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.
