Answer :
Answer:
It will take 10.3 minutes to fill the bathtub.
Step-by-step explanation:
Let us call the volume of the bathtub [tex]B[/tex], and [tex]V_1[/tex],[tex]V_2[/tex], and [tex]V_3[/tex] the flow rates of cold water, hot water, and the drain respectively.
The cold water faucet fills the bathtub in 12 minutes; therefore,
(1). [tex]12V_1 = B[/tex].
Similarly, for the hot water faucet
(2). [tex]18V_2 = B[/tex]
and for the drain
(3). [tex]24V_3 = B[/tex].
If the drain and the faucets are turned on together, then we have:
[tex]V_1t+V_2t-V_3t = B[/tex] (this says water flow after time [tex]t[/tex] must equal [tex]B[/tex])
[tex](V_1+V_2-V_3)= \dfrac{B}{t}[/tex]
Now, from equations (1), (2), and (3) we substitute the values of [tex]V_1[/tex], [tex]V_2[/tex], and [tex]V_3[/tex] to get:
[tex]\dfrac{B}{12} +\dfrac{B}{18}-\dfrac{B}{24}= \dfrac{B}{t}[/tex]
[tex]\dfrac{1}{12} +\dfrac{1}{18}-\dfrac{1}{24}= \dfrac{1}{t}[/tex]
by simplifying the left side we get:
[tex]\dfrac{7}{72} = \dfrac{1}{t}[/tex]
[tex]t = \dfrac{72}{7}[/tex]
[tex]\boxed{t = 10.3\:minutes}[/tex]
Thus, if both faucets are on and the drain is open, it will take 10.3 minutes to fill the bathtub.