Section 8.1 - Determining Optimum Area And Perimeter

Explanation
Among the given options, diagram d has the greatest area because it has the longest side length among all the rectangles. Since the perimeter of each rectangle is the same, having a longer side length means that diagram d has a larger area.

Explanation
To find the maximum area of a rectangle with a given perimeter, we need to consider the case where the rectangle is a square, as a square has the maximum area for a given perimeter. In this case, the perimeter is 60km, so each side of the square would be 15km. Therefore, the area of the square is 15km multiplied by 15km, which equals 225 km^2.

Explanation
To maximize the area of the garden, Sylvia should use the dimensions of length = 2.75m and width = 2.75m. The area of a rectangle is calculated by multiplying its length and width together. In this case, the area would be 2.75m x 2.75m = 7.5625 square meters. By choosing these dimensions, Sylvia can achieve the greatest area possible within her budget and the available fencing materials.

Explanation
To maximize the area of the paddleball court, the dimensions should be such that the perimeter is maximized. Since the court consists of a wall outlined by 40m of paint, the perimeter of the court should be 40m. The dimensions that satisfy this condition are a length of 10m and a width of 20m. This results in a perimeter of 40m and hence maximizes the area of the court.

Explanation
The area of a rectangle is determined by multiplying its length and width. Since all the rectangles have an area of 49 square units, it means that they all have the same length and width. The perimeter of a rectangle is calculated by adding up the lengths of all its sides. In this case, since all the rectangles have the same length and width, the rectangle with the greatest perimeter would be the one with the longest sides. Looking at the diagrams, it can be observed that diagram c has the longest sides compared to the other rectangles, therefore it has the greatest perimeter.

Explanation
To find the greatest possible area for the quilt, we need to determine the dimensions that will maximize the area. Since Raquel has 540cm of fabric to border the quilt, we can assume that the border will be the same width on all sides. Let's call the width of the border x cm. This means that the dimensions of the quilt will be (540-2x) cm by (540-2x) cm. The area of the quilt can be calculated by multiplying these dimensions, which gives us (540-2x)^2 cm^2. To maximize the area, we need to find the maximum value of this expression. This can be done by taking the derivative and setting it equal to zero. However, since we have a multiple-choice question, we can simply plug in the given answer options and see which one gives the maximum area. By plugging in the dimensions for each answer option, we find that the greatest possible area is 18,225 cm^2.

Explanation
The minimum perimeter of a rectangle can be achieved when the length and width of the rectangle are equal. In this case, the area of the rectangle is 625 mm^2, so both the length and width would be the square root of 625, which is 25 mm. The perimeter is calculated by adding all four sides of the rectangle, which would be 25 + 25 + 25 + 25 = 100 mm. Therefore, the minimum perimeter of a rectangle with an area of 625 mm^2 is 100 mm.

Explanation
To maximize the area of the rectangular pen, Brandon should use the dimensions length = 2.2 m and width = 2.2 m. The area of a rectangle is calculated by multiplying its length and width. By using the same value for both the length and width, Brandon will create a square-shaped pen, which will have the largest area possible.

Explanation
The correct answer is length = 6.25 m, width = 12.5 m. This is because to maximize the area of the dog run, the length and width should be equal and half of the total fencing. In this case, the total fencing is 25m, so half of that is 12.5m. Therefore, the length and width should both be 12.5m/2 = 6.25m.