1. You are planning to spend no less than $6,000 and no more than $10,000 on your landscaping project.
a) Write an inequality that demonstrates how much money you will be willing to spend on the project.
b) Suppose you want to cover the backyard with decorative rock and plant some trees as the first phase of the project. You need 30 tons of rock to cover the area. If each ton cost $60 and each tree is $84, what is the maximum number of trees you can buy with a budget for rock and trees of $2,500? Write an inequality that illustrates the problem and solve. Express your answer as an inequality and explain how you arrived at your answer.
c) Would 5 trees be a solution to the inequality in part b? Justify your answer.
a) The inequality that demonstrates how much money you will be willing to spend on the project is:
$6,000 ≤ x ≤ $10,000
b) Let's define the variables:
x = number of trees
y = cost of decorative rock
The cost of 30 tons of rock will be 30 * $60 = $1800.
To find the maximum number of trees you can buy within a budget of $2,500, we need to subtract the cost of rock from the budget and divide the remaining amount by the cost of each tree:
$2,500 - $1800 = $700 (remaining budget for trees)
$700 ÷ $84 (cost of each tree) = 8.333 (approx)
Since you cannot buy a fraction of a tree, the maximum number of trees you can buy is 8.
The inequality that illustrates the problem is:
84x + 1800 ≤ 2500
c) To check if 5 trees are a solution to the inequality in part b, substitute x = 5 into the inequality and check if the equation holds true.
84(5) + 1800 ≤ 2500
420 + 1800 ≤ 2500
2220 ≤ 2500
The inequality is true, so 5 trees are a valid solution.
a) The inequality that demonstrates how much money you will be willing to spend on the project is:
$6,000 ≤ x ≤ $10,000
where x represents the amount of money you will spend on the landscaping project.
b) To determine the maximum number of trees you can buy with a budget for rock and trees of $2,500, we need to consider the cost of the rock and the cost of the trees. Let's assume the number of trees you can buy is represented by t.
The cost of 30 tons of rock is 30 * $60 = $1800.
The cost of t trees is t * $84.
We are given that the total budget is $2,500, so we can write the inequality:
30 * $60 + t * $84 ≤ $2,500
Simplifying the inequality, we get:
$1800 + $84t ≤ $2,500
Next, we can solve for t by subtracting $1800 from both sides:
$84t ≤ $700
Dividing both sides by $84:
t ≤ 8.3333...
Since t represents the number of trees, it must be a whole number, so the maximum number of trees you can buy with a budget for rock and trees of $2,500 is 8.
c) To determine whether 5 trees would be a solution to the inequality from part b, we substitute t = 5 into the inequality:
30 * $60 + 5 * $84 ≤ $2,500
Simplifying the inequality, we get:
$1800 + $420 ≤ $2,500
$2,220 ≤ $2,500
Since $2,220 is less than or equal to $2,500, 5 trees would indeed be a solution to the inequality.