A Friend Has Given You Leftover Landscaping Bricks
A Friend Has Given You Left Over Landscaping Bricks
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, each measuring 8 inches long. The garden bed should be slightly more than twice as long as it is wide. You have a budget of $125 for additional materials if needed. Bricks cost $1.98 each. Using algebra, you will determine the dimensions of the garden bed, the number of bricks used on each side, the lengths of each side, and whether you can construct an additional layer within your budget.
Paper For Above instruction
Part 1: Writing an equation for the perimeter of the garden bed
Let w represent the width of the garden bed in terms of the number of bricks, and l represent the length in bricks. Since the bricks are each 8 inches long, the actual physical dimensions will be based on these variables. The problem states that the length should be slightly more than twice the width, which can be expressed as:
l = 2w + k
where k is a small positive value (e.g., 1 or 2 bricks) to make the length more than twice the width.
The perimeter (P) of the rectangular garden bed in terms of bricks is given by:
P = 2(l + w)
Substituting l from the previous expression:
P = 2((2w + k) + w) = 2(3w + k) = 6w + 2k
This equation represents the total number of bricks used along the perimeter, considering the length and width in bricks.
Part 2: Calculating how many bricks are used on each side
Suppose the total number of bricks used is less than or equal to 62. Using the perimeter equation, the number of bricks on each side is as follows:
- Width side: w bricks
- Length side: l = 2w + k
The total bricks along the perimeter are:
Total bricks = 2(w + l) = 2(w + 2w + k) = 2(3w + k) = 6w + 2k
Given that there are 62 bricks available, we set:
6w + 2k ≤ 62
Part 3: Determining the length of each side
Assuming the smallest positive value of k (say, 1 brick) to make the length slightly more than twice the width:
k = 1
From the inequality:
6w + 2(1) ≤ 62
6w ≤ 60
w ≤ 10
Therefore, the maximum width in bricks is 10 bricks. Correspondingly, the length in bricks is:
l = 2w + k = 2(10) + 1 = 21 bricks
In inches, the length and width are:
- Width: 10 bricks × 8 inches = 80 inches
- Length: 21 bricks × 8 inches = 168 inches
Part 4: Writing an inequality that represents how many bricks can be purchased within your budget
The cost of each brick is $1.98. The total cost for n bricks is:
Cost = 1.98n
To stay within the budget of $125:
1.98n ≤ 125
n ≤ \(\frac{125}{1.98}\)
Calculating:
n ≤ 63.13
Since only whole bricks can be purchased, the maximum number of bricks purchasable is 63.
Because we only have 62 bricks, we are within the budget, and the current arrangement is feasible.
Part 5: Can you make another complete layer of bricks on top and stay within your budget?
Constructing an additional layer involves adding another 'border' of bricks around the existing perimeter. The total number of bricks needed increases accordingly, with each new layer contributing additional bricks equal to the perimeter of the new, larger rectangle.
The perimeter increases by 2 bricks on each side for each new layer. Specifically, the number of bricks for the second layer would be:
Total bricks for second layer = (perimeter in bricks) + 8
This pattern continues with each new layer requiring additional bricks proportional to the rectangular perimeter. The total bricks required for one additional layer can be expressed as:
Bricks for second layer = 2(l + w) + 8
If the initial perimeter uses 6w + 2k bricks, then the second layer adds an additional 8 bricks, making the total:
Total bricks after second layer = 6w + 2k + 8
Cost-wise, adding another layer will add $1.98 × 8 = $15.84 to the total cost. Since the initial bracelet budget is $125, and the current number of bricks (62) costs:
Cost for current bricks: 62 × 1.98 = $122.76
The additional layer costing $15.84 would push the total well above the budget, totaling:
$122.76 + $15.84 = $138.60, which exceeds the $125 budget.
Therefore, proceeding to add another complete layer would not be feasible within the current budget constraints.
In conclusion, with 62 bricks and a budget of $125, it is possible to construct the first layer of the garden bed with dimensions approximately 80 inches wide and 168 inches long. However, adding a second complete layer would exceed the budget.
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