There are two different vats. The first vat has 20 gallons is being filled at a rate of 1.51.gallons per minute. The second vat has 120 gallons and is leaking fluid at a rate of 2.5 gallons per minute. After how many seconds will both vats have the same amount of liquid?

Question

There are two different vats.  The first vat has 20 gallons is being filled at a rate of 1.51.gallons per minute.  The second vat has 120  gallons and is leaking fluid at a rate of 2.5  gallons per minute.  After how many seconds will both vats have the same amount of liquid?

GPT-4o mini · Answer

Let $ t $ be the time in minutes after which both vats will contain the same amount of fluid. For the first vat: - It starts with 20 gallons and is filling at 1.51 gallons per minute. - Therefore, the amount of fluid in the first vat after $ t $ minutes, $ V_1(t) $, is: $$ V_1(t) = 20 + 1.51t $$ For the second vat: - It starts with 120 gallons and is leaking at a rate of 2.5 gallons per minute. - Therefore, the amount of fluid in the second vat after $ t $ minutes, $ V_2(t) $, is: $$ V_2(t) = 120 - 2.5t $$ We want to find $ t $ such that both vats have the same amount of fluid: $$ V_1(t) = V_2(t) $$ Setting the equations equal to each other: $$ 20 + 1.51t = 120 - 2.5t $$ Now, combine terms: $$ 1.51t + 2.5t = 120 - 20 $$ $$ 4.01t = 100 $$ Now, solve for $ t $: $$ t = \frac{100}{4.01} \approx 24.94 \text{ minutes} $$ To convert minutes to seconds: $$ t \approx 24.94 \times 60 \approx 1496.4 \text{ seconds} $$ Thus, after approximately $ \boxed{1496.4} $ seconds, both vats will have the same amount of liquid.