Hello and Happy Halloween! Trying to get this problem and could use some help.
Find the dimensions of the rectangle with minimum perimeter if its area is 400 square meters. Find the least perimeter.
I know that a rectangle has two sides that are equal and two other sides that are equal but I am unsure about what type of equation to use. I thought maybe I would try guess and check but I would rather have an equation.
"two sides that are equal and two other sides that are equal" is a good start. Call the length of one type of side x, and the other y.
Then the area is x * y
And the perimeter is 2x + 2y
Check this on a small 5 * 3 rectangle, to be sure you understand it.
We are told the area is 400. Does this give you enough to go further?
Hello and Happy Halloween to you too! I'd be happy to help you with this problem.
To find the dimensions of the rectangle with the minimum perimeter, we can start by assigning variables to the sides of the rectangle. Let's call one side of the rectangle x, and the other side y. Since the area of the rectangle is given as 400 square meters, we have the equation:
x * y = 400
Now, we want to find an equation that relates the perimeter of the rectangle to its sides. The perimeter of a rectangle is given by the formula:
P = 2x + 2y
We want to minimize the perimeter P, given the constraint that the area xy is equal to 400. To do this, we can use the method of optimization.
First, solve the equation for y in terms of x. We have:
y = 400 / x
Now substitute this expression for y into the equation for the perimeter:
P = 2x + 2*(400 / x)
Simplify this equation:
P = 2x + 800 / x
To find the minimum perimeter, we can take the derivative of this equation with respect to x and set it equal to zero.
dP/dx = 2 - 800 / x^2
Setting this derivative equal to zero, we have:
2 - 800 / x^2 = 0
To solve for x, rearrange the equation:
800 / x^2 = 2
Cross-multiply:
800 = 2x^2
Divide both sides by 2:
400 = x^2
Take the square root of both sides:
x = √400 = 20
Now that we have the value of x, substitute it back into the equation for y:
y = 400 / x = 400 / 20 = 20
So the dimensions of the rectangle with the minimum perimeter and an area of 400 square meters are 20 meters by 20 meters.
To find the least perimeter, substitute these values back into the equation for the perimeter:
P = 2x + 2y = 2*20 + 2*20 = 40 + 40 = 80 meters
Therefore, the least perimeter of the rectangle is 80 meters.
I hope this explanation helps you solve the problem! Let me know if you have any further questions.