this is similar to a problem in the textbook. time spent running (minutes) 0 15 30 45 60 75 speed (mph) 13…

this is similar to a problem in the textbook. time spent running (minutes) 0 15 30 45 60 75 speed (mph) 13 12 12 8 7 0 shown in the table is a runners speed at various times. use this data to sketch a curve relating the two variables. draw rectangles on this graph to help you underestimate the total distance traveled by the runner during the first half hour. what underestimate did you obtain? miles draw rectangles on this graph to help you overestimate the total distance traveled by the runner during the first half hour. what overestimate did you obtain? miles draw rectangles on this graph to help you underestimate the total distance traveled by the runner during the whole time. what underestimate did you obtain? miles draw rectangles on this graph to help you overestimate the total distance traveled by the runner during the whole time. what overestimate did you obtain? miles you can view a streaming video of the solution of a version of this problem at m119tube solution.

this is similar to a problem in the textbook. time spent running (minutes) 0 15 30 45 60 75 speed (mph) 13 12 12 8 7 0 shown in the table is a runners speed at various times. use this data to sketch a curve relating the two variables. draw rectangles on this graph to help you underestimate the total distance traveled by the runner during the first half hour. what underestimate did you obtain? miles draw rectangles on this graph to help you overestimate the total distance traveled by the runner during the first half hour. what overestimate did you obtain? miles draw rectangles on this graph to help you underestimate the total distance traveled by the runner during the whole time. what underestimate did you obtain? miles draw rectangles on this graph to help you overestimate the total distance traveled by the runner during the whole time. what overestimate did you obtain? miles you can view a streaming video of the solution of a version of this problem at m119tube solution.

Answer

Explanation:

Step1: Recall distance - speed formula

Distance = Speed×Time. The time intervals are $\Delta t = 15$ minutes $=\frac{15}{60}=\frac{1}{4}$ hours.

Step2: Calculate underestimates and overestimates for first - half hour

Underestimate for first half - hour:

The time intervals are from $t = 0$ to $t = 15$ and $t = 15$ to $t = 30$ minutes. For an underestimate, we take the lower speed in each interval. The speeds are $12$ mph and $12$ mph. $d_{under1}=\ 12\times\frac{1}{4}+12\times\frac{1}{4}=3 + 3=6$ miles.

Overestimate for first half - hour:

For an overestimate, we take the higher speed in each interval. The speeds are $13$ mph and $12$ mph. $d_{over1}=13\times\frac{1}{4}+12\times\frac{1}{4}=\frac{13 + 12}{4}=\frac{25}{4}=6.25$ miles.

Step3: Calculate underestimates and overestimates for whole time

The time intervals are $n = 5$ intervals of $\Delta t=\frac{1}{4}$ hours each.

Underestimate for whole time:

We take the lower speed in each interval. The speeds are $12,12,8,7,0$. $d_{under2}=12\times\frac{1}{4}+12\times\frac{1}{4}+8\times\frac{1}{4}+7\times\frac{1}{4}+0\times\frac{1}{4}=\frac{12 + 12+8 + 7+0}{4}=\frac{39}{4}=9.75$ miles.

Overestimate for whole time:

We take the higher speed in each interval. The speeds are $13,12,12,8,7$. $d_{over2}=13\times\frac{1}{4}+12\times\frac{1}{4}+12\times\frac{1}{4}+8\times\frac{1}{4}+7\times\frac{1}{4}=\frac{13 + 12+12 + 8+7}{4}=\frac{52}{4}=13$ miles.

Answer:

Underestimate for first half - hour: 6 miles Overestimate for first half - hour: 6.25 miles Underestimate for whole time: 9.75 miles Overestimate for whole time: 13 miles