given constant perimeter (people),Irene made a rectangle with an area (tables) of 196 sq. units. she finds that this is the largest area (greatest number of tables) possible for the given perimeter. Irene drew a rectangle with an area of 196 sq. units. she finds that this i the largest area possible for any rectangle with the same perimeter. what are the dimensions of the rectangle and what is the perimeter of the retangle
Considering all possible rectangles with a given perimeter, the square encloses the greatest area.
Proof:
Consider a square of dimensions "x "by "x", the area of which is x^2.
Adjust the dimensions by adding "a" to one side and subtracting "a" from the other side.
This results in an area of (x + a)(x - a) = x^2 - a^2.
Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.
To determine the dimensions and perimeter of the rectangle, we need to use the given information that the rectangle has a constant perimeter and an area of 196 square units.
Let us assume that the length of the rectangle is L units and the width is W units.
The perimeter of a rectangle is given by the formula: Perimeter = 2L + 2W.
In this case, as the perimeter is constant, we can write the equation: 2L + 2W = constant (eq.1)
The area of a rectangle is given by the formula: Area = Length × Width.
In this case, the area of the rectangle is 196 square units, so we can write the equation: L × W = 196 (eq.2)
To find the dimensions of the rectangle that will give the largest area, we will solve these equations simultaneously.
One way to solve these equations is to rewrite eq.1 to express L in terms of W.
Let's rearrange eq.1 to get: L = (constant - 2W)/2.
Now we can substitute this value of L in eq.2:
((constant - 2W)/2) × W = 196.
Simplifying this equation, we get: (constant - 2W) × W = 392.
Expanding the brackets, we have: constant × W - 2W^2 = 392.
Rearranging the equation to a quadratic form, we get: 2W^2 - constant × W + 392 = 0.
This is a quadratic equation. To find the dimensions and perimeter of the rectangle, we need to know the value of the constant.
Can you provide me with the value of the constant (perimeter)?