A rectangular photograph measuring six centimeters by four centimeters is to be displayed in a card mount in a rectangular frame, as shown in the diagram. Write an equation that can be used to find 𝑥, the width of the mount, if it’s area is 64 centimeters squared.
The area of the mount is 64 centimeters squared. The area of the mount would be the area of the large rectangle minus the area of the photograph. It’s the area shaded in yellow here. The area of the mount will be equal to the area of the large rectangle minus the area of the photograph. So we need to write an equation for each piece. We know that the area of the mount is 64 centimeters squared because our question gave us that information. The formula for finding area is length times width. So we’ll need to use the area formula to find the area of the large rectangle and the area of the photograph.
The photograph measures six centimeters by four centimeters. The area of the photograph will be six times four. First, we’ll just clean up the shading a little bit so it’s easier to see what the length and the width of the large rectangle would be. What would be the length of this rectangle? We know that the middle portion is four centimeters because that’s the space that the photograph takes up. We know that the space from the top of the rectangle to the top of the photograph is 𝑥, and the space from the bottom of the photograph to the bottom of the large rectangle is 𝑥. This makes the width of our rectangle four plus 𝑥 plus 𝑥.
We can find the length of our rectangle in a similar way. The middle portion is six and the space on either side of the photograph is equal to 𝑥. 64 has to be equal to six plus 𝑥 plus 𝑥 times four plus 𝑥 plus 𝑥 minus six times four. But we don’t wanna leave that as our final answer. Let’s see if we can simplify and clean this up a little bit.
We can bring down the 64. Instead of saying six plus 𝑥 plus 𝑥, we can say six plus two 𝑥. We can do the same thing for our four. Instead of saying four plus 𝑥 plus 𝑥, we say four plus two 𝑥. We go ahead and multiply six times four equals 24.
One equation that could be used to find 𝑥 would be 64 equals six plus two 𝑥 times four plus two 𝑥 minus 24.