A dry goods company buys a new, modern mixer to supplement the output of their old, current 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? Assume that emptying and reloading the mixers can be done instantly.
The old mixer has a capacity of 50 gallons and can thoroughly mix a full load of soil and fill in 40 minutes. This means that the old mixer can mix $\frac{50}{40}=\frac{5}{4}$ gallons of soil per minute.
Similarly, the new mixer has a capacity of 20 gallons and can thoroughly mix a full load of soil and fill in 15 minutes. This means that the new mixer can mix $\frac{20}{15}=\frac{4}{3}$ gallons of soil per minute.
To find the combined output of the two mixers at any given time, we can add their individual outputs. Therefore, the function C(x) represents the output of the two mixers combined, where t represents the time in minutes since starting, is given by:
$C(t) = \left(\frac{5}{4}\right) t + \left(\frac{4}{3}\right) t$
Simplifying, we have:
$C(t) = \frac{5}{4}(t) + \frac{4}{3}(t)$
Multiplying each term by their respective denominators to get rid of fractions, we have:
$C(t) = \frac{5}{4} \cdot \frac{4}{4}(t) + \frac{4}{3} \cdot \frac{3}{3}(t)$
Simplifying further, we have:
$C(t) = \frac{20}{16}(t) + \frac{12}{9}(t)$
$C(t) = \frac{5}{4}(t) + \frac{4}{3}(t)$
Therefore, the function $C(t) = \frac{5}{4}t + \frac{4}{3}t$ represents the output of the two mixers combined.