Question 1019887


the answer is that the group was supposed to make 32 vacuum cleaners a day.


the total time it was supposed to take was 24 days.


working at 32 vacuum cleaners a day for 24 days would allow them to make a total of 768 vacuum cleaners in 24 days.


in fact, they made 844 vacuum cleaners in 23 days.


this constituted working for 5 days at 32 vacuum cleaners a day and then working for 18 days at 38 vacuum cleaners a day.


5 * 32 + 18 * 38 = 160 + 684 equals a total of 844 vacuum cleaners in 23 days.


your solution is that they were supposed to make 32 vacuum cleaners a day.


you are given that they were supposed to make 768 vacuum cleaners in a given time.


the basic formula if R * T = Q


R is the rate per day.
T is the number of days.
Q is equal to 768 vacuum cleaners.


the formula becomes R * T = 768


you are given that they worked for 5 days at the required rate.


you are also given that they worked for the remaining number of days at a rate per day that was 6 vacuum cleaners more than required.


if they worked for 5 days at the given rate, then the remaining number of days to finish the job would be T - 5 days.


since they had provided 844 cleaners a day earlier, than the number of days they worked at the increased rate must have been T - 6 days.


for 5 days they worked at the given rate.
then, for T-6 days, they worked at the increased rate.
at the end, they produced 844 vacuum cleaners.


your formula becomes:


5R + (R + 6) * (T - 6) = 844.


use the distributive law of multiplication to simplify this equation to get:


5R + RT - 6R + 6T - 36 = 844.


you were also given that they were supposed to make 768 vacuum cleaners in the given number of days at the given rate per day.


the formula for that was R * T = 768.


solve that equation for R to get R = 768 / T.


in the equation of 5R + RT - 6R + 6T - 36 = 844, replace R with (768/T) to get:


(5 * 768/T) + (768/T * T) - (6 * 768/T) + 6T - 36 = 844.


since (5 * 768/T) - (6 * 768/T) = - 768/T, and since (768/T * T) = 768, then you get:


- (768/T) + 768 + 6T - 36 = 844.


multiply both sides of this equation by T to get:


-768 + 768T + 6T^2 - 36T = 844T


subtract 844T from both sides of this equation to get:


-768 + 768T + 6T^2 - 36T - 844T = 0


combine like terms and reorder the terms in descending order to get:


6T^2 - 112T - 768 = 0


use the quadratic formula to solve for T.


you will get:


T = - (5 and 1/3) or T = 24.


since T can't be negative, then T must be equal to 24.


since R * T = 768, and T = 24, then solve for R to get R = 768 / 24 = 32.


you solution appears to be R = 32 and T = 24


that means that 32 vacuum cleaners per day times 24 days equals 768 vacuum cleaners total.


in fact, if they had worked at the rate of 32 vacuum cleaners a day for 24 days, they would have completed 768 vacuum cleaners in those 24 days.


as it was, they worked at that rate for 5 days and then worked for the remaining days at a rate of 32 + 6 = 38 vacuum cleaners a day.


they were supposed to work for 24 days.


as it was they worked for 23 days and produced a total of 844 vacuum cleaners.


23 - 5 = 18 days that they worked at the increased rate.


5 days at 32 vacuum cleaners a day plus 18 days at 38 vacuum cleaners a day is equal to 5 * 32 + 18 * 38 which is equal to a total of 160 + 684 = 844 vacuum cleaners in 23 days.


the solution looks good.


they were supposed to make 32 vacuum cleaners a day.


the graph of your equation looks like this.


in this graph, T has been replaced with x.


you are looking to solve for the values of x that will make the equation equal to 0.


the equation of y = 6T^2 - 112T -768 which has been graphed as y = 6x^2 - 112x - 768 and you are looking for the y = 0 crossing points on the x-axis.


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