28. Problem:Find the length of each side of the triangle.

Solution:
Strategy: Try to write two expressions for the perimeter.
Develop a mathematical model:
(1) The perimeter is 483 ft.
(2) The perimeter is x + 5x + (6x – 3)
These two expressions for the perimeter must be equal.
So we have the equation
                x + 5x + (6x – 3) = 483 ft.
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for x
x + 5x + (6x – 3) = 483 ft.
                              Add like terms
12x –3 = 483
                              Add 3 to both sides of the equation
12x = 486
                              Divide both sides of the equation by 12
x = 40.5 ft.
Conclusion:
The length of one side is 40.5 ft.
The length of a second side is 5x = 202.5 ft.
The length of the third side is 6x – 3 = 240 ft.

32. Problem: A premedical student at a local university was complaining that she had just paid $86.11 for her human anatomy book, including tax. Find the price of the book before taxes if the tax rate at this university is 9%.
Begin by stripping away everything except the essence of the problem. This yields:
        The total cost of a book (price of a book plus 9% tax) is $86.11.
        What is the price of the book?
Solution:
Strategy: Try to write two expressions for the total cost.
Develop a mathematical model:
Let p represent the price of the book (an unknown and the value we want to determine)
Tax on the book the is 0.09p (9% of the price)
(1) Total cost is the price of the book plus the tax p + 0.09p
(2) Total cost is also $86.11
These two representations for the total cost must be equal.
So we have the equation
                   p + 0.09p = $86.11
This is the mathematical model for the problem.
Solve the Equation:
Solve the equation (in the model) for p.
p + 0.09p = $86.11
                                       (Use the Distributive Law)
p(1 + 0.09) = $86.11
                                       (Add the numbers inside parenthesis)
1.09p = $86.11
                                       (divide both sides of the equation by 1.09)

Conclusion:
The price of the book, before taxes, is $79.00.

33. Problem: INVESCO Field at Mile High, home to the Denver Broncos, has 11,675 more seats than Heinz Field, home to the Pittsburgh Steelers. Together these two stadiums can seat a total of 140,575 NFL fans. How many seats does each stadium have.
Solution: Strip away all non-essential verbiage to restate the problem more succinctly as:

INVESCO has 11,675 more seats than Heinz.  Total seating in the two stadiums is 140,575.
What is the seating capacity of each ?

Strategy: Try to write two expressions for the total seating.
Develop a mathematical model:
Let x be the seating capacity at Heinz.
Then seating at INVESCO is x + 11,675
(1) Total seating is 140,575
(2) Total seating is x + (x + 11,675)
These two expressions for the total seating must be equal.
So we have the equation
                       x + (x + 11,675) = 140,575
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for x
x + (x + 11,675) = 140,575
                                                Add like terms
2x + 11675 = 140,575
                                       Add - 11675 to both sides of the equation
2x = 128,900
                          Divide both sides of the equation by 2
x = 64,450
Conclusion:
Seating capacity of Heinz Field is 64,450
Seating capacity of INVESCO Field is
x + 11,675 = 64,450 + 11,675 = 76,125

36.  Problem: The three tallest hospitals in the world are Guy’s Tower in London, Queen Mary Hospital in Hong Kong, and Galter Pavilion in Chicago. These buildings have a total heigh of 1320 feet. Guy’s Tower is 67 feet taller that Galter Pavilion and the Queen Mary Hospital is 47 feet taller than Galter Pavilion. Find the heights of the three hospitals.
Solution: Strip away all non-essential verbiage to restate the problem more succinctly as:
The total of the heights of three buildings is 1320 feet.
Guy’s Tower is 67 ft. taller than Galter Pavilion
Queen Mary Hospital is 47 ft. taller than Galter Pavilion.
What is the height of each ?
Strategy: Try to write two expressions for the total height.
Develop a mathematical model:
Note that two of the heights are related to the height of Galter Pavilion.
Let h be the height of Galter Pavilion.
Then h + 67 is the height of Guy’s Tower
and h + 47 is the height of Quenn Mary Hospital.
(1) Total height is 1320
(2) Total height is h + (h + 67) + (h + 47)
These two expressions for the total height must be equal.
So we have the equation
                     h + (h + 67) + (h + 47) = 1320
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for h
h + (h + 67) + (h + 47) = 1320
                                      Add like terms
3h + 114 = 1320
                                 Add - 114 to both sides of the equation
2x = 1206
                           Divide both sides of the equation by 2
x = 402
Conclusion:
Galter Pavilion is 402 ft. tall
Guy’s Tower is 402 + 67 = 469 ft. tall
Queen Mary Hospital is 402 + 47 = 449 ft. tall.

46.  Problem: Two frames are needed with the same outside perimeter: one frame in the shape of a square and one in the shape of an equilateral triangle. Each side of the triangle is 6 centimeters longer than each side of the square. Find the dimensions of each frame.
Solution: Recall that an equilateral triangle is a triangle with equal sides.
Strategy: Try to write two expressions for the perimeter.
Develop a mathematical model:
Note that length of the triangle’s sides are related to the length of a side of the square.
Let x be the length of a side of the square.
Then x + 6 is the length of a side of the triangle.
Consider the square to conclude the perimeter is 4x.
Consider the triangle to conclude the perimeter is 3(x + 6)
(1) Perimeter is 4x
(2) Perimeter is 3(x + 6)
These two expressions for the total height must be equal.
So we have the equation
                    4x = 3(x + 6)
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for x
4x = 3(x + 6)
                        Use the Distributive Law to remove parenthesis
4x = 3x + 18
                       Add – 3x to both sides of the equation
x = 18

Conclusion:
The square has sides of length 18 cm.
The triangle has sides of length x + 6 = 18 + 6 = 24 cm.

56.   Problem: The blue whale is the largest of whales. Its average weight is 3 times the difference of the average weight of a humpback whale and 5 tons. It the total of the average weights of a blue whale and a humpback whale is 117 tons, find the average weight of each type of whale.
Solution: Begin by stripping away all non-essential words. Note that all weights are average weights – because we are not discussing two particular whales. The fact that these are whales is not important, (they might just as well be really big goldfish). So we have fish A and fish B (the fact that they are fish isn’t important either) .
So we have the weight of A (blue) is 3 times the difference of the weight of B (humpback) and 5 tons. We also know that the combined weight of A and B is 117 tons.
What is the weight of A and of B?
Strategy: Try to write two expressions for the combined weight.
Develop a mathematical model:
Let w be the weight of B.
Then the weight of A is 3(w – 5)
(1) Combined weight is 117
(2) Combined weight is 3(w – 5)
These two expressions for the combined weight must be equal.
So we have the equation
                w + 3(w – 5) = 117
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for w
w + 3(w – 5) = 117
                                   Use the Distributive Law to remove parenthesis
w + 3w – 15 = 117
                                   Add 15 to both sides of the equation and combine like terms
4w = 132
                      Divide both sides of the equation by 3
w = 33
Conclusion:
The average weight of the humpback whale is 33 tons.
The average weight of the blue whale is 3(w – 5) = 3(33 – 5) = 84

56.   The other interpretation of “difference”
Problem: The blue whale is the largest of whales. Its average weight is 3 times the difference of the average weight of a humpback whale and 5 tons. It the total of the average weights of a blue whale and a humpback whale is 117 tons, find the average weight of each type of whale.
Solution: Begin by stripping away all non-essential words. Note that all weights are average weights – because we are not discussing two particular whales. The fact that these are whales is not important, (they might just as well be really big goldfish). So we have fish A and fish B (the fact that they are fish isn’t important either) .
So we have the weight of A (blue) is 3 times the difference of the weight of B (humpback) and 5 tons. We also know that the combined weight of A and B is 117 tons.
What is the weight of A and of B?
Strategy: Try to write two expressions for the combined weight.
Develop a mathematical model:
Let w be the weight of B.
Then the weight of A is 3(5 - w)
(1) Combined weight is 117
(2) Combined weight is 3(5 - w)
These two expressions for the combined weight must be equal.
So we have the equation
            w + 3(5 - w) = 117
This is the mathematical model for the problem.
Solve the equation:
Solve the equation (in the model) for w
w + 3(5 - w) = 117
                                   Use the Distributive Law to remove parenthesis
w – 3w + 15 = 117
                                  Add -15 to both sides of the equation and combine like terms
-2w = 102
                                  Divide both sides of the equation by -2
w = - 51
Conclusion:
This is impossible because a weight cannot be a negative number.

Comment: The phrase
“difference of the average weight of a humpback whale and 5 tons”
can be interpreted in two ways (w - 5 or 5 – w). In this problem one interpretation leads to an impossible answer and can terfore be discarded. In other examples both interpretations may lead to reasonable answers.

The statement in the problem should have been more precise such as:
“difference of the average weight of a humpback whale minus 5 tons”

The point of this brief discussion is that it is important that a statement about subtraction be phrased in such a manner that the minuend and subtrahend are clearly identified.