As a result of a heavy rain, the volume of water in a reservoi...

Question

As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A large water tank is filled with 780,000 gallons of water over a 12-hour period due to a malfunctioning valve. Was there a moment where the inflow rate exceeded 60,000 gallons per hour? Awesome. So it appears for this particular problem, we're trying to determine whether or not, yes or no, was there a moment where this inflow rate Exceeded 60,000 gallons per hour. So yes or no, was there a moment in time where this inflow rate exceeded 60,000 gallons per hour? So that's what we're trying to figure out. So A is yes and B is no. So now that we know that we're trying to solve yes or no, was this inflow rate exceeding 60,000 gallons per hour. Our first step to solve this problem is we need to let FFT, we need to let FFT denote the amount of water in our tank in units of gallons. So once again, FAT will denote the amount of water in our tank in gallon units, and at time T hours where T in this case is going to be in the interval. So let's note that T is within our interval of square bracket 0.12 square bracket. Whereas we should recall, square brackets indicate a closed interval. Since we're told that it's from 0, since it starts from rest, it starts from doing nothing, so it starts at 0, and then it goes over a 12 hour period. So that means it ends at 12. So that's why it's 0.12 for this closed interval. Therefore, we could go ahead and write that F of 12. When we're plugging in a value of 12 in for X, we could say that F of 12 minus F of 0 is gonna be equal to 780,000 gallons of water. Awesome. So once again, we can say F 12 minus F 0 is equal to 780,000 gallons of water. So now we need to recall and use the mean value theorem, and as we should recall, the mean value theorem states that if F of T, so let's make a note of this, that states that if F of T is continuous on square bracket 0.12 square bracket, and it's differentiable on parentheses 0.12, where parentheses indicate an open interval. Then there will exist at least 1 time C in parentheses 0.12 such that F of C is equal to f of 12 minus F of 0, all divided by 12 minus 0. So let's repeat that one more time. So the mean value theorem states that if F of T is continuous on this closed interval of 0.12 and differentiable on this open interval of 0.12, then there will exist at least one time C within our open interval of 0.12 such that that F C will equal F of 12 minus F of 0, all divided by 12 minus 0. So, plugging in our given variables, we will find that F of C is going to be equal to 780,000 gallons of water, which we can write our units later, so we can ride it without units to make it easy. So we need to take, since we know that F of 12 minus F of 0 is equal to 780,000, and then we need to divide it by 12 means 12 minus 0 is 12. So when we Plug that into our calculator, we will get an answer of 65,000 gallons. Per hour, which I'll just say gallon for short for gallons per hour. So we're gonna say that F C is equal to 65,000 gallons of water per hour. So since 65,000. is greater than 60,000. There must be at least one moment during this 12 hour period when where the instantaneous inflow rate exceeded 60,000 gallons per hour. So therefore, without a doubt, yes, there was a moment during the during this whole situation that our promises said that the inflow rate exceeded 60,000 gallons per hour. So that means our final answer has to be yes. Hooray, we did it. So this corresponds with multiple choice answer letter A, which once again is yes. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Rate of Change

Mean Value Theorem

Unit Conversion

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