April 2021 (in North America and South America)
April 2021 (outside of North American and South America)
©2021 University of Waterloo
The total cost to rent a car is $180.00.
If 4 people rent a car, the cost per person is
Since the members of the group equally share the total cost to rent the vehicle, the smaller the group, the greater the cost per person.
To rent an SUV, the smallest group size required is 5 passengers and the total cost is $200.00.
Thus, the maximum possible cost per person to rent an SUV is
Let the total cost to rent a van be
When renting a van, the maximum possible cost per person occurs when the number of
passengers is 9 (the fewest number possible), and so this maximum cost is
The minimum possible cost per person occurs when the number of passengers is 12 (the
greatest number possible), and so this minimum cost is
Then,
Trapezoid
The slope of line segments
The length of
The height of the trapezoid is equal to the vertical distance between
The line passing through
The slope of the line through
Solution 1
Since
Equating slopes, we get
Thus, point
Solution 2
The line passing through
Rearranging, we get
Since this line has
Solution 3
The line passing through
This line passes through
Sides
Solution 1
Let the coordinates of
Since
Since
Equating slopes, we get
Substituting
When
Solution 2
The line passing through
The line passing through
These two lines intersect at
Let
Assume
In this case, if the height of
The area of
That is,
There are two possibilities:
In the first case,
Recall that
The line passing through
If
Similarly, if
Since
Since
Since
From the definition of sequence
Since
From the definition of sequence
This gives two equations in two unknowns,
Adding these two equations, we get
Substituting, we get
Using algebraic manipulation, and the definitions
The expression
Thus,
The expression
This simplifies to 0 if there are an even number of terms, that is, if
Summarizing, we have
Since
When
When
Since
Since
Thus, we want the smallest positive integer
When
When
In
Using the Pythagorean Theorem, we get
The value of
When
By the Pythagorean Theorem,
Thus, we get
Since
When
The product
If
There are exactly two Pythagorean triples
In the first case,
In the second case,
Therefore,
In the table below, we use the positive factor pairs of 72 to determine all possible integer values of
Factor pair | ||||||
---|---|---|---|---|---|---|
1 | 72 | 13 | 84 | 85 | ||
2 | 36 | 14 | 48 | 50 | ||
3 | 24 | 15 | 36 | 39 | ||
4 | 18 | 16 | 30 | 34 | ||
6 | 12 | 18 | 24 | 30 | ||
8 | 9 | 20 | 21 | 29 |
It is worth noting that instead of factoring the equation
Since
When
We begin by assuming that
Substituting, we get
The product
If
If
Using the Pythagorean Theorem, we get
For each of the three possibilities,
It can similarly be shown that
Since
This is not possible since
Assuming
Thus,
Beginning with the fact that the perimeter of the triangle is 510 cm, we get the following equivalent equations
Substituting
We continue our analysis of the remaining 5 factor pairs in the table below.
As before, we make the assumption that
Factor pair | Value(s) of |
|||
---|---|---|---|---|
No |
||||
No |
||||
No |
Therefore, the values of