Odie goes out for a 90 minute run. As is his pattern, he starts out too fast and then slows down. The table to the right gives his speed in mph after a given number of minutes. Using 6 trapezoids and 3 midpoint Riemann sums, approximate the distance that Odie travels.

To approximate the distance that Odie travels during the 90-minute run using trapezoidal rule and midpoint Riemann sums, we can follow these steps:

Calculate the average speed at each interval:

[tex] \textsf{Average Speed} = \dfrac{\textsf{Speed}_i + \textsf{Speed}_{i+1}}{2} [/tex]

Multiply the average speed by the time interval:

[tex] \textsf{Distance}_i = \textsf{Average Speed}_i \cdot \textsf{Time Interval}_i [/tex]

Sum up all the distances to get the total distance:

[tex] \textsf{Total Distance} = \sum_{i=1}^{n} \textsf{Distance}_i [/tex]

Let's perform the calculations for both trapezoidal rule and midpoint Riemann sums.

Trapezoidal Rule:

[tex] \textsf{Distance}_i = \dfrac{\textsf{Speed}_i + \textsf{Speed}_{i+1}}{2} \cdot \textsf{Time Interval}_i [/tex]

[tex] \textsf{Distance}_1 = \dfrac{11 + 10}{2} \cdot 15 = 157.5 [/tex]

[tex] \textsf{Distance}_2 = \dfrac{10 + 10}{2} \cdot 15 = 150 [/tex]

[tex] \textsf{Distance}_3 = \dfrac{10 + 9}{2} \cdot 15 = 142.5 [/tex]

[tex] \textsf{Distance}_4 = \dfrac{9 + 7}{2} \cdot 15 = 120 [/tex]

[tex] \textsf{Distance}_5 = \dfrac{7 + 6}{2} \cdot 15 = 97.5 [/tex]

[tex] \textsf{Distance}_6 = \dfrac{6 + 0}{2} \cdot 15 = 45 [/tex]

[tex] \textsf{Total Distance (Trapezoidal)} = 157.5 + 150 + 142.5 \\+ 120 + 97.5 + 45 \\\\ = 712.5 [/tex]

Midpoint Riemann Sums:

[tex] \textsf{Distance}_i = \textsf{Speed}_{\textsf{midpoint}_i} \cdot \textsf{Time Interval}_i [/tex]

[tex] \textsf{Midpoint}_1 = \dfrac{11 + 10}{2} = 10.5 [/tex]

[tex] \textsf{Midpoint}_2 = \dfrac{10 + 10}{2} = 10 [/tex]

[tex] \textsf{Midpoint}_3 = \dfrac{10 + 9}{2} = 9.5 [/tex]

[tex] \textsf{Midpoint}_4 = \dfrac{9 + 7}{2} = 8 [/tex]

[tex] \textsf{Midpoint}_5 = \dfrac{7 + 6}{2} = 6.5 [/tex]

[tex] \textsf{Midpoint}_6 = \dfrac{6 + 0}{2} = 3 [/tex]

[tex] \textsf{Distance}_1 = 10.5 \cdot 15 = 157.5 [/tex]

[tex] \textsf{Distance}_2 = 10 \cdot 15 = 150 [/tex]

[tex] \textsf{Distance}_3 = 9.5 \cdot 15 = 142.5 [/tex]

[tex] \textsf{Distance}_4 = 8 \cdot 15 = 120 [/tex]

[tex] \textsf{Distance}_5 = 6.5 \cdot 15 = 97.5 [/tex]

[tex] \textsf{Distance}_6 = 3 \cdot 15 = 45 [/tex]

[tex] \textsf{Total Distance (Midpoint)} = 157.5 + 150 + 142.5 \\+ 120 + 97.5 + 45 \\\\ = 712.5 [/tex]

Therefore, both the Trapezoidal Rule and Midpoint Riemann Sums yield a total distance of 712.5 miles for Odie's 90-minute run.