A dry goods company buys a new, modern mixer to supplement the output of their old mixer. The old mixer has a capacity of 50 gallons and can thoroughly mix a full load of soil and fill in 40 minutes. The new mixer has a capacity of 20 gallons and can thoroughly mix a full load of soil and fill in 15 minutes. Both mixers are loaded and started at the same time at the beginning of the day. Which function c(x) represents the output of the two mixers combined, where t represents the time in minutes since starting? Assuming that emptying and reloading the mixers can be done instantly. Choose the correct answer from the following. Show your work.
a) c(t) = 5/4t + 4/3t
b) c(t) = 14/11t
c) c(t) = 10/11t + 4/11t
d) c(t) = 7/4t + 14/3t
The old mixer has a capacity of 50 gallons and can mix a full load in 40 minutes. This means that in 1 minute, it can mix 50/40 = 5/4 gallons of soil.
The new mixer has a capacity of 20 gallons and can mix a full load in 15 minutes. This means that in 1 minute, it can mix 20/15 = 4/3 gallons of soil.
To find the combined output of the two mixers at any given time t, we need to add their individual outputs. So the function c(t) representing the output of the two mixers combined is:
c(t) = (5/4)t + (4/3)t
Multiplying through by 12 to clear the fractions gives:
c(t) = (15/4)t + (16/4)t
Combining like terms, we get:
c(t) = (31/4)t
Therefore, the function representing the output of the two mixers combined is c(t) = (31/4)t.
The correct answer is: d) c(t) = 7/4t + 14/3t