Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square

Find the length and width of a rectangle whose perimeter is...

Key Concepts

Systems of Equations

A system of equations consists of multiple equations that share common variables. In mathematical problems, systems of equations are often used to find the values of unknown quantities that satisfy all equations simultaneously. This method is particularly useful when dealing with problems where multiple conditions, such as area and perimeter, are given.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation that typically takes the form ax² + bx + c = 0. Solving quadratic equations is key in many geometric problems involving rectangular dimensions, especially when the relationships between the sides lead to quadratic expressions. Techniques such as factoring, completing the square, or using the quadratic formula are applied to find the solutions.

Perimeter

The perimeter of a shape is the total distance around its boundary. For a rectangle, the perimeter is calculated by summing the lengths of all four sides, effectively doubling the sum of its length and width. This concept is critical in problems where boundary measurements are provided or need to be determined.

Rectangle

A rectangle is a quadrilateral with four right angles. It is a fundamental shape in geometry, often used as the basis for problems involving spatial reasoning, perimeter, and area calculations. Understanding the properties of a rectangle, such as the fact that opposite sides are equal, is essential in solving related problems.

Area

The area of a shape represents the amount of two-dimensional space it occupies. In the case of a rectangle, the area is found by multiplying its length by its width. This mathematical relationship is widely used in geometry to solve for unknown dimensions when the area is known.

Transcript

00:01 So we are given the perimeter and area of a rectangle and asked to solve what the length and the width for that rectangle would look like.

00:12 So they give us the length in terms of x and they give us the width in terms of y.

00:18 So the perimeter that they give us is going to be 40 feet long.

00:23 So we can create a equation that would satisfy that.

00:28 So 2 times x plus y equals 40.

00:33 Again, this is the perimeter.

00:35 So we have x plus y or the length plus the width times two because there's four sides of the rectangle and would equal to 40.

00:43 So that's the perimeter.

00:45 And then what we can do is actually divide that all by two to make it easier for us to solve later.

00:53 So if we divide that by two, we would see it's x plus y equals 20.

00:58 So that's going to be the new equation we're going to use moving forward.

01:02 And then they also give us an area of the rectangle, which should be 96 square feet.

01:09 So we see that x times y, which is how you would find the area, equals 96.

01:16 So those are our two equations.

01:19 So from there, what we want to do is we can use one of the variables in terms of the other.

01:25 So we can see that we can find x or y from the x.

01:30 X times y equals 96 pretty easily so what we would do is divide by y to get rid of it and have x equals 96 over y and then since we have what that would equal we can plug that into our other equation so we would have x or 96 over y plus y equals 20 so then we need to get rid of the y on the denominator.

02:03 So we would multiply everything by y.

02:06 So we would end up getting y squared plus 96 equals 20.

02:16 And then this would be 20 y because again it's both sides times y...

Please give Ace some feedback