012.  `query 12

Question:  `qproblem 1. box of length 30 centimeters capacity 50 liters

What is the proportionality for this situation, what is the proportionality constant and what is the specific equation that relates capacity y to length x? 

Your solution: 

Confidence Assessment:

Given Solution: 

** The proportionality for volume is y = k x^3, where y is capacity in liters when x is length in cm.

Since y = 50 when x = 30 we have

50 = k * 30^3 so that

k = 50 / (30^3) = 50 / 27,000 = 1/540 = .0019 approx.

Thus y = (1/540) * x^3. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qWhat is the storage capacity of a box of length 100 centimeters?

Your solution: 

Confidence Assessment:

Given Solution: 

** The proportionality is y = 1/540 * x^3 so if x = 100 we have

y = 1/540 * 100^3 = 1900 approx.

A 100 cm box geometrically similar to the first will therefore contain about 1900 liters.  **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qWhat length is required of a geometrically similar box to obtain a storage capacity of 100 liters?

Your solution: 

Confidence Assessment:

Given Solution: 

** If y = 100 then we have

100 = (1/540) * x^3 so that

x^3 = 540 * 100 = 54,000.

Thus x = (54,000)^(1/3) = 38 approx.

The length of a box that will store 100 liters is thus about 38 cm. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qHow long would a geometrically similar box have to be in order to store all the water in a swimming pool which contains 450 metric tons of water? A metric ton contains 1000 liters of water.

Your solution: 

Confidence Assessment:

Given Solution: 

** 450 metric tons is 450 * 1000 liters = 450,000 liters.  Thus y = 450,000 so we have the equation

540,000 = (1/540) x^3

which we solve in a manner similar to the preceding question to obtain

x = 624, so that the length of the box is 624 cm. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 2. cleaning service scrub the surface of the Statute of width of finger .8 centimeter vs. 20-centimeter width actual model takes .74 hours.

How long will it take to scrub the entire statue?

Your solution: 

Confidence Assessment:

Given Solution: 

** y = k x^2 so

.74 = k * .8^2.  Solving for k we obtain

k = 1.16 approx. so

y = 1.16 x^2.

The time to scrub the actual statue will be

y = 1.16 x^2 with x = 20. 

We get

y = 1.16 * 20^2 = 460 approx..

It should take 460 hrs to scrub the entire statue. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 3. illumination 30 meters is 5 foot-candles.  What is the proportionality for this situation, what is the value of the proportionality constant and what equation relates the illumination y to the distance x?

Your solution: 

Confidence Assessment:

Given Solution: 

** The proportionality should be

y = k x^-2,

where y is illumination in ft candles and x the distance in meters.

We get

5 = k * 30^-2, or

5 = k / 30^2 so that

k = 5 * 30^2 = 4500.

Thus y = 4500 x^-2.

We get an illumination of 10 ft candles when y = 10.  To find x we solve the equation

10 = 4500 / x^2.  Multiplying both sides by x^2 we get

10 x^2 = 4500.  Dividing both sides by 10 we have

x^2 = 4500 / 10 = 450 and

x = sqrt(450) = 21 approx..

For illumination 1000 ft candles we solve

1000 = 4500 / x^2,

obtaining solution x = 2.1 approx..

We therefore conclude that the comfortable range is from about x = 2.1 meters to x = 21 meters. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 5.

Does a 3-unit cube weigh more or less than 3 times a 1-unit cube? Why is this?

Your solution: 

Confidence Assessment:

Given Solution: 

** A 3-unit cube is equivalent to 3 layers of 1-unit cubes, each layer consisting of three rows with 3 cubes in each row.

Thus a 3-unit cube is equivalent to 27 1-unit cubes.

If the weight of a 1-unit cube is 35 lbs then we have the following:

Edge      equiv. # of       weight

Length  1-unit cubes

   1               1                   35

   2               8           8 * 35 = 360

   3               27         27 * 35 = 945

   4              64          64 * 35 = 2240

   5             125        125 * 35 = 4375

Each weight is obtained by multiplying the equivalent number of 1-unit cubes by the 35-lb weight of such a cube.  **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 6. Give the numbers of 1-unit squares required to cover 6-, 7-, 8-, 9- and 10-unit square, and also an n-unit square.

Your solution: 

Confidence Assessment:

Given Solution: 

**

To cover a 6-unit square requires 6 rows each containing 6 1-unit squares for a total of 36 one-unit squares.

To cover a 7-unit square requires 7 rows each containing 7 1-unit squares for a total of 49 one-unit squares.

To cover a 8-unit square requires 8 rows each containing 8 1-unit squares for a total of 64 one-unit squares.

To cover a 9-unit square requires 9 rows each containing 9 1-unit squares for a total of 81 one-unit squares.

To cover a 10-unit square requires 10 rows each containing 10 1-unit squares for a total of 100 one-unit squares.

To cover an n-unit square requires n rows each containing n 1-unit squares for a total of n*n=n^2 one-unit squares. **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 8. Relating volume ratio to ratio of edges.

Your solution: 

Confidence Assessment:

Given Solution: 

** right idea but you have the ratio upside down.

The volume ratio of a 5-unit cube to a 3-unit cube is (5/3)^3 = 125 / 27 = 4.7 approx..

The edge ratio is 5/3 = 1.67 approx.

VOlume ratio = edgeRatio^3 = 1.678^3 = 4.7 approx..

From this example we see how volume ratio = edgeRatio^3.

If two cubes have edges 12.7 and 2.3 then their edge ratio is 12.7 / 2.3 = 5.5 approx..

The corresponding volume ratio would therefore be 5.5^3 = 160 approx..

If edges are x1 and x2 then edgeRatio = x2 / x1.  This results in volume ratio

volRatio = edgeRatioo^3 = (x2 / x1)^3.  **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 9. Relating y and x ratios for a cubic proportionality.

What is the y value corresponding to x = 3 and what is is the y value corresponding to x = 5?

Your solution: 

Confidence Assessment:

Given Solution: 

** If y = a x^3 then

if x1 = 3 we have y1 = a * 3^3 and

if x2 = 5 we have y2 = a * 5^3.

This gives us ratio y2 / y1 = (a * 5^3) / (a * 3^3) = (a / a) * (5^3 / 3^3) = 1 * 125 / 27 = 125 / 27.

In general if y1 = a * x1^3 and y2 = a * x2^3 we have

}

y2 / y1 = (a x2^3) / (a x1^3) = (a / a) * (x2^3 / x1^3) = (x2/x1)^3.

This tells you that to get the ratio of y values you just cube the ratio of the x values.  **

Self-critique (if necessary):

Self-critique Rating:

Question:  `qproblem 10. Generalizing to y = x^p.

Your solution: 

Confidence Assessment:

Given Solution: 

** If y = a x^2 then

y2 / y1 = (a x2^2) / (a x1^2) = (a / a) * (x2^2 / x1^2) = (x2/x1)^2.

This tells you that to get the ratio of y values you just square the ratio of the x values.

If y = f(x) = a x^p then

y1 = f(x1) = a x1^p  and

y2 = f(x1) = a x2^p  so that

y2 / y1 = f(x2) / f(x1) = (a x2^p) / (a x1^p) = (a / a) ( x2^p / x1^p ) = x2^p / x1^p = (x2 / x1)^p.  **

Add comments on any surprises or insights you experienced as a result of this assignment.

this was a pretty easy assignment to comprehend, I did like the ratio stuff looks like it will come in handy ** this stuff is very important in most areas of study **

Self-critique (if necessary):

Self-critique Rating:

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily.  If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

Question:  A rectangular box whose longest diagonal is 50 cm will hold 100 kilograms of gravel.  What would be the longest diagonal of a geometrically similar box with a capacity of 25 kilograms of gravel?

Question:  The ratio of the volumes of two geometrically similar solids is the cube of the ratio of a given linear dimension.  What must be the ratio of the diameters of two spheres if one has twice the volume of the other?

Question:  If y1 = a x1^4 and y2 = a x2^4, then if y2 / y1 = 4, what is the ratio x2 / x1?

Self-critique rating: