Draw a diagram using the given information. Shade the region that represents the wooden frame.
Area: 20x^2+5x Factored Form: 5x(4x+1)
Practice makes perfect
We are asked to find the area of the wooden part a rectangular frame. To answer this question let's use the given bullet points.
How can drawing a diagram help you solve the problem?
In most cases, when dealing with a geometry exercise the first thing you should do is draw a diagram. It helps you visualize the problem and find the solution. Let's draw the diagram for our exercise! This means drawing a smaller rectangle inside a bigger one.
Now, let's write the side lengths of each rectangle on the diagram. The frame, or the bigger rectangle, has side lengths 5x and 7x+1. The rectangular opening, or the smaller rectangle, has side lengths 3x and 5x.
Remember that we only want to find the area of the wooden part of the frame. Let's shade this area.
Our diagram gathers all of the given information. This will help us later on!
How can you express the area of the wooden part of the frame as a difference of areas?
The wooden part of the frame is equivalent to the shaded region on our diagram.
The shaded region is the bigger rectangle without the smaller rectangle. Therefore, we can express the area of the wooden part of the frame as the area of the bigger rectangle minus the area of the smaller rectangle.
W=B-S
Here, W represents the area of the wooden part of the frame, B is the area of the bigger rectangle, and S is the area of the smaller rectangle.
What is the area of the wooden part of the frame?
Finally, we will calculate the area of the wooden part of the frame. To do so, we have to calculate the area of the bigger rectangle B and the area of the smaller rectangle S. Let's recall the formula for the area of a rectangle.
A=l w
In this formula l is the length of a rectangle and w is the width of a rectangle. Let's start with calculating the area of the bigger rectangle. We can read its dimensions from our diagram.
The length of the bigger rectangle is 7x+1 and its width is 5x. We will substitute these expressions into the formula and simplify.
Recall that the area of the wooden part of the frame W is equal to the difference of B and S.
W=B-S=35x^2+5x-15x^2=20x^2+5x
The area of the wooden part of the frame is 20x^2+5x. We have to rewrite our answer in factored form. Let's find the Greatest Common Factor (GCF) of the expression. To do so, we will factor each term in the expression.
20x^2 & = 2* 2* 5* x* x
5x & = 5*x
The GCF is 5* x, or 5x. Next, we will factor out the GCF.
