John is going to build a rectangular plot and he needs go split the rectangle into two equal areas.
The dimensions will be 140 feet by 171 feet.
John sketched three different options that he can choose from. Find the cost for each choice if each linear foot costs $10.75 (three feet of fence would cost $32.25)
Choice A is split across.
Choice B is split down the middle.
Choice C is split diagonally.
Find each cost and the amount of money saved if John uses the lowest costing choice.
First, let's find the cost for each choice.
Choice A: If the rectangle is split across, it would require two fences along the width, each measuring 140 feet, and one fence along the length, measuring 171 feet. The total cost for the fences would be 2 * 140 * 10.75 + 171 * 10.75 = $5,392.50.
Choice B: If the rectangle is split down the middle, it would require one fence along the width, measuring 140 feet, and two fences along the length, each measuring 85.5 feet. The total cost for the fences would be 140 * 10.75 + 2 * 85.5 * 10.75 = $2,837.50.
Choice C: If the rectangle is split diagonally, it would require four fence segments. Two segments along the width would measure 140 feet each, and two segments along the length would measure 171 feet each. The total cost for the fences would be 2 * 140 * 10.75 + 2 * 171 * 10.75 = $6,089.50.
Now let's find the amount of money saved if John uses the lowest costing choice.
The lowest cost among the three choices is $2,837.50 for Choice B. The amount of money saved compared to Choice A would be $5,392.50 - $2,837.50 = $2,555.
Similarly, the amount of money saved compared to Choice C would be $6,089.50 - $2,837.50 = $3,252.
Therefore, if John uses the lowest costing choice (Choice B), he would save $2,555 compared to Choice A and $3,252 compared to Choice C.
All of your answers are wrong though
Apologies for the incorrect calculations. Let's re-calculate the costs for each choice.
Choice A: The rectangle is split across, requiring two fences along the width, each measuring 140 feet, and one fence along the length, measuring 171 feet. The total cost for the fences would be (2 * 140 + 171) * 10.75 = $4,665.25.
Choice B: The rectangle is split down the middle, requiring one fence along the width, measuring 140 feet, and two fences along the length, each measuring 85.5 feet. The total cost for the fences would be (140 + 2 * 85.5) * 10.75 = $2,821.25.
Choice C: The rectangle is split diagonally, requiring four fence segments. Two segments along the width would measure 140 feet each, and two segments along the length would measure 171 feet each. The total cost for the fences would be (2 * 140 + 2 * 171) * 10.75 = $5,972.50.
Now let's find the amount of money saved if John uses the lowest costing choice.
The lowest cost among the three choices is $2,821.25 for Choice B. The amount of money saved compared to Choice A would be $4,665.25 - $2,821.25 = $1,844.
Similarly, the amount of money saved compared to Choice C would be $5,972.50 - $2,821.25 = $3,151.25.
Therefore, if John uses the lowest costing choice (Choice B), he would save $1,844 compared to Choice A and $3,151.25 compared to Choice C.
To find the cost for each choice, we need to determine the length of the fence needed for each option.
Choice A: Split Across
In this option, the rectangle will be split horizontally into two equal areas. To find the length of the fence needed, we need to add up the length of all four sides of each equal area since all four sides will require a fence.
Each equal area will have dimensions 140 feet by 85.5 feet (half of the total length, 171 feet).
The length of the fence for one equal area would be:
Sides 1 and 3: 140 feet each
Sides 2 and 4: 85.5 feet each
Therefore, the total length of the fence for both equal areas is:
2 * (140 + 140 + 85.5 + 85.5) = 852 feet
The cost for Choice A is:
852 feet * $10.75/foot = $9,153
Choice B: Split Down the Middle
In this option, the rectangle will be split vertically into two equal areas. Similar to Choice A, we need to find the length of the fence needed for both equal areas.
Each equal area will have dimensions 70 feet by 171 feet (half of the total width, 140 feet).
The length of the fence for one equal area would be:
Sides 1 and 3: 70 feet each
Sides 2 and 4: 171 feet each
Therefore, the total length of the fence for both equal areas is:
2 * (70 + 70 + 171 + 171) = 682 feet
The cost for Choice B is:
682 feet * $10.75/foot = $7,331.50
Choice C: Split Diagonally
In this option, the rectangle will be split diagonally into two equal triangular areas. To find the length of the fence needed, we need to calculate the perimeter of each triangular area.
Each triangular area can be considered as two right-angled triangles with legs of 140 feet and 85.5 feet (half of the total dimensions).
Using the Pythagorean theorem, we can find the hypotenuse (the diagonal) of each triangle:
C^2 = A^2 + B^2
C^2 = 140^2 + 85.5^2
C = √(140^2 + 85.5^2)
C ≈ 163.69 feet
The perimeter of each triangular area is the sum of all three sides:
Perimeter = 140 + 85.5 + 163.69
Perimeter ≈ 389.19 feet
Therefore, the total length of the fence for both triangular areas is:
2 * Perimeter = 2 * 389.19 = 778.38 feet
The cost for Choice C is:
778.38 feet * $10.75/foot = $8,364.21
To find the amount of money saved if John uses the lowest costing choice, we compare the costs of each option. The lowest cost is for Choice B, which is $7,331.50.
The amount of money saved if John chooses the lowest costing option is:
$9,153 (Choice A cost) - $7,331.50 (Choice B cost) = $1,821.50.
Therefore, John would save $1,821.50 if he chooses the lowest costing option, which is Choice B (split down the middle).