A city park planner is deciding between two proposals for a new rectangular playground. One proposal 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.
You were off by 3 and it ruined the whole response
a. To find the area of the second proposal playground, we can use the formula:
Area = Length × Width
Given that the length of the playground is 11.5 meters and the width is 19.5 meters, we can calculate the area as follows:
Area = 11.5 m × 19.5 m
= 224.25 m²
Now, to convert this area from square meters to square feet, we can use the conversion factor of 1m = 3.28ft:
Area = 224.25 m² × (3.28 ft/m)²
≈ 224.25 m² × 10.76 ft²/m²
≈ 2415.39 ft²
So, the area of the second proposal playground is approximately 2415.39 square feet.
b. To calculate the amount the city will save on mulch by using the first proposal over the second, we need to find the difference in areas between the two proposals.
Let A₁ be the area of the first proposal (1930 ft²) and A₂ be the area of the second proposal (2415.39 ft²). The algebraic expression for the amount the city will save on mulch is:
Savings = A₂ - A₁
= 2415.39 ft² - 1930 ft²
= 485.39 ft²
So, the city will save 485.39 square feet of rubber mulch by using the first proposal over the second.
c. To determine if the city will save at least $3000 by using the first proposal, we need to calculate the cost of rubber mulch for each proposal.
Assuming c represents the cost in dollars to cover 1 square foot with rubber mulch, the cost for the first proposal would be:
Cost₁ = 1930 ft² × c
To find the cost range for the rubber mulch, we can use the given range of $11 to $17 per square foot.
Lowest cost₁ = 1930 ft² × $11/ft²
Highest cost₁ = 1930 ft² × $17/ft²
Since the cost range for the first proposal is not known, we cannot determine if the city will save at least $3000 by using it.
a. To find the area of the second playground, we can use the formula for the area of a rectangle, which is length multiplied by width. Given that the length of the second playground is 11.5 meters and the width is 19.5 meters, we can calculate the area as follows:
Area = length × width
= 11.5 m × 19.5 m
However, we need to convert the measurements from meters to feet since the area of the first playground is given in square feet. Given that 1 meter is equal to 3.28 feet, we can convert the measurements as follows:
Length in feet = 11.5 m × 3.28 ft/m
Width in feet = 19.5 m × 3.28 ft/m
Now we can calculate the area in square feet:
Area = Length in feet × Width in feet
b. The amount the city will save on mulch by using the first proposal over the second can be calculated by finding the difference in area between the two proposals and then multiplying it by the cost per square foot of rubber mulch.
The algebraic expression for the amount the city will save is:
Amount saved = (Area of the first proposal - Area of the second proposal) × cost per square foot of rubber mulch
c. To determine if the city will save at least $3000 by using the first proposal, we need to compare the amount saved (from part b) to $3000. If the amount saved is equal to or greater than $3000, then the city will save at least $3000.
a. Area = L*W = (11.5*3.28) * (19.5*3.28) = 2413 sq. Ft.
b. Amt. = 2413C - 1930C = 483C.
c. 483C =< 3000.
C =< $6.21/sq. ft.
To save $3,000, the cost per sq. ft. must be 6.21 max.