Question 1096665: Mario and Kenneth are garbage collectors.The time that Mrio takes in unloading the garbage from the dump truck alone is 20 minutes less than the time that Kenneth takes in unloading the same amount of garbage. If both of them work together in unloading the garbage it will take them 90 minutes. How long will it take each of them to unload the garbage?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! unfortunately, you don't get a nice tidy answer, but you do get an answer.
the general formula is rate * time = quantity.
the quantity is equal to 1 unloaded garbage truck.
therefore the formula becomes rate * time = 1
if you let R = rate and T = time, then the formula becomes R * T = 1
mario takes 20 minutes less than ken to unload the garbage truck.
assuming that ken takes T minutes, then:
for mario, the formula becomes R * (T-20) = 1
solve for his rate to get R = 1 / (T-20)
for ken, the formula becomes R * T = 1
solve for his rate to get R = 1 / T
when they work together, their rates are additive, and the time it takes them to unload the garbage truck is 90 minutes.
therefore (1/(T-20) + 1/T) * 90 = 1
multiply both sides of this equation by (T * (T - 20) and you get:
(T + T - 20) * 90 = T * (T - 20)
simplify to get (2T - 20) * 90 = T^2 - 20T
simplify further to get 180T - 1800 = T^2 - 20T
subtract (180T - 1800) from both sides of this equation and flip sides of the equation to get:
T^2 - 20T - 180T + 1800 = 0
simplify to get T^2 - 200T + 1800 = 0
factor this equation to get T = 190.55385138137 or T = 9.4461486186258.
T cannot be 9.446..... because T - 20 would then be negative.
therefore T has to be equal to 190.55385... and T - 20 has to be 170.55385...
mario's rate of work is equal to 1 / (T-20) which is equal to 1/170.55385138137 which is equal to 0.005863251.
ken's rate of work is equal to 1 / T which is equal to 1/190.55385138137 which is equal to 0.00524786
when they work alone, it takes mario 170.55385 minutes to unload the truck and it takes ken 190.55385 minutes to unload the truck, the number of minutes being rounded to 5 decimal places.
that's your solution.
when they work together, the formula becomes (.005863251 + .00524786) * 90 = 1.
evaluate that equation and you get 1 = 1 which confirms the solution is correct and the rates of each person are as shown.
the answer is messy, but the process is good and the answer is confirmed to be good.
|
|
|
