Use the formula D=rt where D is distance, r is rate, and t is time.

The distance is 40
Julio finishes the 40 yd dash in 4.39 seconds. Rich finishes it at 6.18 seconds

1. Solve D=rt for rate. (1 pt)
D/t=r

2. Use the statistics given in Act Two and your equation from #1. Statistics shown above.
a. At what speed (rate) does Rich run the 40-yard dash? (1 pt)
40/6.18=6.47

b. At what speed (rate) does Julio run the 40-yard dash? (1 pt)
40/4.39=9.11

c. Solve D=rt for time. How long does it take Rich to run with a 10-yard lead? (1 pt)
D(r)=t.
40-10=30yards
30(6.47)=194.1
d. How long does it take Julio to run the 40-yard dash at half-speed? Who will win? (2 pts)
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3. If both runners are at full speed, how many yards of a head start will Rich need to win? (2 pts)
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4. Write and solve your own question using this scenario. (Remember to show all work!) (2 pts)
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yikes c.

if D = r t
then t = D/r (you KNOW that :)
30 yards/6.47 yards/sec = 4.64 sec
You KNOW it would not take him 194 seconds !!!!

d.
Julio runs at 9.11/2
= 4.555 yards/second
40 yards/ 4.555 yards/sec =8.78 seconds
Which you already know is TWICE as long because half the speed :)

now head start
Julio took 4.39 seconds
How far can Rich go in 4.39 seconds?
4.39 * 6.47 = 28.4 yards
40 - 28.4 = 11.6 yards head start to tie

Omg thankyou. XD I dont know why I was having such a hard time with this

d. To find out how long it takes Julio to run the 40-yard dash at half-speed, we need to divide the distance by half of his rate: D = rt. In this case, D = 40 yards and r = 9.11 yards/second.

D/r = t
40 / (9.11/2) = t
40 / 4.55 = t
8.79 = t

So, it takes Julio approximately 8.79 seconds to run the 40-yard dash at half-speed. To determine the winner, we compare this time to Rich's time. Since Rich finishes the 40-yard dash in 6.18 seconds, Rich would win the race.

3. To find out how many yards of a head start Rich needs to win when both runners are at full speed, we need to set up an equation using the formula D = rt. In this case, we know the distance (D) is 40 yards, the rate for Julio (r1) is 9.11 yards/second, and the rate for Rich (r2) is unknown. Let's represent the head start distance as H.

For Julio:
D = r1t
40 = 9.11t

For Rich:
D + H = r2t
40 + H = r2t

To find out the head start distance needed for Rich to win, we need to solve these two equations simultaneously.

By substituting the value of t from the first equation into the second equation, we get:

40 + H = r2(40 / 9.11)
40 + H = 4.39r2

Since Rich finishes in 6.18 seconds, we can further substitute r2 with the given rate.

40 + H = 4.39 * 6.47
40 + H = 28.4033
H = 28.4033 - 40
H = -11.5967

Since it doesn't make sense to have a negative head start distance, we can conclude that Rich does not need any head start to win when both runners are at full speed.

4. Write and solve your own question using this scenario. (Remember to show all work!)

Example question: If Rich runs at a rate of 7.5 yards/second, how long will it take him to finish a 60-yard dash?

To solve this, we can use the formula D = rt, where D = 60 yards and r = 7.5 yards/second.

D = rt
60 = 7.5t
t = 60 / 7.5
t = 8

So, it will take Rich approximately 8 seconds to finish a 60-yard dash at a rate of 7.5 yards/second.

c. To solve D=rt for time, rearrange the equation to isolate t: t = D/r. Insert the values for D and r to get:

t = 40/6.47 = 6.18 seconds

So it takes Rich 6.18 seconds to run the 40-yard dash with a 10-yard lead.

d. To find out how long it takes Julio to run the 40-yard dash at half-speed, divide the distance by half of his usual rate:

t = 40/(9.11/2) = 4.39/2 = 2.195 seconds

Therefore, Julio will win the race as he takes 2.195 seconds to complete the 40-yard dash at half-speed, while Rich takes 6.18 seconds.

3. To determine how many yards of a head start Rich needs to win when both runners are at full speed, we need to compare their times for a distance of 40 yards. Let's assume that Rich needs x yards of head start. The time it takes Rich to run 40 yards will be equal to the time it takes Julio to run (40 - x) yards. We can set up the equation as follows:

40/r = (40 - x)/r

Cross-multiply to solve for x:

40r = 40r - xr
xr = 40r - 40r
xr = 0

This implies that x = 0, which means Rich doesn't need any head start to win when both runners are at full speed.

4. Write and solve your own question using this scenario:

Question: If Rich can run the 40-yard dash in 5 seconds and Julio can run it in 6 seconds, how much faster is Rich than Julio in yards per second?

Solution:
To find the difference in speed, we need to calculate the rates of Rich and Julio using the formula D=rt.

For Rich:
r = D/t = 40/5 = 8 yards per second

For Julio:
r = D/t = 40/6 = 6.667 yards per second

The difference in their speeds is:
8 - 6.667 = 1.333 yards per second

Therefore, Rich is 1.333 yards per second faster than Julio.