The community raised twice the amount of money needed to buy a piece of land for the playground, with a given area. Jon and his dad decided to multiply the length and the width of the planned playground by 2. If the cost of the land is proportional to the area, will there be enough money to cover this change?
To determine if there will be enough money to cover the change in the land area, we need to compare the original cost of the land to the new cost after multiplying the length and width by 2.
Let's assume the original length of the planned playground is L and the width is W. The original area of the land is A = L * W.
The new length after multiplying by 2 is 2L, and the new width is 2W. The new area of the land is A' = (2L) * (2W) = 4 * (L * W).
Given that the cost of the land is proportional to the area, we can assume that the ratio of the original cost to the new cost is the same as the ratio of the original area to the new area.
Let C be the original cost of the land, and C' be the new cost of the land. The ratio of the original cost to the new cost is:
C / C' = A / A'
Substituting the values:
C / C' = (L * W) / (4 * (L * W))
The length and width cancel out, leaving us with:
C / C' = 1 / 4
This means that the new cost C' is 1/4 of the original cost C.
Since the community raised twice the amount of money needed, and the new cost is only 1/4 of the original cost, there will still be enough money to cover the change in the land area.
To determine whether there will be enough money to cover the change in the area, we need to understand the relationship between the cost of the land and the area.
Let's first denote the original length and width of the planned playground as L and W, respectively. The original area of the playground is then A = L * W.
According to the given information, the community raised twice the amount of money needed to buy the land for the playground, with this specific area A.
Now, Jon and his dad decided to multiply the length and width by 2. Therefore, the new length is 2L and the new width is 2W. The new area of the playground is A' = (2L) * (2W) = 4 * A.
To see whether there will be enough money to cover this change, we need to compare the cost of the land for the original area (A) with the cost of the land for the new area (A').
Since the cost of the land is directly proportional to the area, we can set up the following proportion:
Original Cost / Original Area = New Cost / New Area
Let's assume the original cost of the land is C.
Then, we have: C / A = New Cost / 4A
To find out if there will be enough money to cover the change, we need to compare C / A with New Cost / 4A.
If C / A is less than New Cost / 4A, it means there will not be enough money to cover the change. On the other hand, if C / A is greater than or equal to New Cost / 4A, it means there will be enough money to cover the change.
To determine which scenario applies, you need to know the values of the original cost and the new cost of the land. With this information, you can plug the values into the proportion and compare the two sides to determine the outcome.
no, since the area is increased by a factor of 4.
The original area is A = LW
The new area is (2L)(2W) = 4LW = 4A