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mth 173
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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txt problems 31, 32 and 43
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31. The energy E, expended by as swimming dolphin is proportional to the cube of the speed, v, of the dolphin.
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Here's more than you probably want to know about proportionality: This is very much worth knowing, as the concept of proportionality is among the most important mathematical concepts in real-world applications. Another common term applied to this general concept is 'scaling'.
y is proportional to x if there exists a constant c such that y = k x.
k is called the proportionality constant.
So if E is proportional to v^3, this means that there exists a value of k such that E = k * v^3.
One implication of this proportionality:
You can sketch a graph of y = v^3. Then you can label the x and y axes with values of E and v. As long as you use a consistent scale for each graph, you can use any scale that's appropriate to the values of E and v for that dolphin. For example if you know that the dolphin uses 1000 Joules of energy when swimming at 10 meters / second, you could pick any point on the v axis and call it 10 m/s, and any point on the E axis and call it 1000 Joules. Your graph would then be the model for that dolphin. I actually doesn't make any sense to say that the dolphin uses 1000 Joules of energy to swim at that speed (see below) but for the moment I'm sticking with the book's (mis)statement.. See the next paragraph for a correct statement of the problem.
Incidentally the problem is poorly posed, in the sense that the amount of energy used depends on how long the dolphin swims at the given speed. The accurate relationship would be that the rate of energy expenditure is proportional to the cube of the speed. The rate of energy use is called power. So we would more appropriately say the P = k v^3. A dolphin might use 1000 Joules / second to swim at 10 meters / second. 1000 Joules / second is a measure of power (a Joule / second is a watt, so 1000 Joules / second is 1000 watts, or a kilowatt). If you fix the graph described in the preceding by replacing 1000 Joules with 1000 watts, then it will make sense.
So the correct proportinonality would be better expressed as
P = k v^3.
1000 watts at 10 m/s is in the right ballpark, and it turns out that if P = k v^3, with these numbers, we would have k = P / v^3 = 1000 watts / (10 m/s)^3 = 1 watt / (m^3 / s^3). That unit watt / (m^3 / s^3) simplifies but let's not worry about that right now. So if the proportionality is correct, and our information about the dolphin is correct, we have
P = 1 watt / (m^3 / 2^3) * v^3.
If we wanted to find the power output for this dolphin at a speed of 12 m/s, we could just plug 12 m/s into the formula (we would get P = 1440 watts). If we wanted to know how fast a dolphin could go with power output 2000 watts, we could solve the equation for v.
10 m/s is about 23 mph, so we could also say that k = P / v^3 = 1000 watts / (23 mph)^3 = .1 watt / (mph)^3. In this case we would write
P = .1 watt / (mph)^3 * v^3.
If we want to know the power in watts necessary for a dolphin to swim at 30 mph, we could use this equation.
We could alternatively label a graph of y vs. x^3, picking a point on the curve and labelling the corresponding x coordinate as 23 mph and its y coordinate as 1000 watts. The graph would then represent the function P = .1 watt / (mph)^3 * v^3.
We could use other units. This dolphin is using about 1.3 horsepower to move at 23 mph. We could again solve for k and express the equation, this time in terms of horsepower and mph.
So you see that the proportionality constant k will differ if the two quantities are measured in different units. It will also differ for different dolphins (a fat dolphin might, for example, require 1500 watts at 10 m/s or 23 mph), so k would be different for that dolphin)
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32. the number of animal species, N, of a certain body length, l, is inversely proportional to the square of l.
43. You drive at a constant speed from Chicago to Detroit, a distance of 275 miles. About 120 miles from Chicago you pass through Kalamazoo, Michigan. Sketch a graph of your distance from Kalamazoo as a function of time.
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31. The book said it was E=kv^3. Why wouldn't it be E=v^3, the speed is v and E is proportional to the cube of speed. Where is the k coming from.
32. It is an even numbered problem so I couldn't check my answer. I would say it would be 1/N= l^2. Is this correct?
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I'll use the capital L to avoid confusion with the number 1.
The basic proportionality equation is y = k x, with constant k. This indicates that y is proportional to x.
The 'inverse' of L is 1 / L (technically this is the multiplicative inverse of L). The inverse of the square of L is 1 / L^2.
To say that N is inversely proportional to the square of L is therefore to say that N is proportional to 1 / L^2. So we write
N = k * ( 1 / L^2), or just
N = k / L^2.
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43. The book showed a v-shape graph. If you are going at a constant speed and the total miles traveled is 275 and you arrived at Kalamazoo after traveling only 120 miles. Starting from the origin (starting from Chicago) the graph would increase until 275 miles on the y axis. The book had it starting from (0,120) moving down to the x axis to (t,0) and then to (t,275). I'm confused as why the graph looks this way. If not knowing that Kalamazoo is out of the way from a straight shot to Detroit, why would the y value on the graph decrease, then increase again. This is probably simply and I made it more complex than it is but, don't understand it nevertheless.
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In Chicago you're 120 miles from Kalamazoo. To a graph of distance from Kalamazoo vs. time would start at (0, 120 miles).
When you get to Kalamazoo, your distance from Kalamazoo is zero. So the point on your graph will be on the t axis. The graph will have decreased from (0, 120 miles) to this point on the t axis. We aren't told what the constant speed is, so we don't have a definite coordinate to use for t.
Having passed Kalamazoo your distance from Kalamazoo starts increasing, and continues increasing until the distance is 155 miles. So your graph starts rising. Its steepness is the same as before, but it is now sloping upward rather than downward. It continues increasing until the distance is 155 miles.
At no point on this trip are you 275 miles from Kalamazoo.
Incidentally I drive the stretch of road from Kalamazoo to Detroit at least once a year. You don't travel from Chicago to Detroit as a constant speed.
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