In order to estimate the length of the runway, a passenger on an airplane jotted down some velocity data during takeoff from the on-board entertainment screen. From the resulting table given below, calculate O8+U8/2 to find his estimate. Round any intermediate calculations, if needed, to no less than six decimal places, and round your final answer to three decimal places.
 
Time (s)	6	12	18	24	30	36	42	48
v (mph)	34	79	86	150	169	204	223	230
Answer in miles

To solve the problem, we need to first understand the estimation process. The given data represents the time and velocity at different intervals during the takeoff. The method likely used by the passenger to estimate the runway length involves calculating the area under the velocity-time graph. This area represents the distance traveled, which corresponds to the length of the runway.

We'll calculate the distance using the trapezoidal rule, which approximates the area under the curve by dividing it into trapezoids and summing their areas.

### Step 1: Convert velocities from mph to miles per second
Since time is in seconds and velocity is in miles per hour (mph), we should convert the velocity to miles per second (mps).

\[ 1 \text{ mph} = \frac{1 \text{ mile}}{3600 \text{ seconds}} \]

Let's convert each velocity value:

\[
\text{v}(mps) = \frac{\text{v}(mph)}{3600}
\]

Using this formula:
- \( v_1 = \frac{34}{3600} \)
- \( v_2 = \frac{79}{3600} \)
- \( v_3 = \frac{86}{3600} \)
- \( v_4 = \frac{150}{3600} \)
- \( v_5 = \frac{169}{3600} \)
- \( v_6 = \frac{204}{3600} \)
- \( v_7 = \frac{223}{3600} \)
- \( v_8 = \frac{230}{3600} \)

Let's calculate these values first.The velocities converted to miles per second (mps) are as follows:

\[
\begin{aligned}
v_1 &= 0.009444 \text{ mps}, \\
v_2 &= 0.021944 \text{ mps}, \\
v_3 &= 0.023889 \text{ mps}, \\
v_4 &= 0.041667 \text{ mps}, \\
v_5 &= 0.046944 \text{ mps}, \\
v_6 &= 0.056667 \text{ mps}, \\
v_7 &= 0.061944 \text{ mps}, \\
v_8 &= 0.063889 \text{ mps}.
\end{aligned}
\]

### Step 2: Use the trapezoidal rule to estimate the distance

The trapezoidal rule for estimating the area (distance) is given by:

\[
\text{Distance} \approx \frac{\Delta t}{2} \left[ v_1 + 2(v_2 + v_3 + v_4 + v_5 + v_6 + v_7) + v_8 \right]
\]

Given that the time intervals are evenly spaced at 6 seconds (Δt = 6 seconds), we can now plug in the values to find the estimated distance.

\[
\text{Distance} = \frac{6}{2} \left[ 0.009444 + 2(0.021944 + 0.023889 + 0.041667 + 0.046944 + 0.056667 + 0.061944) + 0.063889 \right]
\]

Let's compute this next.The estimated distance, which represents the length of the runway, is approximately **1.738 miles**.

Would you like any further details or explanations on how this calculation was made? Here are five related questions for further exploration:

1. How does the trapezoidal rule compare to other methods of numerical integration?
2. What would happen to the estimate if the time intervals were not evenly spaced?
3. How could you estimate the distance if only the initial and final velocities were known?
4. Why is it important to convert velocity units when calculating distance?
5. How does rounding intermediate calculations affect the accuracy of the final result?

**Tip:** When converting units, always ensure consistency throughout the problem to avoid errors in your final calculations.

Learn how to estimate the runway length using velocity-time data from an airplane's takeoff. This problem involves using the trapezoidal rule for numerical integration to calculate the distance traveled, converted from miles per hour to miles per second. Get step-by-step guidance on solving similar problems.

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