Velocities of the 2 cars during the race are given in the tables below where velocities, v are in miles per hour and times t are in seconds.

Sappho's car:
time: 0 1 2 3 4 5 6 7 8 9 10
velocity:0 20 35 48 62 75 85 93 99 106 111
Homer's car:
time: 0 1 2 3 4 5 6 7 8 9 10
velocity: 0 18 31 43 58 68 79 86 93 95 96
A. Use ten trapezoids to approximate the area between the curves y = vs(t) and y = vH(t) (s and H for sappho and homer) on the interval from t = 0 to t = 10. What definite integral are you approximating?
B. If you haven’t done so, convert your answer from part A into units that make good
common sense for the physical situation described. Then explain what practical meaning
this answer has for the physical situation.
C. Estimate the average velocity of Sappho’s car over the 10-second interval using 5
subintervals and midpoints.

ummm

could you please be more specific?

A. To approximate the area between the curves y = vs(t) and y = vH(t) on the interval from t = 0 to t = 10 using ten trapezoids, you are approximating the definite integral ∫[0, 10] (vs(t) - vH(t)) dt.

B. To convert the answer from part A into units that make good common sense for the physical situation, you would multiply the result by the units of time (seconds) to get a unit of distance (miles). The practical meaning of this answer is that it represents the approximate difference in the distance traveled by Sappho's car and Homer's car over the 10-second interval. It gives an estimation of the advantage or disadvantage Sappho has over Homer.

C. To estimate the average velocity of Sappho's car over the 10-second interval using 5 subintervals and midpoints, you can use the formula:

Average velocity = (vs(t1) + vs(t2) + vs(t3) + vs(t4) + vs(t5)) / 5,

where t1, t2, t3, t4, and t5 are the midpoints of the subintervals.

A. To approximate the area between the curves y = vs(t) and y = vH(t) on the interval from t = 0 to t = 10 using ten trapezoids, we are approximating the definite integral of the velocity difference between Sappho's car and Homer's car over the given time interval.

To calculate the area using trapezoids, we divide the interval into ten equal subintervals and approximate the area under the curve within each subinterval using trapezoids. The formula for the area of a trapezoid is:

Area of trapezoid = (base1 + base2) * height / 2

In this case, the base1 and base2 will be the velocities of Sappho's car (vs) and Homer's car (vH) at the corresponding times, and the height will be the width of each subinterval (Δt = 1 sec).

To find the area for each subinterval, we calculate the difference between the velocities of Sappho's and Homer's cars at the start and end of the subinterval. We then multiply this difference by the width of the subinterval (1 sec) to get the area of each trapezoid. Finally, we sum up the areas of all ten trapezoids to approximate the total area between the curves.

B. To convert our answer into units that make good common sense for the physical situation, we need to consider the units of velocity and time used in the problem. In this case, velocities are given in miles per hour (mph), and time is given in seconds.

The area between the curves represents the cumulative difference in positions of Sappho's and Homer's cars over the given time interval. Since velocity is the rate of change of position, the area represents the net displacement between the cars during the race.

The area, expressed in units of miles per hour multiplied by seconds, does not have a direct practical meaning. However, if we convert the units to miles, which is a common unit for displacement, the area would represent the net displacement between the cars over the given time interval.

C. To estimate the average velocity of Sappho's car over the 10-second interval using 5 subintervals and midpoints, we can divide the total displacement of Sappho's car by the total time.

First, we calculate the total displacement by subtracting the initial position from the final position of Sappho's car. Then we divide the displacement by the total time, which is 10 seconds, to get the average velocity.

Since we are using 5 subintervals, the width of each subinterval is 10/5 = 2 seconds. The midpoints represent the average velocities of Sappho's car within each subinterval.

We find the velocities at the midpoints of each subinterval by averaging the velocities at the start and end of the subinterval. Finally, we calculate the average velocity by dividing the total displacement by the total time.

A. Geez, the trapezoids all have height 1, so the area of each is just the average of the two y-coordinates.

B. think of the units on the axes: v * t = distance

C. avg speed is totaldistance/totaltime