Thursday, April 12, 2018
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Friday, April 13, 2018
(outside of North American and South America)
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Simplifying, we get
Since
So when
Simplifying, we get
The value of the expression
Substituting
When
Simplifying, we get
When
That is, when
Solving
Since
Note: In each of the solutions to (b), (c) and (d), we chose to simplify the expression before substituting. Changing the order to substitution followed by simplification would also allow us to solve these problems.
In
Using the Pythagorean Theorem, we get
In Figure 2,
From the second fact, we know that
Using the Pythagorean Theorem, we get
Since
In
Since
In
In
A cylinder having radius
Cylinder A has radius 12 and height 25, and so its volume is
Before Cylinder B is lowered into Cylinder A, the height of water in Cylinder A is 19, and so initially the volume of water in Cylinder A is
The height of Cylinder B, 30, is greater than the height of Cylinder A, and so it is not possible for water to pour out of Cylinder A and into Cylinder B.
When Cylinder B is lowered to the bottom of Cylinder A, the portion of Cylinder B lying inside Cylinder A has radius 9 and height 25 (the height of Cylinder A).
Thus, the volume that Cylinder B occupies within Cylinder A is
Since water cannot pour into Cylinder B, the space available for water within Cylinder A (and outside Cylinder B) is the difference between the volume of Cylinder A and the volume of Cylinder B lying inside Cylinder A, or
The volume of water in Cylinder A was initially
Therefore, the volume of water that spills out of Cylinder A and onto the ground is
As Cylinder B is lowered into Cylinder A, water spills out of Cylinder A and onto the ground when:
the volume of water in Cylinder A exceeds the volume inside Cylinder A and outside Cylinder B, and
the top of Cylinder B lies above the top of Cylinder A.
(See Figure 1 given in the question.)
As Cylinder B is lowered into Cylinder A, water spills out of Cylinder A and into Cylinder B when:
the top of Cylinder B lies below the top of Cylinder A, and
the volume of water in Cylinder A (and outside Cylinder B) exceeds the volume inside Cylinder A that lies below the top of Cylinder B and outside Cylinder B, and
Cylinder B is not full of water.
(See Figure 2 given in the question.)
In Figure 3 shown, the top of Cylinder B has been lowered to the same level as the top of Cylinder A.
At this point, the volume of space inside Cylinder A and outside Cylinder B is
The initial volume of water in Cylinder A was
(Since the top of Cylinder B is not below the top of Cylinder A, no water has spilled out of Cylinder A and into Cylinder B at this point.)
As Cylinder B is lowered below this level, water will spill out of Cylinder A and into Cylinder B. How much water will spill into Cylinder B?
In Figure 4, the volume of water labelled
This volume of water will spill into Cylinder B (since the top of Cylinder B will be below the top of Cylinder A).
The shape of the water labelled
So the volume of the water labelled
In addition, the water labelled
The shape of the water labelled
The volume of water that spills from Cylinder A into Cylinder B is
The depth,
Note: We could have determined the volume of water that spills into Cylinder B by noticing that the volume labelled
Since both volumes have height 5, their combined volume is equal to that of a cylinder with radius 12 and height 5, and so
Solution 1
We begin by finding the range of values of
Consider lowering Cylinder B into Cylinder A until the water level reaches the top of Cylinder A, as shown in Figure 7 (we know this is possible for some values of
Let
The volume of water,
That is,
From part (a), the initial volume of water is
So if
What if
When
In this case when
That is, when Cylinder B is lowered so that its top is level with the top of Cylinder A, the volume of water that lies directly below Cylinder B is greater that the volume of Cylinder B and so Cylinder B will be completely full of water when it is lowered to the bottom of Cylinder A.
In this question, we require that Cylinder B not be full and so
Next, we will further restrict the range of values of
Consider lowering Cylinder B to the point where the tops of the two cylinders are level with one another (so then
Some water has spilled out of Cylinder A.
When Cylinder B is lowered beyond this point (so then we require
From the solution in part (b), recall that when Cylinder B is lowered to the bottom of Cylinder A, the volume of water that will spill from Cylinder A into Cylinder B is equal to the volume of water inside Cylinder A that lies below the bottom of Cylinder B (as in Figure 8).
This cylinder has radius 12 and height
Assume that when this volume of water has spilled into Cylinder B, it fills Cylinder B to a depth of
Once Cylinder B is lowered to the bottom of Cylinder A, the volume of water in Cylinder B,
Solving for
The depth of water in Cylinder B must be less than the height of Cylinder B (Cylinder B cannot be full), so then
As noted earlier, no water can spill into Cylinder B unless its height is less than that of Cylinder A, and so
When Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full when
Solution 2
Let the volume of Cylinder A be
As we determined in Solution 1,
If
This gives
If
Assume that
Then the volume of water that spills out of Cylinder A onto the ground is
When the tops of the two cylinders are at the same level (Figure 9), no water has spilled into Cylinder B, and so the volume of water in Cylinder A is the initial volume of water less the volume of water that has spilled out onto the ground.
That is,
From this point on, all water stays in Cylinder B or in Cylinder A.
When Cylinder B is on the bottom of Cylinder A (Figure 10), the volume of water outside of Cylinder B (but inside Cylinder A), is the volume of Cylinder A that lies below the top of Cylinder B less the volume of Cylinder B.
That is,
That is,
When Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full when
As a sum of one of more consecutive positive integers, 45 can be written as
The value of
The sum of the positive integers from 1 to
The sum of the positive integers from 4 to
Therefore,
If
That is,
Simplifying, we get
In this question, we are asked to determine the number of such pairs
Since
That is,
The difference between these two integers is
Since the difference between
Thus, the problem of evaluating
At this point, we have shown that each pair of integers
We must now show that the converse is also true; that is, each factor pair with different parity gives a unique pair
Suppose that
We show that each pair
If
Adding the equations
Subtracting the two equations
Since
Therefore, each of
(If we assume that
That is, each factor pair
This confirms that evaluating
Before evaluating
Since
Since there are 3 choices for
Next, we demonstrate that each of these 6 odd factors gives a unique pair
The odd factors
Next we note that since
Adding the two equations to solve this system of equations, we get
This pair
We summarize the results using the other factor pairs in the table below.
Factor Pair | |||||
---|---|---|---|---|---|
1 | 90 | 45 | 44 | 45 | |
3 | 30 | 16 | 13 | ||
5 | 18 | 11 | 6 | ||
9 | 10 | 9 | 0 | ||
6 | 15 | 10 | 4 | ||
2 | 45 | 23 | 21 |
Comparing this table to our answer in part (a), we see that indeed each odd factor of
Finally, we turn our focus to evaluating
The odd factors of
Since there are 5 choices for
We would like to determine the smallest positive integer
If
That is,
From part (c), we know that
Since
To minimize
The smallest positive integer