if the perimeter of a rectangle is 50m what is the largest area possible
Shouldn't it be 12.5?
156.25m^2
for a given perimeter, the largest area is enclosed by a square.
That is, all 4 sides are equal...
Calculate the smallest and largest parameter that can be obtained from a 50 square meter
I had this problem during my 10 years of engineering. I found out that the answer is 12.5m^2 on a square.
To find the largest possible area of a rectangle given its perimeter, we need to apply the concept of optimization.
Let's start by understanding the formula for the perimeter of a rectangle. The perimeter of a rectangle is given by the equation:
Perimeter = 2(length + width)
In this case, we are given that the perimeter is 50m. Substituting this into the formula, we have:
50 = 2(length + width)
To find the largest possible area, we need to maximize the area formula, which is given by:
Area = length * width
To optimize the area, we can use the perimeter constraint equation to isolate one of the variables (length or width) and substitute it in the area formula.
Let's solve for one of the variables using the perimeter equation:
50 = 2(length + width)
25 = length + width
length = 25 - width
Substituting this value of length into the area formula, we have:
Area = (25 - width) * width
Now we have an equation for the area in terms of a single variable, width. To find the maximum area, we need to find the value of width that maximizes this equation.
To find the maximum or minimum of a quadratic equation, we can either complete the square or use calculus. In this case, let's use calculus to find the maximum area.
Take the derivative of the area equation with respect to width:
d(Area)/d(width) = 0
Setting this derivative equal to zero, we can solve for the value of width that maximizes the area.
d(Area)/d(width) = 0
(25 - 2*width) = 0
Solving this equation, we find:
25 - 2*width = 0
2*width = 25
width = 25/2
width = 12.5m
Now we have the value of the width that maximizes the area. To find the corresponding length, we can substitute this value back into the perimeter equation:
50 = 2(length + 12.5)
50 = 2*length + 25
2*length = 50 - 25
2*length = 25
length = 25/2
length = 12.5m
So, to achieve the maximum area given a perimeter of 50m, the rectangle should have a width of 12.5m and a length of 12.5m.
To find the actual largest area, we can substitute these values into the area formula:
Area = length * width
Area = 12.5m * 12.5m
Area = 156.25m^2
Therefore, the largest possible area for a rectangle with a perimeter of 50m is 156.25 square meters.
Well, let's think about this. If we have a rectangle with a given perimeter, we know that the length plus the width of the rectangle equals half of the perimeter. Since the question asks for the largest possible area, we want to find the dimensions that will give us that.
Now, the clown in me wants to say that the largest possible area would be in the shape of a circle, so let's just go ahead and call it a rectangular circle! But, sadly, rectangles don't work that way.
To maximize the area, we want to make the length and width as close to each other as possible. So, if the perimeter is 50m, let's assume the length and width are both 25m.
Now, the area of a rectangle is calculated by multiplying the length by the width. Therefore, the largest possible area for this rectangle would be 25m x 25m, or 625 square meters.
Just remember, even if a clown tries to trick you into thinking otherwise, squares are just special rectangles pretending to be circles!