Posted: February 13th, 2023

A friend has given you left-over landscaping bricks. You decide to make a garden bed and surround it with the bricks. There are 62 bricks, and each brick is 8 inches long. You would like the garden bed to be slightly more than twice as long as it is wide, as shown in the diagram below. You have also given yourself a budget of $125 for additional materials should you need them. Your local home improvement store sells the same bricks for $1.98 per brick. The displayed sides present the number of bricks on each side, where *x* is a number of bricks.

Show and explain all steps in your responses to the following parts of the assignment using the Algebra concepts discussed within the course. All mathematical steps and explanations must be typed up and formatted using the equation editor.

**Part 1: **Write an equation representing the perimeter of the garden bed.

**Part 2: **Calculate how many bricks are used on each side.

**Part 3:** Determine the length of each side.

**Part 4: **Write an inequality that represents how many bricks can be purchased within your budget.

**Part 5: **Will you be able to make another complete layer of bricks on top and stay within your budget?

SOLUTION

Here’s one way to approach the problem:

- First, let’s determine the width of the garden bed. If the garden bed is x inches wide, then it must be 2x + 8 inches long.
- The total number of inches of brick needed is given by the formula: 2(8x + 8x + 8) + 2x(2x + 8) = 32x + 64 + 4x^2 + 16x.
- We know that the number of inches of brick is equal to 8 times the number of bricks, so we can write the equation: 32x + 64 + 4x^2 + 16x = 8 * 62 = 496.
- We can now solve for x by solving the equation obtained in step 3. We get: 4x^2 + 48x – 432 = 0.
- We can use the quadratic formula to find x: x = (-b ± √(b^2 – 4ac)) / 2a, where a = 4, b = 48, and c = -432.
- Plugging in the values for a, b, and c, we get: x = (-48 ± √(48^2 – 4 * 4 * -432)) / 2 * 4 = (-48 ± √(2304 + 5376)) / 8 = (-48 ± √(7680)) / 8 = (-48 ± 88) / 8 = (-48 + 88) / 8 = 40 / 8 = 5.
- So the width of the garden bed is 5 inches, and the length is 2 * 5 + 8 = 18 inches.
- The total cost of the bricks is 62 * $1.98 = $123.56, which is within the budget of $125.

So, the final answer is that the width of the garden bed is 5 inches, the length is 18 inches, and the total cost of the bricks is $123.56.

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