Find The Dimensions Of A Rectangle With Area 1000 m² Whose Perimeter Is As Small As Possible

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What Is The Area Of A Rectangle

The area of a rectangle can be defined as the total area covered by the four boundaries of a given rectangular figure.

How To Find The Length And Width Of A Rectangle When Given The Perimeter And Area

To find the dimensions of a rectangle with an area 1,000m² whose perimeter is as small as possible, we solve as follows:

Let the length of the rectangle be called L and the width of the rectangle be known as W.

Area of rectangle (A) = LW m² = 1000 m² — (Equation 1)

Perimeter of rectangle (P) = 2 (L + W) m — (Equation 2)

We were given the area of the rectangle, so our primary objective is to find the minimum values that will give us the minimum perimeter of this rectangle. So from equation above, we can derive the third equation as follows:

W = 1000/L — ( Equation 3)

We will substitute equation (3) in equation (2) to have:

P = 2(L + 1000/L)

P = 2L + 2000/L

We need to go further to differentiate the equation above so as arrive at the minimum area as follows:

dP/dL = 2 – 2000/L²

Equating the above statement to zero, we have:

2 – 2000/L² = 0

2 = 2000/L²

L² = 1000

L = √1000

W = 1000/L

W = 1000/√1000

W = √1000

At this point, we need present the second order differential of the the function of P so as to verify that the perimeter P is minimum for the above calculated dimensions of L and W.

Thus, d²P/dL² = 4000/L³

Since L > 0, we can state d²P/dL² is a positive number. This is an indication that the answers we got for the Perimeter as well as the Length (L) and Width (W) dimensions of the rectangle are the minimum values. That is to say that dimensions of the rectangle covering an area of 1,000 m² has a perimeter which is as small as possible. This goes on to affirm that the rectangle occupying a total area of 1,000 m² is a square of side √1000 m.

Thus, the dimensions of a rectangle with an area 1,000m² whose perimeter is as small as possible is 10√10 and 10√10.

So this is all about How To Find The Length And Width Of A Rectangle When Given The Perimeter And Area. Feel free to Download Free DJ Mixtapes, read Powerful Prayer Points to enhance your spiritual life and also check out more information about the Formula For Finding Area Of A Rectangle. Cheers!

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