Section 1.5 – Applications of Quadratic Equations
Objectives
• Use the four-step problem solving strategy to solve application problems involving
quadratic equations.
• Solve quadratic application problems involving unknown numerical quantities.
• Solve quadratic application problems involving geometric shapes.
• Solve quadratic application problems involving projectile motion.
• Solve quadratic application problems involving distance, rate, and time.
• Solve quadratic application problems involving two people working together.
Preliminaries
The four-step problem solving strategy can be quickly summarized by the following:
1.
2.
3.
4.
Warm-up
Determine a mathematical expression that relays the information described in the following
problems. Remember to define your variables.
1. A car is traveling M mph for H hours. Represent the number of miles traveled.
2. The speed of a boat in still water is S. The current of the water is C. Represent the
boat’s speed going upstream.
3. An alloy contains 40% gold. Represent the number of grams of gold present in G grams
of the alloy.
4. An item originally priced at P dollars will be discounted 15%. Represent the new price.
5. The width of a rectangle is half of its length. Express the perimeter of the rectangle in
terms of width only.
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Class Notes and Examples
For each question below, solve the problem algebraically. Clearly define all variables used.
Check your solution. You may want to refer to the strategies listed in the front of these notes.
The columns are set up so you can check your solution on the right-hand side.
1.5.1
The sum of the square of a number and the square of 7 more than the number is 169.
What is/are the number(s)?
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1.5.2
A stone thrown downward from a height of 274.4 meters. The distance it travels in t
seconds is given by the function 𝑠(𝑡) = 4.9𝑡 2 + 49𝑡. How long will it take the stone to
hit the ground?
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1.5.3
Amy travels 450 miles in her car at a certain speed. If the car had gone 15 mph faster,
the trip would have taken 1 hour less. Determine the speed of Amy’s car.
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1.5.4
The area of a square is numerically 60 more than the perimeter. Determine the length
of the side of the square.
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1.5.5
The length of a rectangle is 7 centimeters longer than the width. If the diagonal of the
rectangle is 17 centimeters, determine the length and width.
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1.5.6
It takes Julia 16 minutes longer to chop vegetables for soup than it takes Bob. Working
together, they are able chop the vegetables in 15 minutes. How long will it take each of
them if they work by themselves?
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1.5.7
Maria traveled upstream along a river in a boat a distance of 39 miles and then came
right back. If the speed of the current was 1.3 mph and the total trip took 16 hours,
determine the speed of the boat relative to the water.
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Section 1.5 Self-Assessment (Answers on page 253)
1.
(Multiple Choice) The height of a leaf above the ground after it falls from a tree is given
by the formula 𝑠(𝑡) = −3.5𝑡 2 + 37.8, where time is given in seconds and height is given
in feet above the ground. When would the leaf land on the top a person’s head if the
person was seated on the ground under the tree and their head was 2.9 feet above the
ground?
The amount of time is:
(A)
(B)
(C)
(D)
(E)
Between 3.5 and 3.6 seconds
Between 3.4 and 3.5 seconds
Between 3.3 and 3.4 seconds
Between 3.2 and 3.3 seconds
Between 3.1 and 3.2 seconds
2.
The length of a rectangle is 17 inches longer than the width. If the diagonal of the
rectangle is 25 inches, determine the length and width.
3.
(Multiple Choice) A train that is traveling 45 mph leaves a train station and goes west.
Another train leaves the same train station two hours later on a parallel track traveling
west at 75 mph. How long will it take the fast train to catch up with the slow train?
(A) 6 hours
4.
(B) 5 hours
(C) 3 hours
(D) 1.5 hours
(E) None of these
Arnold and Samantha are working together to paint the walls in a bedroom. Arnold can
paint the walls 2 hours faster than Samantha when each of them works alone. If it takes
them 8 hours to paint the walls when they work together, determine how long it would
take Samantha to paint the walls if she works alone. (Round your answer to 2 decimal
places.)
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