Let's break this problem down step by step.
### Part A: Left-Hand Sum
To estimate the distance traveled using the left-hand sum with 4 subdivisions, we will use the velocity values at the beginning of each interval. We are given the velocity [tex]\( v \)[/tex] and time [tex]\( t \)[/tex] in the table:
- At [tex]\( t = 0 \)[/tex] seconds, [tex]\( v = 0 \)[/tex] ft/sec
- At [tex]\( t = 3 \)[/tex] seconds, [tex]\( v = 9 \)[/tex] ft/sec
- At [tex]\( t = 6 \)[/tex] seconds, [tex]\( v = 29 \)[/tex] ft/sec
- At [tex]\( t = 9 \)[/tex] seconds, [tex]\( v = 40 \)[/tex] ft/sec
- At [tex]\( t = 12 \)[/tex] seconds, [tex]\( v = 77 \)[/tex] ft/sec
Each interval is 3 seconds long [tex]\((t_{i+1} - t_i = 3)\)[/tex].
The left-hand sum uses the velocities at the start of each interval, hence:
[tex]\[
\text{Left-Hand Sum} = v_0 \cdot (t_1 - t_0) + v_1 \cdot (t_2 - t_1) + v_2 \cdot (t_3 - t_2) + v_3 \cdot (t_4 - t_3)
\][/tex]
Substituting the values:
[tex]\[
\text{Left-Hand Sum} = (0 \cdot 3) + (9 \cdot 3) + (29 \cdot 3) + (40 \cdot 3)
\][/tex]
[tex]\[
= 0 + 27 + 87 + 120
\][/tex]
Adding these together:
[tex]\[
\text{Left-Hand Sum} = 234 \text{ feet}
\][/tex]
### Part B: Right-Hand Sum
To estimate the distance traveled using the right-hand sum with 4 subdivisions, we will use the velocity values at the end of each interval:
The right-hand sum uses the velocities at the end of each interval, hence:
[tex]\[
\text{Right-Hand Sum} = v_1 \cdot (t_1 - t_0) + v_2 \cdot (t_2 - t_1) + v_3 \cdot (t_3 - t_2) + v_4 \cdot (t_4 - t_3)
\][/tex]
Substituting the values:
[tex]\[
\text{Right-Hand Sum} = (9 \cdot 3) + (29 \cdot 3) + (40 \cdot 3) + (77 \cdot 3)
\][/tex]
[tex]\[
= 27 + 87 + 120 + 231
\][/tex]
Adding these together:
[tex]\[
\text{Right-Hand Sum} = 465 \text{ feet}
\][/tex]
### Determine the Underestimation
To determine which estimate is an underestimation, compare the two sums:
[tex]\[
\text{Left-Hand Sum} = 234 \text{ feet}
\][/tex]
[tex]\[
\text{Right-Hand Sum} = 465 \text{ feet}
\][/tex]
Since 234 feet is less than 465 feet, the left-hand sum (Part A) is the underestimation.
### Summary
1. The distance estimated using the left-hand sum is [tex]\( 234 \text{ feet} \)[/tex].
2. The distance estimated using the right-hand sum is [tex]\( 465 \text{ feet} \)[/tex].
3. The underestimation is the left-hand sum (Part A).
[tex]\[
\boxed{234 \text{ feet}}
\][/tex]
[tex]\[
\boxed{465 \text{ feet}}
\][/tex]
The underestimation is [tex]\( \boxed{A} \)[/tex].
