A rectangular garden is 30 ft by 40 ft. Part of the garden is removed in order to install a walk

A rectangular garden is 30 ft by 40 ft. Part of the garden...

A rectangular garden is $30 \mathrm{ft}$ by $40 \mathrm{ft}$. Part of the garden is removed in order to install a walkway of uniform width around it. The area of the new garden is one-half the area of the old garden. How wide is the walkway?

Key Concepts

Rectangular Area

This concept refers to calculating the area of a rectangle by multiplying its length by its width. It is fundamental in geometry and is used to determine the size of rectangular regions, which is essential when comparing different portions or modifications of a shape.

Uniform Border in Geometric Shapes

A uniform border or frame is a region of constant width added around or removed from a geometric shape. In problems like this, a constant width is subtracted from each dimension of a rectangle, reducing its overall area. Understanding this concept is vital for modeling changes to shapes in a real-world context.

Algebraic Modeling of Area Problems

Algebraic modeling involves translating a geometric situation into an equation that represents the area or other properties. In this case, the dimensions of the inner rectangle (after the border is removed) are expressed in terms of the uniform width and set equal to a fraction of the original area, illustrating how geometry and algebra interact.

Quadratic Equations

Quadratic equations commonly arise in problems where a variable affects a product of two linear expressions, such as the modified dimensions of a rectangle with an added or removed border. Solving these equations is essential to find the unknown width when given an area constraint, making them a key tool in algebra and geometry problem solving.

Transcript

00:01 All right, so the question that we're going to be looking at today is we have a rectangular garden that is 30 by 40 feet.

00:07 And part of this garden has to be removed to install a walkway around it, which is equal lengths around all sides.

00:15 So the area of the new garden is one half of the area of the old garden.

00:19 And we need to find out how wide the walkway is.

00:21 First thing we're going to write down is we're going to find is the area of our rectangular garden, just normally without the walkway.

00:30 So in the question it stated it was 30 feet by 40 feet.

00:40 So that means that our area of our rectangular garden normally is 1 ,200 feet.

00:51 So the new area is going to be half of that, as stated in the question, new area.

00:59 So that's going to be 600 feet.

01:05 We're going to let x equals the width of the walkway.

01:13 Of the walkway.

01:19 It's going to be the width of the walkway.

01:23 This subtracts 2x from both of the dimensions.

01:26 So the 30 feet and the 40 feet are both going to be subtracted by 2x.

01:30 In order to write out this formula, our new formula is going to look like 30 minus 2x.

01:37 And 40 minus 2x is equal to 600.

01:49 So that's our formula right there...

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