Word Problem on Quantity of a Substance

Topic : Word Problem on Quantity of a Substance

Question : A soda production line fills bottles with "a" liters of product. A bottle is placed under a nozzle and a valve is opened which dispenses the soda under a pressure. The soda is dispensed as the valve opens and accelerates at a constant rate "r" from 0 liters per second to "b" liters per second in "c" seconds. After "c" seconds, the soda is dispensed at the constant rate of "b" liters per second. When the valve closes, the soda flow decelerates.
from "b" liters per second to 0 liters per second at the constant rate "-r" in "c" seconds.
Find the time "T" it takes to fill a bottle of "a" liters with b.

Answer :

During the the first c seconds the rate of filling forms an arithmetic progression.
r,2r, 3r,……b. b is also a multiple of r and as the rate is reached after c seconds
b=cr or r =b/c ………(i)

Also the quantity filled in litres during this period is the sum to c terms of the AP.
Quantity filled in first c seconds =c/2{2r + (c-1)r}, which when simplified
= c(c+1)r/2

Same way we can take the amount filled in the last c seconds when the filling rate
Decelerates as c(c +1)r/2.

Total amount filled in the first c and the last c seconds that is
2c seconds = c(c+1)r. Substituting r = b/c from (i) we get
Amount filled in 2c seconds = b(c+1)


Now to find the amount filled during the constant speed we subtract this from a.

a – b(c+1) = a-bc-b litres.

This is filled at a constant speed b litres per second so the time taken
= (a-bc-b)/b = a/b –c -1 seconds.

When the 2c seconds (first c + last c) to this time
Total time taken to fill = a/b –c -1 + 2c = a/b + c – 1 seconds.