{t = 0}\), we're at the y-axis

Strategic insight: \(\mathrm{V(0)}\) is the y-intercept—the point where the graph crosses the vertical axis.

3. TRANSLATE (read) the y-intercept from the graph

Looking at the graph where \(\mathrm{t = 0}\) (the y-axis):

• The line intersects the y-axis at the gridline marked 15
• Therefore, \(\mathrm{V(0) = 15}\) gallons

Answer: 15 gallons

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing that 'initial volume at noon' means \(\mathrm{V(0)}\) because t represents 'hours after noon'

Some students may think:

• 'Initial' might mean some other special time value
• They might look for a labeled point on the graph
• They might try to use the slope or other point on the line

This leads to confusion and guessing among various values visible on the graph.

Second Most Common Error:

Poor TRANSLATE reasoning (graph reading): Misreading the y-intercept value

Students might:

• Read the value imprecisely, estimating it as 14 or 16
• Confuse the y-intercept with another nearby value
• Not align their reading properly with the gridlines

This may lead them to select an answer close to but not exactly 15.

The Bottom Line:

This problem tests whether students can connect real-world language ('initial,' 'at noon') to mathematical representations (\(\mathrm{t = 0}\), y-intercept). The key insight is understanding the domain of the function: since t counts hours after noon, noon itself is \(\mathrm{t = 0}\). Once that translation is made, the rest is straightforward graph reading.