Question

problem 4
r(t) and k(t) model the savings account balances of rafael and
katie after t years.
select all the statements that are true.
☑️ katie has a lower average rate of change in the last two
years.
☑️ katie’s balance is always less than
rafael’s.
□ ( r(2) = 100 )
□ rafael’s balance is increasing from year 0 to year
6.
☑️ rafael has a higher average rate of change in the first four
years.
chart: balance ($) vs time (years) with r(t) (green, piecewise) and k(t) (purple, curve) from t=0 to t=6, grid, labels, points at (1,300), (2,100) for k(t), (4,480) intersection, etc.

Answer:

To determine the true statements, we analyze each option using the graph of \( r(t) \) (Rafael) and \( k(t) \) (Katie):

1. “Katie has a lower average rate of change in the last two years.”

• Time interval: Years 4 to 6 (last two years).
• For \( k(t) \) (purple line): At \( t = 4 \), balance is \( 480 \); at \( t = 6 \), balance is \( 370 \) (approx.).

Rate of change: \( \frac{370 - 480}{6 - 4} = \frac{-110}{2} = -55 \) (decreasing, negative rate).

• For \( r(t) \) (green line): At \( t = 4 \), balance is \( 480 \); at \( t = 6 \), balance is \( 540 \) (approx.).

Rate of change: \( \frac{540 - 480}{6 - 4} = \frac{60}{2} = 30 \) (increasing, positive rate).

• A negative rate (Katie) is “lower” (more negative) than a positive rate (Rafael). So this statement is true.

2. “Katie’s balance is always less than Rafael’s.”

• At \( t = 1 \): Katie’s balance is \( 300 \); Rafael’s is \( 200 \) (approx.). Here, Katie’s balance (\( 300 \)) is greater than Rafael’s (\( 200 \)).
• Thus, the statement “always less” is false.

3. “\( r(2) = 100 \)”

• From the graph, at \( t = 2 \), Rafael’s balance (\( r(2) \)) is \( 220 \) (approx.), not \( 100 \).
• So this statement is false.

4. “Rafael’s balance is increasing from year 0 to year 6.”

• From \( t = 0 \) to \( t = 1 \): Rafael’s balance is \( 200 \) (constant, no increase).
• From \( t = 1 \) to \( t = 2 \): Rafael’s balance is \( 220 \) (constant, no increase).
• Thus, his balance is not “always increasing” from 0 to 6. This statement is false.

5. “Rafael has a higher average rate of change in the first four years.”

• Time interval: Years 0 to 4 (first four years).
• For \( r(t) \): At \( t = 0 \), balance is \( 200 \); at \( t = 4 \), balance is \( 480 \).

Rate of change: \( \frac{480 - 200}{4 - 0} = \frac{280}{4} = 70 \).

• For \( k(t) \): At \( t = 0 \), balance is \( 150 \); at \( t = 4 \), balance is \( 480 \).

Rate of change: \( \frac{480 - 150}{4 - 0} = \frac{330}{4} = 82.5 \).

• Wait—this contradicts the checked box. Did we misread? Wait, the graph: At \( t = 0 \), Katie’s balance is \( 150 \), Rafael’s is \( 200 \). At \( t = 4 \), both are \( 480 \). Wait, maybe the “first four years” is from \( t = 0 \) to \( t = 4 \). Wait, Rafael’s rate: \( \frac{480 - 200}{4} = 70 \); Katie’s rate: \( \frac{480 - 150}{4} = 82.5 \). But the box is checked. Maybe the interval is different? Wait, maybe “first four years” is \( t = 0 \) to \( t = 4 \), but maybe the graph’s \( r(t) \) at \( t = 0 \) is \( 200 \), \( t = 4 \) is \( 480 \); \( k(t) \) at \( t = 0 \) is \( 150 \), \( t = 4 \) is \( 480 \). Wait, maybe the user’s initial check is incorrect, but based on the graph:

Wait, the original problem has this as checked. Let’s re-express:

• Rafael’s first four years: From \( t = 0 \) (200) to \( t = 4 \) (480): \( \frac{480 - 200}{4} = 70 \).
• Katie’s first four years: From \( t = 0 \) (150) to \( t = 4 \) (480): \( \frac{480 - 150}{4} = 82.5 \).

But the checked box says “Rafael has a higher rate”—this would be false. But the problem shows it as checked. Maybe the interval is \( t = 2 \) to \( t = 4 \)? Let’s check \( t = 2 \) to \( t = 4 \):

• Rafael: \( t = 2 \) (220) to \( t = 4 \) (480): \( \frac{480 - 220}{2} = 130 \).
• Katie: \( t = 2 \) (100) to \( t = 4 \) (480): \( \frac{480 - 100}{2} = 190 \). Still higher for Katie.

Wait, maybe the initial check is a mistake. But based on the graph, the only true statement is:

Final Answer

Only the first statement is true:

• Katie has a lower average rate of change in the last two years.

(Note: The “Rafael has a higher average rate...” is false based on calculation, but if the graph is misinterpreted, maybe the intended answer includes it. However, from the graph, the only definitively true statement is the first one.)