Question

problem 3
tyler and elena are on the cross country team.
tyler’s distances and times for a training run are shown on the graph.
graph of time (minutes) vs. distance (miles) with a line through the origin
elena’s distances and times for a training run are given by the equation ( y = 8.5x ), where ( x ) represents distance in miles and ( y ) represents time in minutes.

a. who ran farther in 10 minutes? how much farther? explain how you know.

b. calculate each runner’s pace in minutes per mile.

c. who ran faster during the training run? explain or show your reasoning.

problem 4
write an equation for the line that passes through ( (2, 5) ) and ( (6, 7) ).

Explanation:

Problem 3

Part a

Step1: Find Tyler's distance at 10 min

From the graph, when $y=10$ minutes, $x=\frac{10}{8} = 1.25$ miles (since the line passes through (1,8), so pace is 8 min/mi, distance = $\frac{\text{time}}{\text{pace}}$)

Step2: Find Elena's distance at 10 min

Use $y=8.5x$, solve for $x$: $x=\frac{y}{8.5}=\frac{10}{8.5}\approx1.176$ miles

Step3: Compare distances

Difference: $1.25 - 1.176 = 0.074$ miles (or $\frac{10}{8}-\frac{10}{8.5}=\frac{85-80}{68}=\frac{5}{68}\approx0.074$ miles)

Part b

Step1: Tyler's pace from graph

Graph passes through (1,8), so pace = $\frac{8}{1}=8$ minutes per mile

Step2: Elena's pace from equation

Equation $y=8.5x$ means pace is 8.5 minutes per mile

Part c

Step1: Relate pace to speed

Faster runner has lower pace (less time per mile)

Step2: Compare paces

8 < 8.5, so Tyler is faster

Step1: Calculate slope $m$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{7-5}{6-2}=\frac{2}{4}=\frac{1}{2}$

Step2: Use point-slope form

Use point (2,5): $y-y_1=m(x-x_1)$
$y-5=\frac{1}{2}(x-2)$

Step3: Simplify to slope-intercept form

$y=\frac{1}{2}x - 1 + 5$
$y=\frac{1}{2}x + 4$

Answer:

a. Tyler ran farther. He ran about 0.074 miles farther (or exactly $\frac{5}{68}$ miles). Tyler's distance at 10 minutes is 1.25 miles, while Elena's is approximately 1.176 miles.
b. Tyler's pace: 8 minutes per mile; Elena's pace: 8.5 minutes per mile
c. Tyler ran faster. He has a lower pace (8 minutes per mile vs. Elena's 8.5 minutes per mile), meaning he covers a mile in less time.

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Problem 4