The perimeter of a rectangle is 27 centimeters, and its area is 35 square centimeters. Find the

The perimeter of a rectangle is 27 centimeters, and its area...

Key Concepts

Rectangle Properties

Rectangles are quadrilaterals with opposite sides equal and four right angles. They are a fundamental shape in geometry, defined by their length and width, and are analyzed using various formulas relating to area and perimeter.

Perimeter Calculation

The perimeter is the total distance around the boundary of a shape. For rectangles, it is calculated as 2 times the sum of the length and width, which provides a linear equation constraint when the perimeter is known.

Area Calculation

The area represents the amount of space enclosed within a shape. In the case of rectangles, it is found by multiplying the length by the width. This relationship often leads to a quadratic equation when combined with the perimeter information.

System of Equations

A system of equations is a set of two or more equations with multiple variables. In the context of geometrical problems like rectangles, the simultaneous equations derived from the area and perimeter formulas must be solved together to determine the unknown dimensions.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree and can emerge when substituting one variable in terms of another in problems involving both area and perimeter. Solving these equations, through methods like factoring or using the quadratic formula, is essential for finding valid dimensions of the rectangle.

Transcript

00:01 In this problem, we're being asked to find the width and the length of a rectangle, whose perimeter is 27 centimeters, and the area is 35 centimeters.

00:09 Well, the formula to find the perimeter of a rectangle is 2w plus 2l is equal to the perimeter.

00:16 Well, in this case, we know that's equal to 27.

00:18 And to find the area of a rectangle, we multiply the width times the length.

00:23 And again, we know that's equal to 35.

00:25 Perfect.

00:26 So now we have a system of equations.

00:28 We just need to solve.

00:30 So in this case, what i'm going to do is i'm going to take our first equation for the perimeter, and i'm going to go ahead and solve for w.

00:38 So i'm going to subtract 2l from both sides of our inequality.

00:43 So we'd have 2w is equal to 27 minus 2l, and then we'll divide both sides by 2.

00:50 So that leaves us with w equals 27 over 2 minus l.

00:55 So now i'm going to substitute this expression in for w in our second equation.

00:59 Equation.

01:01 So we'll have 27 over 2 minus l times l is equal to 35.

01:07 Okay, so now we just need to solve.

01:10 So first thing i'm going to distribute, 27 over 2 times l is 27 over 2 l, and negative l times l is negative l squared.

01:20 All right, now i see we have a quadratic equation.

01:23 So we need to set this equal to zero.

01:25 So the first thing i'm going to do is subtract 35 from each side of the equation, and it's a same time i'm going to put this in standard form, meaning we're going to have negative l squared plus 27 over 2l minus 35 is equal to 0.

01:41 Now, a couple of things i'm going to do.

01:44 First off, i want to get rid of our fraction, so i'm going to multiply each side of our equation by 2, but i also want the first term to be positive.

01:51 So i'm actually going to multiply everything by negative 2.

01:54 Well, negative 2 times negative l squared is positive 2l squared.

01:59 Negative 2 times 27 over 2l is equal.

02:02 To negative 27l and negative 2 times negative 35 is positive 70 and negative 2 times zero that's still zero perfect okay so now what we need to do is solve this equation so i'm going to go ahead and solve by factor so i notice we don't have a greatest common factor and the leading coefficient is greater than one so i'm going to actually solve this kind of using a grouping method so first thing i'm do is multiply our first and our last coefficients.

02:34 Well, two times 70 is equal to 140.

02:38 Okay.

02:39 So now we have to find two numbers that multiply to 140 that will add to our middle coefficient of negative 27.

02:48 So if we were to find this, you could go ahead and figure out which two numbers are going to be it.

02:57 Well, i know one times 140 is equal to 140, but that won't work.

03:01 I know that two times 70 is equal to 140, but that won't work.

03:01 I know that two times 70 is equal to 140, but that's not going to be helpful.

03:07 140 is not divisible by three.

03:09 140 divided by four is equal to 35.

03:13 So we're getting closer because using the numbers four and 35, we could get a 31, but that's not quite it...

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