Benito and Tyler are painting opposite sides of a 150-foot fence. Tyler has already paint 39/2 ft of his side when Benito gets started. Benito is painting at a rate of 15 ft/min, while Tyler is moving slower at 11 ft/min.
a. Write and solve an equation to determine how long it will take for the two sides of the fence to have an equal number of feet painted? How many feet will be on Benito's side of the fence when the two sides have an equal number of feet painted?
b. Tyler claims that because he started first, he will finish painting his side of the fence before Benito finishes painting his side. is this true? Explain using mathematical evidence.
c. The painter who finishes first gets to rest while the other painter finishes. How long will the painter who finishes first get to rest? Explain.
a. To determine how long it will take for the two sides of the fence to have an equal number of feet painted, we can create an equation using the distances painted by Benito and Tyler.
Let t be the time (in minutes) it takes for both sides to have an equal number of feet painted.
The distance painted by Benito is given by the equation:
Distance painted by Benito = 15 ft/min * t
The distance painted by Tyler is given by the equation:
Distance painted by Tyler = 39/2 ft + 11 ft/min * t
To find when both sides have an equal number of feet painted, we equate the two expressions and solve for t:
15t = 39/2 + 11t
Multiplying both sides of the equation by 2 to get rid of the fraction, we get:
30t = 39 + 22t
Bringing the terms with t on one side, we have:
30t - 22t = 39
Simplifying, we get:
8t = 39
Dividing both sides by 8, we find:
t = 39/8 = 4.875 minutes
Therefore, it will take approximately 4.875 minutes for the two sides of the fence to have an equal number of feet painted.
To find how many feet will be on Benito's side of the fence at that time, we substitute the value of t into the equation for Benito's distance:
Distance painted by Benito = 15 ft/min * 4.875 min = 73.125 ft
So, when both sides have an equal number of feet painted, Benito will have painted approximately 73.125 feet of his side of the fence.
b. Tyler's claim that he will finish painting his side of the fence before Benito finishes painting his side might not be true. To determine if this claim is valid, we compare the times it will take for each of them to complete painting their respective sides.
Since Tyler started painting earlier, his time can be represented by the value of t we solved for earlier (t = 4.875 minutes). We can calculate the distance he will paint within this time:
Distance painted by Tyler = 39/2 ft + 11 ft/min * 4.875 min = 39/2 + 53.625 = 93.625 ft
On the other hand, the time it will take for Benito to finish painting his side can be calculated by equating his distance to the total length of the fence:
Distance painted by Benito = 15 ft/min * t
150 ft = 15 ft/min * t
Simplifying, we find:
t = 150 ft / 15 ft/min = 10 minutes
Therefore, Benito will take 10 minutes to finish painting his side of the fence, while Tyler will take approximately 4.875 minutes. Tyler's claim is false. Benito will finish painting his side first.
c. The time the painter who finishes first gets to rest can be determined by subtracting the time taken by the faster painter from the time taken by the slower painter.
The time taken by Tyler is approximately 4.875 minutes, and the time taken by Benito is 10 minutes. Thus, the rest time for the faster painter is:
Rest time = Time taken by Benito - Time taken by Tyler
Rest time = 10 min - 4.875 min
Rest time = 5.125 minutes
Therefore, the painter who finishes first will get to rest for approximately 5.125 minutes.
(a) 15t = 39/2 + 11t
(b) Solve (a) for t. If t <= 10 then Benito has not painted his whole side.
(c) the winner will get to rest for x/s hours, where x is the amount of fence still unpainted by the loser, and y is the loser's speed.