006. `query 6

Question: `qQuery class notes #06 If x is the height of a sandpile and y the volume, what proportionality governs geometrically similar sandpiles? Why should this be the proportionality?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a the proportionality is y = k x^3. Any proportionality of volumes is a y = k x^3 proportionality because volumes can be filled with tiny cubes; surface areas are y = k x^2 because surfaces can be covered with tiny squares. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qIf x is the radius of a spherical balloon and y the surface area, what proportionality governs the relationship between y and x? Why should this be the proportionality?

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a Just as little cubes can be thought of as filling the volume to any desired level of accuracy, little squares can be thought of as covering any smooth surface. Cubes 'scale up' in three dimensions, squares in only two. So the proportionality is y = k x^2.

 

Surfaces can be covered as nearly as we like with tiny squares (the more closely we want to cover a sphere the tinier the squares would have to be). The area of a square is proportional to the square of its linear dimensions. Radius is a linear dimension. Thus the proportionality for areas is y = k x^2.

 

By contrast, for volumes or things that depend on volume, like mass or weight, we would use tiny cubes to fill the volume. Volume of a cube is proportional to the cube of linear dimensions. Thus the proportionality for a volume would be y = k x^3. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qExplain how you would use the concept of the differential to find the volume of a sandpile of height 5.01 given the volume of a geometrically similar sandpile of height 5, and given the value of k in the y = k x^3 proportionality between height and volume.

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a The class notes showed you that the slope of the y = k x^3 graph is given by the rate-of-change function y' = 3 k x^2. Once you have evaluated k, using the given information, you can evaluate y' at x = 5. That gives you the slope of the line tangent to the curve, and also the rate at which y is changing with respect to x. When you multiply this rate by the change in x, you get the change in y.

 

The differential is 3 k x^2 `dx and is approximately equal to the corresponding `dy. Since `dy / `dx = 3 k x^2, the differential looks like a simple algebraic rearrangement `dy = 3 k x^2 `dx, though what's involved isn't really simple algebra. The differential expresses the fact that near a point, provided the function has a continuous derivative, the approximate change in y can be found by multiplying the change in x by the derivative). That is, `dy = derivative * `dx (approx)., or `dy = slope at given point * `dx (approx), or `dy = 3 k x^2 `dx (approx).

 

 

 

The idea is that the derivative is the rate of change of the function. We can use the rate of change and the change in x to find the change in y.

 

The differential uses the fact that near x = 5 the change in y can be approximated using the rate of change at x = 5.

 

Our proportionality is y = k x^3. Let y = f(x) = k x^3. Then y' = f'(x) = 3 k x^2. When x = 5 we have y' = f'(5) = 75 k, whatever k is. To estimate the change in y corresponding to the change .01 in x, we will multiply y ' by .01, getting a change of y ' `dx = 75 k * .01.

}

SPECIFIC EXAMPLE: We don't know what k is for this specific question. As a specific example suppose our information let us to the value k = .002, so that our proportionality is y = .002 x^3. Then the rate of change when x is 5 would be f'(5) = 3 k x^2 = 3 k * 5^2 = 75 k = .15 and the value of y would be y = f(5) = .002 * 5^3 = .25. This tells us that at x = 5 the function is changing at a rate of .15 units of y for each unit of x.

 

Thus if x changes from 5 to 5.01 we expect that the change will be

 

change in y = (dy/dx) * `dx =

rate of change * change in x (approx) =

.15 * .01 = .0015,

 

so that when x = 5.01, y should be .0015 greater than it was when x was 5. Thus y = .25 + .0015 = .2515. This is the differential approximation. It doesn't take account of the fact that the rate changes slightly between x=5 and x = 5.01. But we don't expect it to change much over that short increment, so we expect that the approximation is pretty good.

 

Now, if you evaluate f at x = 5.01 you get .251503. This is a little different than the .2515 approximation we got from the differential--the differential is off by .000003. That's not much, and we expected it wouldn't be much because the derivative doesn't change much over that short interval. But it does change a little, and that's the reason for the discrepancy.

The differential works very well for decently behaved functions (ones with smooth curves for graphs) over sufficiently short intervals.**

 

STUDENT QUESTION

 

so for my function y'=3 k x^2 = 3 k * 5^2 = 75k:

 

If I were to assume assume k=35 then y'=2625.

 

Not sure where to go from here.

 

INSTRUCTOR RESPONSE

 

Briefly that means that y would be changing 2625 times as fast as x, so if x changes by .01, y will change by 2625 * .01 = 26.25.

 

This is based on your assumption that k = 35.  That's an OK assumption, but it isn't related to any of the quantities in the problem.

 

More detail:

 

y ' is how fast y is changing, relative to x.

According to your assumed value k = 35, when x = 5 you do get y ' = 2625. This is based on an assumed value of k, which is not part of the information for this problem.

If y ' = 2625, this means that y is changing 2625 times as fast as x.

So if x changes by .01, y will change by about 2625 times that much and the change in y will be

`dy = 2625 * `dx = 2625 * .01 = 26.25.

As noted above this would not be an exact expression, because y ' itself changes as x changes. However since x changes only from 5 to 5.01, y ' won't change by much, and the approximation will be a good one.

More generally, when x = 5 we our expression for y ' is just y ' = 75 k. The value of k would be dictated by the particular situation.

This would mean that y is changing 75 k times as fast as x.

So to get the approximate change in y you would multiply the change in x by 75 k:

`dy = 75 k * `dx.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qWhat would be the rate of depth change for the depth function y = .02 t^2 - 3 t + 6 at t = 30? (instant response not required)

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a You saw in the class notes and in the q_a_ that the rate of change for depth function y = a t^2 + b t + c is y ' = 2 a t + b. This is the function that should be evaluated to give you the rate.

 

Evaluating the rate of depth change function y ' = .04 t - 3 for t = 30 we get y ' = .04 * 30 - 3 = 1.2 - 3 = -1.8.

 

COMMON ERROR: y = .02(30)^2 - 2(30) + 6 =-36 would be the rate of depth change

 

INSTRUCTOR COMMENT: This is the depth, not the rate of depth change. **

 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qmodeling project 3 problem a single quarter-cup of sand makes a cube 1.5 inches on a side. How many quarter-cups would be required to make a cube with twice the scale, 3 inches on a side? Explain how you know this.

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a You can think of stacking single cubes--to double the dimensions of a single cube you would need 2 layers, 2 rows of 2 in each layer.

 

Thus it would take 8 cubes 1.5 inches on a side to make a cube 3 inches on a side.

 

Since each 1.5 inch cube containts a quarter-cup, a 3 inch cube would contain 8 quarter-cups.

 

COMMON ERROR:

 

It would take 2 quarter-cups.

 

INSTRUCTOR COMMENT: 2 quarter-cups would make two 1.5 inch cubes, which would not be a 3-inch cube but could make a rectangular solid with a square base 1.5 inches on a side and 3 inches high. **

 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qWhat value of the parameter a would model this situation? How many quarter-cups does this model predict for a cube three inches on a side? How does this compare with your previous answer?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a The proportionality would be

 

y = a x^3,

 

with y = 1 (representing one quarter-cup) when x = 1.5. So we have

 

1 = a * 1.5^3, so that

a = 1 / 1.5^3 = .296 approx.

 

So the model is y = .2963 x^3.

 

Therefore if x = 3 we have

 

y = .296 * 3^3 = 7.992, which is the same as 8 except for roundoff error. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qWhat would be the side measurement of a cube designed to hold 30 quarter-cups of sand? What equation did you solve to get this?

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a You are given the number of quarter-cups, which corresponds to y. Thus we have

 

30 = .296 x^3 so that

 

x^3 = 30 / .296 = 101, approx, and

 

x = 101^(1/3) = 4.7, approx..**

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qquery problem 2. Someone used 1/2 cup instead of 1/4 cup. The best-fit function was y = .002 x^3. What function would have been obtained using 1/4 cup?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a In this case, since it takes two quarter-cups to make a half-cup, the person would need twice as many quarter-cups to get the same volume y.

 

He would have obtained half as many half-cups as the actual number of quarter-cups.

 

To get the function for the number of quarter-cups he would therefore have to double the value of y, so the function would be y = .004 x^3. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qquery problem 4. number of swings vs. length data. Which function fits best?

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a If you try the different functions, then for each one you can find a value of a corresponding to every data point. For example if you use y = a x^-2 you can plug in every (x, y) pair and solve to see if your values of a are reasonably consistent. Try this for the data and you will find that y = a x^-2 does not give you consistent a values-every (x, y) pair you plug in will give you a very different value of a.

 

The shape of the graph gives you a pretty good indication of which one to try, provided you know the shapes of the basic graphs.

 

For this specific situation the graph of the # of swings vs. length decreases at a decreasing rate.

 

The graphs of y = a x^.p for p = -.3, -.4, -.5, -.6 and -.7 all decrease at a decreasing rate. In this case you would find that the a x^-.5 function works nicely, giving a nearly constant value of a.

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qproblem 7. time per swing model. For your data what expression represents the number of swings per minute?

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a The model that best fits the data is a x^-.5, and with accurate data we find that a is close to 55.

 

The model is pretty close to

 

# per minute frequency = 55 x^-.5.

 

As a specific example let's say we obtained counts of 53, 40, 33 and 26 cycles in a minute at lengths of 1, 2, 3 and 4 feet, then using y = a x^-.5 gives you a = y * x^.5. Evaluating a for y = 53 and x = 1 gives us a = 53 * 1^.5 = 53; for y = 40 and x = 2 we would get a = 40 * 2^.5 = 56; for y = 34 and x = 3 we get a = 33 * 3^.5 = 55; for y = 26 and x = 4 we get a = 26 * 4^.5 = 52. Since our value of a are reasonably constant the y = a x^.5 model works pretty well, with a value of a around 54.

 

The value of a for accurate data turns out to be about 55.**

 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qIf the time per swing in seconds is y, then what expression represents the number of swings per minute?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a To get the number of swings per minute you would divide 60 seconds by the number of seconds in a swing (e.g., if a swing takes 2 seconds you have 30 swings in a minute). So you would have f = 60 / y, where f is frequency in swings per minute.

 

COMMON ERROR: y * 60

 

INSTRUCTOR COMMENT: That would give more swings per minute for a greater y. But greater y implies a longer time for a swing, which would imply fewer swings per minute. This is not consistent with your answer. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qIf the time per swing is a x ^ .5, for the value determined previously for the parameter a, then what expression represents the number of swings per minute? How does this expression compare with the function you obtained for the number of swings per minute vs. length?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a Time per swing turns out to be a x^.5--this is what you would obtain if you did the experiment very accurately and correctly determined the power function. For x in feet a will be about 1.1.

 

Since the number of swings per minute is 60/(time per swing), you have f = 60 / (a x^.5), where f is frequency in swings / minute.

 

Simplifying this gives f = (60 / a) * x^(-.5).

 

60/a is just a constant, so the above expression is of form f = k * x^-.5, consistent with earlier statements.

 

60 / a = 60 / 1.1 = 55, approx., confirming our frequency model F = 55 x^-.5. **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qquery problem 8. model of time per swing what are the pendulum lengths that would result in periods of .1 second and 100 seconds?

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a You would use your own model here.

 

This solution uses T = 1.1 x^.5. You can adapt the solution to your own model.

 

According to the model T = 1.1 x^.5 , where T is period in seconds and x is length in feet, we have periods T = .1 and T = 100. So we solve for x:

 

For T = .1 we get:

 

.1 = 1.2 x^.5 which gives us

 

x ^ .5 = .1 / 1.2 so that

x^.5 = .083 and after squaring both sides we get

x = .083^2 = .0069 approx., representing .0069 feet.

 

We also solve for T = 100:

 

100 = 1.2 x^.5, obtaining

x^.5 = 100 / 1.2 = 83, approx., so that

x = 83^2 = 6900, approx., representing a pendulum 6900 ft (about 1.3 miles) long. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qquery problem 9. length ratio x2 / x1.

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qWhat expressions, in terms of x1 and x2, represent the frequencies (i.e., number of swings per minute) of the two pendulums?

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a The solution is to be in terms of x1 and x2.

 

If lengths are x2 and x1, you would substitute x2 and x1 for L in the frequency relationship f = 60 / (1.1 `sqrt(L)) to get 60 / (1.1 `sqrt(x1) ) and 60 / (1.1 `sqrt(x2)).

 

Alternative form is f = 55 L^-.5. Substituting would give you 55 * x1^-.5 and 55 * x2^-.5.

 

If you just had f = a L^-.5 (same as y = a x^-.5) you would get f1 = a x1^-.5 and f2 = a x2^-.5 **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qWhat expression, in terms of x1 and x2, represents the ratio of the frequencies of the two pendulums?

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a We need to do this in terms of the symbols x1 and x2. If f = a x^-.5 then f1 = a x1^-.5 and f2 = a x2^-.5.

 

With these expressions we would get

 

f2 / f1 = a x2^-.5 / (a x1^-.5) =

x2^-.5 / x1^-.5 =

(x2 / x1)^-.5 =

1 / (x2 / x1)^.5 =

(x1 / x2)^.5.

 

Note that it doesn't matter what a is, since a quickly divides out of our quotient. For example if a = 55 we get

 

f2 / f1 = 55 x2^-.5 / (55 x1^-.5) =

x2^-.5 / x1^-.5 =

(x2 / x1)^-.5 =

1 / (x2 / x1)^.5 =

(x1 / x2)^.5.

 

This is the same result we got when a was not specified. This shouldn't be surprising, since the parameter a divided out in the third step. **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `qquery problem Challenge Problem for Calculus-Bound Students: how much would the frequency change between lengths of 2.4 and 2.6 feet

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a STUDENT SOLUTION: Note that we are using frequency in cycles / minute.

 

I worked to get the frequency at 2.4 and 2.6

 

y = 55.6583(2.4^-.5) = 35.9273 and y = 55.6583(2.6^-.5)= 34.5178.

 

subtracted to get -1.40949 difference between 2.4 and 2.6.

 

This, along with the change in length of .2, gives average rate -1.409 cycles/min / (.2 ft) = -7.045 (cycles/min)/ft , based on the behavior between 2.4 ft and 2.6 ft.

 

This average rate would predict a change of -7.045 (cycles/min)/ft * 1 ft = -7/045 cycles/min for the 1-foot increase between 2 ft and 3 ft.

 

The change obtained by evaluating the model at 2 ft and 3 ft was -7.2221 cycles/min.

 

The answers are different because the equation is not linear and the difference between 2.4 and 2.6 does not take into account the change in the rate of frequency change between 2 and 2.4 and 2.6 and 3

 

for 4.4 and 4.6

 

y = 55.6583(4.4^-.5) y = 55.6583(4.6^-.5)

y = 26.5341 y = 25.6508

 

Dividing difference in y by change in x we get -2.9165 cycles/min / ft, compared to the actual change -2.938 obtained from the model.

 

The answers between 4-5 and 2-3 are different because the equation is not linear and the frequency is changing at all points. **

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: What is the midpoint between the points (3, 8) and (7, 12)?

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

 

Given Solution:

 

`a You are given two points. The points each have two coordinates. You have to average the x coordinates to get the x coordinate of the midpoint, then average the y coordinates to get the y coordinate of the midpoint.

 

For example if the points are (3, 8) and (7, 12), the average of the x coordinates is (3 + 7) / 2 = 5 and the average of the y coordinates is (8 + 12) / 2 = 10 so the coordinates of the midpoint are (5, 10). **

 

 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `q1.1.20 (was 1.1.18 are (0,4), (7,-6) and (-5,11) collinear?

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Rating:

Given Solution:

`a The distance between (-5,11) and (7,-6) is approximately 20.81:

 

d3 = sqr rt [(7+5)^2 + -(6 +11)^2]

d3 = sqr rt 433

d3 = 20.81

 

Using the distance formula the distances between (-5,11) and (0,4) is 8.6 and the distance between (0,4) and (7, -6) is 12.2.

 

'Collinear' means 'lying along the same straight line'.

 

If three points are collinear then the sum of the distances between the two closer pairs of points will equal the distance between the furthest two.

 

Since 8.6 + 12.2 = 20.8 the points are on the same straight line. **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: `q1.1.24 find x | dist (2,-1) to (x,2) is 5What value of x makes the distance 5?