College Algebra Word Problems: Step-by-Step Guidance

Q1. A rectangular piece of metal is 5 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. Which equation indicates that the volume of the box is 74 in³?

Background

Topic: Quadratic Equations and Volume of Rectangular Prisms

This question tests your ability to model a geometric situation with a quadratic equation and relate it to the volume of a box.

Key Terms and Formulas

• Volume of a box: $V = l \times w \times h$

• Let $x$ be the width of the original piece of metal. Then the length is $x + 5$.

• After cutting 2 in. squares from each corner, the new dimensions are $(x - 4)$ and $(x + 1)$, with height 2 in.

Step-by-Step Guidance

1. Let $x$ be the width of the original piece of metal. The length is $x + 5$.

2. After cutting 2 in. squares from each corner, the new width is $x - 4$ and the new length is $(x + 5) - 4 = x + 1$.

3. The height of the box is 2 in. (the side of the square cut out).

4. Set up the equation for the volume: $2(x + 1)(x - 4) = 74$.

Try solving on your own before revealing the answer!

Final Answer: $2(x + 1)(x - 4) = 74$

This equation models the volume of the open box after the corners are cut and the sides are folded up.

Q2. The product of two consecutive even integers is 440. Find the integers.

Background

Topic: Quadratic Equations and Integer Properties

This question tests your ability to set up and solve a quadratic equation involving consecutive even integers.

Key Terms and Formulas

• Let $x$ be the first even integer. The next consecutive even integer is $x + 2$.

• The product is $x(x + 2)$.

Step-by-Step Guidance

1. Let $x$ be the first even integer. The next is $x + 2$.

2. Set up the equation: $x(x + 2) = 440$.

3. Expand and rearrange: $x^2 + 2x - 440 = 0$.

4. Solve this quadratic equation for $x$ to find the integers.

Try solving on your own before revealing the answer!

Final Answer: 20, 22 and -20, -22

Both positive and negative pairs of consecutive even integers satisfy the equation.

Q3. The sum of the squares of two consecutive integers is 61. Find the integers.

Background

Topic: Quadratic Equations and Integer Properties

This question tests your ability to model a situation with consecutive integers and their squares.

Key Terms and Formulas

• Let $x$ be the first integer. The next is $x + 1$.

• The sum of their squares: $x^2 + (x + 1)^2$.

Step-by-Step Guidance

1. Let $x$ be the first integer. The next is $x + 1$.

2. Set up the equation: $x^2 + (x + 1)^2 = 61$.

3. Expand: $x^2 + x^2 + 2x + 1 = 61$.

4. Combine like terms and rearrange to form a quadratic equation.

Try solving on your own before revealing the answer!

Final Answer: 5, 6 and -5, -6

Both positive and negative pairs of consecutive integers satisfy the equation.

Q4. The difference of the squares of two positive consecutive even integers is 84. Find the integers.

Background

Topic: Quadratic Equations and Integer Properties

This question tests your ability to use the difference of squares formula with consecutive even integers.

Key Terms and Formulas

• Let $x$ be the first even integer. The next is $x + 2$.

• Difference of squares: $(x + 2)^2 - x^2$.

Step-by-Step Guidance

1. Let $x$ be the first even integer. The next is $x + 2$.

2. Set up the equation: $(x + 2)^2 - x^2 = 84$.

3. Expand: $x^2 + 4x + 4 - x^2 = 84$.

4. Simplify and solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: 20, 22

These are the two positive consecutive even integers whose squares differ by 84.

Q5. The length of each side of a square is 3 in. more than the length of each side of a smaller square. The sum of the areas of the squares is 117 in². Find the lengths of the sides of the two squares.

Background

Topic: Quadratic Equations and Area

This question tests your ability to model a geometric relationship with a quadratic equation.

Key Terms and Formulas

• Let $x$ be the side of the smaller square. The larger square has side $x + 3$.

• Area of a square: $A = s^2$.

Step-by-Step Guidance

1. Let $x$ be the side of the smaller square. The larger square has side $x + 3$.

2. Set up the equation: $x^2 + (x + 3)^2 = 117$.

3. Expand: $x^2 + x^2 + 6x + 9 = 117$.

4. Combine like terms and solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: 6 in (small), 9 in (large)

The sides of the squares are 6 inches and 9 inches, respectively.

Q6. Cynthia Besch wants to buy a rug for a room that is 19 ft wide and 34 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 450 square feet of carpeting. What dimensions should the rug have?

Background

Topic: Quadratic Equations and Area of Rectangles

This question tests your ability to model a geometric situation with a quadratic equation and solve for the dimensions of a rectangle.

Key Terms and Formulas

• Let $x$ be the width of the uniform strip. The rug's dimensions are $(19 - 2x)$ by $(34 - 2x)$.

• Area of a rectangle: $A = l \times w$.

Step-by-Step Guidance

1. Let $x$ be the width of the uniform strip. The rug's dimensions are $(19 - 2x)$ by $(34 - 2x)$.

2. Set up the equation: $(19 - 2x)(34 - 2x) = 450$.

3. Expand and rearrange to form a quadratic equation in $x$.

4. Solve for $x$, then substitute back to find the rug's dimensions.

Try solving on your own before revealing the answer!

Final Answer: 15 ft by 30 ft

These are the dimensions of the rug that will leave a uniform strip and use exactly 450 square feet of carpeting.