Okay, we're going to record some of our progress, I think. Our problem set, well, we're still writing. I'll post what we've got so far, though:
The volume of a perfectly half-spherical lake is changing at a rate relative to time as rain falls. At some time t the depth at the exact center of the lake is 30 meters, and the rain is falling at a rate such that the change in the depth at the center is .04 centimeters/day.
Some ecologists are worried that if the volume of the lake changes faster than 2.5 cubic meters per day, the fragile ecosystem will be damaged. Given the initial conditions, will the ecosystem be in danger?
In similar conditions, a smaller lake of the same shape has a depth of 15 meters at the exact center. If the rain falls and changes the depth at the center at the same rate as above, and the ecosystem is damaged when the volume of water changes at .5 cubic meters per day, should the scientists worry?
What is the depth at the time when the volume is 25000 cubic meters?
Scientists measure the rate of change in the volume to be 3.5 cubic meters per day, and they know the depth at the bottom of the lake is 27 meters. Their instruments have since broken, so they are unable to ascertain the rate the aforementioned depth is changing. Find that rate.
We tested the first one extensively enough to find a decent answer, but we'll work some more before we post anything definitive.
Oh, and we also have to script. We'll do that tomorrow, I think.
--Nathaniel/Rebecca/Tristan
Nathaniel: Well, let’s start out today looking at New England: it’s relatively rain free, but the sun sure isn’t showing its face; I’d look forward to some rain in the morning if I had to commute up in Maine, but other than that, everything should be fine.Points to the rain stuff moving down from Canada. Now, the rest of the country can expect a wonderful day, full of blooming flowers and sunny skies. Philadelphia’s hitting a high of 64, not the hottest day on record, but certainly nice. Down in Orlando, a few showers may dampen the day, but they’ll be scattered and temperatures will still be in the high seventies. Nathaniel, cont.: Over in Austin, it’s a sweltering 100 degrees, a startling, record breaking high; the heat won’t last, though, and it should be down to a relatively chill 82 pretty soon. There’s nothing like that over in Denver, where it’s only 62; I’ll bet they’re jealous. San Francisco hit a high of 70, and Chicago’s still 62, but it looks like it’ll be warming up real quick next week, reaching as high as the low eighties. Nathaniel, still cont.: Now, it’s Illinois that’s really where we have problems. You see, there’s a lake there, and the rain’s pouring so hard that we’ve had word from some prominent ecologists that there could be some serious danger. Touches ear, then: Actually, I’ve just gotten word. One of those ecologists, Dr. Johnson, has a problem for us. Dr. Johnson, over to you. The camera shifts to Rebecca, who looks rather worried. Rebecca: Reads problem, looking over at her student, Tristan. Okay, Tristan, give me a hand with this one, won’t you? Begins to write up the basic set up for the problem, that sort of thing. Her face is set, determined to ascertain the validity of the problem. Tristan: Begins to talk through the problem as Rebecca works it out on the board.
The volume of a perfectly half-spherical lake is changing at a rate relative to time as rain falls. At some time t the depth at the exact center of the lake is 30 meters, and the rain is falling at a rate such that the change in the depth at the center is .04 centimeters/day.
Some ecologists are worried that if the volume of the lake changes faster than 2.5 cubic meters per day, the fragile ecosystem will be damaged. Given the initial conditions, will the ecosystem be in danger?
In similar conditions, a smaller lake of the same shape has a depth of 15 meters at the exact center. If the rain falls and changes the depth at the center at the same rate as above, and the ecosystem is damaged when the volume of water changes at .5 cubic meters per day, should the scientists worry?
What is the depth at the time when the volume is 25000 cubic meters?
Scientists measure the rate of change in the volume to be 3.5 cubic meters per day, and they know the depth at the bottom of the lake is 27 meters. Their instruments have since broken, so they are unable to ascertain the rate the aforementioned depth is changing. Find that rate.
We tested the first one extensively enough to find a decent answer, but we'll work some more before we post anything definitive.
Oh, and we also have to script. We'll do that tomorrow, I think.
--Nathaniel/Rebecca/Tristan
Nathaniel: Well, let’s start out today looking at New England: it’s relatively rain free, but the sun sure isn’t showing its face; I’d look forward to some rain in the morning if I had to commute up in Maine, but other than that, everything should be fine. Points to the rain stuff moving down from Canada. Now, the rest of the country can expect a wonderful day, full of blooming flowers and sunny skies. Philadelphia’s hitting a high of 64, not the hottest day on record, but certainly nice. Down in Orlando, a few showers may dampen the day, but they’ll be scattered and temperatures will still be in the high seventies.
Nathaniel, cont.: Over in Austin, it’s a sweltering 100 degrees, a startling, record breaking high; the heat won’t last, though, and it should be down to a relatively chill 82 pretty soon. There’s nothing like that over in Denver, where it’s only 62; I’ll bet they’re jealous. San Francisco hit a high of 70, and Chicago’s still 62, but it looks like it’ll be warming up real quick next week, reaching as high as the low eighties.
Nathaniel, still cont.: Now, it’s Illinois that’s really where we have problems. You see, there’s a lake there, and the rain’s pouring so hard that we’ve had word from some prominent ecologists that there could be some serious danger. Touches ear, then: Actually, I’ve just gotten word. One of those ecologists, Dr. Johnson, has a problem for us. Dr. Johnson, over to you.
The camera shifts to Rebecca, who looks rather worried.
Rebecca: Reads problem, looking over at her student, Tristan. Okay, Tristan, give me a hand with this one, won’t you? Begins to write up the basic set up for the problem, that sort of thing. Her face is set, determined to ascertain the validity of the problem.
Tristan: Begins to talk through the problem as Rebecca works it out on the board.