A Landscaper Who Just Completed A Rectangular Flower Garden ✓ Solved

Alandscaperwho Just Completed A Rectangular Flower Garden Measuring

1. A landscaper, who just completed a rectangular flower garden measuring 10 feet in length by 8 feet in width, orders 1 cubic yard of premixed cement, all of which is to be used to create a border of uniform width around the garden. If the border is to have a depth of 3 inches, how wide will the border be? (1 cubic yard = 27 cubic feet) The width of the border is .........feet

2. Complete the square in the expression. Then factor the perfect square trinomial. x² + 2x

3. Express that x differs from -2 by more than 4 as an inequality involving an absolute value. Solve for x. Write the inequality expressing the given facts.

Sample Paper For Above instruction

The problem involves multiple mathematical concepts including volume calculation, quadratic completion, factoring, and inequalities involving absolute value. Each part will be addressed thoroughly to demonstrate understanding and application of these concepts.

Part 1: Determining the Border Width Around a Rectangular Garden

The initial dimensions of the garden are 10 feet in length and 8 feet in width. A border of uniform width (say x feet) surrounds the garden. The total dimensions including the border will become (10 + 2x) feet in length and (8 + 2x) feet in width because the border adds x feet on each side.

Given the depth of the border is 3 inches, which converts to 0.25 feet (since 12 inches = 1 foot), the volume of cement used will create a border with a specific cross-sectional volume. The volume of cement is provided as 27 cubic feet (since 1 cubic yard = 27 cubic feet).

The volume of the border is obtained by calculating the volume of the larger rectangle excluding the original garden volume. The volume V of the border is approximately:

V = Area of border region * depth

The area of the entire rectangle (garden + border) is (10 + 2x)(8 + 2x). The area of the original garden is 10 * 8 = 80 sq ft. The area of the border region alone is:

Area_border = (10 + 2x)(8 + 2x) - 80

Therefore, the volume of the border cement is:

V = [ (10 + 2x)(8 + 2x) - 80 ] * 0.25

We set this equal to 27 cubic feet and solve for x:

0.25 * [ (10 + 2x)(8 + 2x) - 80 ] = 27

Dividing both sides by 0.25:

(10 + 2x)(8 + 2x) - 80 = 108

Expanding the product:

(10)(8) + (10)(2x) + (8)(2x) + (2x)(2x) - 80 = 108

80 + 20x + 16x + 4x² - 80 = 108

Simplify:

4x² + 36x = 108

Rewrite as a quadratic equation:

4x² + 36x - 108 = 0

Divide the entire equation by 4 to simplify:

x² + 9x - 27 = 0

Use the quadratic formula x = [-b ± √(b² - 4ac)] / 2a, where a=1, b=9, c=-27:

x = [-9 ± √(81 - 41(-27))] / 2

x = [-9 ± √(81 + 108)] / 2

x = [-9 ± √189] / 2

The square root of 189 can be simplified to √189 = √9 * √21 = 3√21:

x = [-9 ± 3√21] / 2

Thus, the possible values for x are:

x = (-9 + 3√21) / 2 or x = (-9 - 3√21) / 2

Since width cannot be negative, only the positive solution is valid in the context of border width:

x ≈ (-9 + 3 * 4.583) / 2 ≈ (-9 + 13.75) / 2 ≈ 4.75 / 2 ≈ 2.375 feet

Therefore, the width of the border is approximately 2.375 feet.

Part 2: Completing the Square in a Quadratic Expression

The quadratic expression given is x² + 2x.

To complete the square, take half of the coefficient of x (which is 2), square it, and add it inside the expression:

(x + 1)² = x² + 2x + 1

Therefore, x² + 2x can be written as:

(x + 1)² - 1

Part 3: Expressing and Solving an Absolute Value Inequality

The problem states that x differs from -2 by more than 4, which can be mathematically stated as:

|x + 2| > 4

To solve this inequality, we consider the definition of absolute value:

|x + 2| > 4 if and only if x + 2 > 4 or x + 2

Solving these inequalities separately:

x + 2 > 4 => x > 2

x + 2 x

Therefore, the solution set is:

x 2

This indicates that x must lie outside the interval [-6, 2] for the inequality to hold true.

In conclusion, understanding the geometric context of volume calculation allows precise determination of border width, and algebraic techniques such as completing the square and solving inequalities deepen comprehension of quadratic and absolute value expressions. These methods are foundational in mathematical problem-solving, applicable in diverse real-world scenarios involving measurement, optimization, and logical analysis.

References

• Larson, R., & Hostetler, R. (2017). Algebra and Trigonometry (12th ed.). Cengage Learning.
• Blitzer, R. (2015). Algebra and Trigonometry. Pearson.
• Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals. Wiley.
• Edwards, C. H., & Penney, K. A. (2015). Calculus and Analytic Geometry. Pearson.
• Devlin, K. (2011). The Math Gene: How Mathematical Thinking Evolved and Why Children Are Gifted. Basic Books.
• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
• Polya, G. (1957). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.
• Krug, R. (2012). Mathematical Methods for the Physical Sciences. Princeton University Press.
• Stillwell, J. (2010). Mathematics and Its History. Springer.
• Knuth, D. E. (1997). The Art of Computer Programming. Addison-Wesley.