Mrs. Del Pup bought 20 yards of border for her classroom bulletin board. a. What is the greatest rectangular area she can frame with the border?b. What are the dimensions of the rectangle with greatest area?c. If each dimension of the above rectangle were increased by 2 yards, what would be the dimensions and area of the new rectangle?d. How much more is the new area?
Mathematics · High School · Thu Feb 04 2021
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a. The greatest rectangular area Mrs. Del Pup can frame with a 20-yard border is achieved when the rectangle is a square because a square has the maximum area for a given perimeter among all rectangles.
To find the side of this square, we divide the total border length by 4 (since a square has four equal sides):
Side of the square = Total border length / 4 = 20 yards / 4 = 5 yards
So the greatest area she can frame is: Area = Side length × Side length = 5 yards × 5 yards = 25 square yards
b. The dimensions of the rectangle with the greatest area, which is a square in this case, are: Length = 5 yards, Width = 5 yards
c. If each dimension of this square were increased by 2 yards, the new dimensions would be:
New Length = Old Length + 2 yards = 5 yards + 2 yards = 7 yards
New Width = Old Width + 2 yards = 5 yards + 2 yards = 7 yards
The area of the new rectangle (which is also a square now) would be: Area = New Length × New Width = 7 yards × 7 yards = 49 square yards
d. To find out how much more the new area is compared to the original maximum area:
Difference in area = New area - Original area = 49 square yards - 25 square yards = 24 square yards
So, the new area is 24 square yards more than the original maximum area.
Extra: To elaborate on these concepts for a school student:
- Perimeter is the total distance around the edge of a shape. For a rectangle, it is calculated by adding together the lengths of all four sides. - Area is the amount of space inside a two-dimensional shape. For rectangles, the area is calculated by multiplying the length by the width. - In the case of a square, all four sides are of equal length, so its area is just the side length squared. - When solving problems about maximizing area for a given perimeter, a square is often the best option because geometrical principles show that among all the rectangles with the same perimeter, the square has the largest area.