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A car starts moving at time [tex][tex]$t = 0$[/tex][/tex] and goes faster and faster. Its velocity is shown in the following table.

[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$t$[/tex] (seconds) & 0 & 3 & 6 & 9 & 12 \\
\hline
Velocity (ft/sec) & 0 & 9 & 29 & 40 & 77 \\
\hline
\end{tabular}
\][/tex]

A. Estimate how far the car traveled during the first 12 seconds using the left-hand sums with 4 subdivisions.

Answer: [tex][tex]$\square$[/tex][/tex] feet

B. Now estimate how far the car traveled during the first 12 seconds using the right-hand sums with 4 subdivisions.

Answer: [tex][tex]$\square$[/tex][/tex] feet

Determine which of the two is an underestimate: [tex][tex]$a$[/tex][/tex] (choose A or B)

Sagot :

GinnyAnsweridn
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].
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