Dimensions of a Rectangle A rectangle has an area of 180 cm^2 and a perimeter of 54 cm . What ar

Dimensions of a Rectangle A rectangle has an area of 180...

Key Concepts

Rectangle

A rectangle is a quadrilateral with four right angles. Its opposite sides are parallel and equal in length, which makes it one of the fundamental shapes in geometry. The properties of a rectangle are used to derive formulas for area and perimeter, which are essential in solving practical problems.

Area of a Rectangle

The area of a rectangle is determined by multiplying the length by the width. This measurement represents the total space enclosed within the boundaries of the rectangle and is a fundamental concept in geometry and various applications such as land measurement and design.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the shape, calculated by adding together twice the length and twice the width. This concept is crucial when determining the boundary length required for tasks such as fencing or framing.

System of Equations

Problems involving multiple geometric properties, such as area and perimeter, often require the formulation and solving of a system of equations. In these cases, different formulas are set equal to given values, and the resulting equations are solved simultaneously to find the unknown dimensions.

Quadratic Equations

When one variable is substituted from one equation into another to solve a system, the process can result in a quadratic equation. Solving quadratic equations is an important algebraic skill that helps find potential dimensions that satisfy the given geometric conditions.

Transcript

00:01 Okay, so we have a rectangle, which i'll try to draw, and what we know about it is that the area is going to be equal to 180 centimeters squared, and the perimeter is going to be equal to 54 centimeters.

00:28 Okay, so in order to get some variables out of this, we're going to label the base of the rectangle right here, it's b, and we know that this, the top side is going to be the same.

00:38 Length also be and these two sides are going to be the height h so then we know that the area is going to be equal to the base times the height so it's going to be equal to b times h and you know that the perimeter is going to be equal to the sum of the lengths of the sides so that's going to be b plus h plus b plus and we can simplify that to 2b plus 2h so now we have a system of equations here you have 2b plus 2h equals 54 and bh equals 180 now we need to solve this system so the first thing we're going to do is take that top expression and we're going to divide both sides by 2 so then now we're going to move b to the other side so we can solve for h h equals 27 minus b so then now we have this expression and we can substitute it back into this equation right here.

02:04 So we're going to hear that here.

02:06 We're going to have b times 27 minus b equals 180.

02:17 Because we substituted our value of h into here.

02:21 So this is equation.

02:22 So now if we expand both sides of this, and what we have here is just a polynomial.

02:34 It's a basic quadratic equation.

02:36 And we can just solve that directly.

02:38 So first we're going to put it into the, we're going to move everything to one side, so it looks real nice.

02:50 180.

02:53 And so what i did there is i just moved this b squared over here and this 27b over here...