Word Problems
Translate real-world situations into algebraic expressions and equations using polynomials and radicals to solve practical problems.
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Word problems require you to translate everyday situations into mathematical expressions. This lesson focuses on recognizing when to use polynomial and radical operations in real-world contexts.
Problem-Solving Strategy
- Read carefully: Identify what the problem is asking
- Define variables: Assign letters to unknown quantities
- Translate: Convert words into mathematical expressions
- Solve: Use algebraic techniques to find the answer
- Check: Verify your answer makes sense in context
Common Word Problem Phrases
| Phrase | Mathematical Operation |
|---|---|
| "product of" | Multiplication |
| "the square of" | Raise to power 2 |
| "increased by" | Addition |
| "decreased by" | Subtraction |
| "the square root of" | Radical (sqrt) |
Worked Examples
Study these examples to see how polynomials and radicals apply to real situations.
Example 1: Area Problem
Problem: A rectangular garden has a length that is 5 feet more than twice its width. If the width is w feet, write an expression for the area.
Step 1: Define the dimensions
Width = w feet
Length = 2w + 5 feet
Step 2: Write the area formula
Area = Length x Width = w(2w + 5)
Step 3: Expand
Area = 2w2 + 5w square feet
Example 2: Distance Formula Application
Problem: Find the distance between points (1, 2) and (4, 6) using the distance formula d = sqrt((x2-x1)2 + (y2-y1)2).
Step 1: Identify coordinates
(x1, y1) = (1, 2) and (x2, y2) = (4, 6)
Step 2: Substitute into formula
d = sqrt((4-1)2 + (6-2)2)
d = sqrt(32 + 42)
d = sqrt(9 + 16)
d = sqrt(25) = 5 units
Example 3: Profit Function
Problem: A company's revenue is R(x) = 50x - x2 and cost is C(x) = 10x + 100, where x is the number of items. Find the profit function.
Step 1: Recall Profit = Revenue - Cost
P(x) = R(x) - C(x)
Step 2: Substitute and simplify
P(x) = (50x - x2) - (10x + 100)
P(x) = 50x - x2 - 10x - 100
P(x) = -x2 + 40x - 100
Practice Problems
Solve these 10 word problems using polynomials and radicals.
Problem 1
A square has a side length of (x + 3) units. Write an expression for its area.
Show Hint
Area of a square = side2. Expand (x + 3)2.
Show Answer
Area = x2 + 6x + 9 square units
Problem 2
A picture frame has outer dimensions of (x + 4) by (x + 2). The picture inside has dimensions of x by (x - 2). Write an expression for the area of the frame only.
Show Hint
Frame area = Outer area - Inner area
Show Answer
Outer: (x+4)(x+2) = x2 + 6x + 8
Inner: x(x-2) = x2 - 2x
Frame: 8x + 8 square units
Problem 3
Find the distance between points (-2, 3) and (4, -1).
Show Hint
Use d = sqrt((x2-x1)2 + (y2-y1)2)
Show Answer
d = sqrt((4-(-2))2 + (-1-3)2) = sqrt(36 + 16) = sqrt(52) = 2sqrt(13) units
Problem 4
The height of a ball thrown upward is given by h(t) = -16t2 + 48t + 5, where h is in feet and t is in seconds. What is the height at t = 2 seconds?
Show Hint
Substitute t = 2 into the polynomial and simplify.
Show Answer
h(2) = -16(4) + 48(2) + 5 = -64 + 96 + 5 = 37 feet
Problem 5
A rectangular pool is 3 feet longer than it is wide. A walkway of uniform width x surrounds it. If the pool is 10 feet wide, write a polynomial for the total area including the walkway.
Show Hint
Pool: 10 by 13. With walkway: (10 + 2x) by (13 + 2x)
Show Answer
Total area = (10 + 2x)(13 + 2x) = 4x2 + 46x + 130 square feet
Problem 6
The diagonal of a rectangle can be found using d = sqrt(l2 + w2). Find the diagonal of a rectangle with length 8 cm and width 6 cm.
Show Hint
Substitute l = 8 and w = 6, then simplify the radical.
Show Answer
d = sqrt(64 + 36) = sqrt(100) = 10 cm
Problem 7
A company's weekly profit P in dollars is modeled by P(x) = -2x2 + 120x - 800, where x is the price per item. What is the profit when x = 25?
Show Hint
Substitute x = 25 and calculate step by step.
Show Answer
P(25) = -2(625) + 120(25) - 800 = -1250 + 3000 - 800 = $950
Problem 8
The period T of a pendulum is given by T = 2pi * sqrt(L/g), where L is length in meters and g = 10 m/s2. If L = 40 meters, express T in simplified radical form (use pi in your answer).
Show Hint
T = 2pi * sqrt(40/10) = 2pi * sqrt(4)
Show Answer
T = 2pi * sqrt(4) = 2pi * 2 = 4pi seconds
Problem 9
Two numbers have a sum of 20. If one number is x, express the product of the two numbers as a polynomial.
Show Hint
If one number is x, the other is (20 - x). Product = x(20 - x).
Show Answer
Product = x(20 - x) = 20x - x2 or -x2 + 20x
Problem 10
A ladder leans against a wall. If the foot of the ladder is 5 feet from the wall and the ladder is 13 feet long, how high up the wall does the ladder reach?
Show Hint
Use the Pythagorean theorem: 52 + h2 = 132
Show Answer
h2 = 169 - 25 = 144, so h = sqrt(144) = 12 feet
Check Your Understanding
Reflect on these questions to solidify your word problem skills.
1. What is the first step you should take when solving a word problem?
Show Answer
Read carefully and identify what the problem is asking for.
2. When does the distance formula involve simplifying a radical?
Show Answer
When the sum under the radical is not a perfect square.
Next Steps
- Practice translating more word problems into algebraic expressions
- Review problems you found challenging
- Continue to Common Mistakes to avoid typical errors