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. Would 5 trees be a solution to the inequality in part b? Justify your answer.
r = tons of rock
t = number of trees
r≥30 to cover yard
c = 60 r + 84 t ≤2500
if I use the minimum amount of rock (30 tons) then I can use the maximum number of trees so
r = 30
60 * 30 + 84 t ≤ 2500
1800 + 84 t ≤ 2500
84 t ≤ 700
t ≤ 8
so 5 trees is a solution
To solve this problem, we can set up an inequality to represent the given situation. Let's assume the maximum number of trees we can buy is represented by the variable 't', and the cost of the rock is 30 tons multiplied by $60, which gives us $1800.
The total cost of the trees will be 't' trees multiplied by $84, which gives us $84t.
So, the total cost of both rock and trees must not exceed the budget of $2,500, which can be expressed as:
1800 + 84t ≤ 2500
Now, let's solve this inequality to find the maximum number of trees 't' we can buy.
1800 + 84t ≤ 2500
First, subtract 1800 from both sides:
84t ≤ 2500 – 1800
84t ≤ 700
Now, divide both sides by 84:
t ≤ 700/84
Simplifying further:
t ≤ 8.33 (rounded to two decimal places)
Since the number of trees cannot be a decimal, we have to round down to the nearest whole number. Therefore, the maximum number of trees we can buy is 8.
Now, to determine if 5 trees would be a solution to the inequality, we substitute 't' with 5 in the inequality:
1800 + 84(5) ≤ 2500
1800 + 420 ≤ 2500
2220 ≤ 2500
Since 2220 is less than or equal to 2500, 5 trees would indeed be a solution to the inequality.