A city park planner is deciding between two proposals for a new rectangular playground. One is for a playground with an area of 1930 square feet.
A) The second proposal is for a playground that is 11.5 meters long and 19.5 meters wide. What is the area of this playground? Use 1m = 3.28 ft. Use the correct number of significant digits.
B) The playground will be covered with rubber mulch. Let c represent the cost in dollars to cover 1 square foot with rubber mulch. Write and simplify an algebraic expression for the amount the city will save on mulch by using the first proposal over the second.
C) Rubber mulch costs between $11 and $17 per square foot. Will the city save at least $3000 by using the first proposal? Explain
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A) To find the area of the second proposal, we need to multiply its length by its width. Given that the length is 11.5 meters and the width is 19.5 meters, the area can be calculated as follows:
Area = Length x Width
Area = 11.5 m x 19.5 m
However, we need to convert the area to square feet as the first proposal is given in square feet. Let's use the conversion factor provided: 1m = 3.28ft. To convert meters to feet, we multiply by the conversion factor.
Area = (11.5 m x 3.28 ft/m) x (19.5 m x 3.28 ft/m)
Area ≈ 37.72 ft x 63.96 ft
Area ≈ 2416.5792 ft²
Since we are asked for the correct number of significant digits, we can round the area to four significant digits:
Area ≈ 2416 ft²
Therefore, the area of the second proposal is 2416 square feet.
B) The amount the city will save on mulch by using the first proposal over the second can be calculated by subtracting the cost of mulching the second proposal from the cost of mulching the first proposal.
Let's assume that the cost of covering 1 square foot with rubber mulch is represented by c dollars. The cost of covering the first proposal (with an area of 1930 ft²) will be:
Cost of covering first proposal = c x 1930
The cost of covering the second proposal (with an area of 2416 ft²) will be:
Cost of covering second proposal = c x 2416
To find the amount the city will save on mulch, we subtract the cost of the second proposal from the cost of the first proposal:
City's savings = Cost of covering first proposal - Cost of covering second proposal
City's savings = (c x 1930) - (c x 2416)
City's savings = c(1930 - 2416)
City's savings = c(-486)
Therefore, the algebraic expression for the amount the city will save on mulch by using the first proposal over the second is -486c dollars.
C) To determine if the city will save at least $3000 by using the first proposal, we need to compare the savings calculated in part B to $3000.
We have the expression for the city's savings: -486c dollars. Since we want to determine if the savings are at least $3000, we can write the inequality:
-486c ≥ $3000
To solve for c, divide both sides of the inequality by -486:
c ≤ $3000 / -486
Using a calculator to evaluate the right side of the inequality, we find:
c ≤ -6.17
Since the cost of mulch cannot be negative, it means that the city cannot save at least $3000 by using the first proposal.
In conclusion, the city will not save at least $3000 by using the first proposal.