Hello there! Could You please help me with this Geometry word problem?
Here's the problem: Helen painted a picture that was 10 inches longer than it was wide. When she framed the picture, the outside dimensions (that is, the length and the width) were each two inches longer. If the area of the picture with the frame was 40 sq inches more than the area of the picture without its frame, what was the length of the original painting?
Any help you could provide would be greatly appreciated!
Hello! I'd be happy to help you solve this geometry word problem step-by-step.
Let's start by assigning variables to the dimensions of the original painting. Let's say that the original width is "w" inches.
According to the problem, the original painting was 10 inches longer than it was wide. Therefore, the length can be expressed as "w + 10" inches.
Now let's consider the dimensions of the framed picture. The outside dimensions of the frame are each two inches longer than the original dimensions. So, the framed length would be "w + 10 + 2 = w + 12" inches, and the framed width would be "w + 2" inches.
The problem tells us that the area of the picture with the frame was 40 square inches more than the area of the picture without its frame. We can write this as an equation:
Area of framed picture = Area of picture without frame + 40
The area of a rectangle is calculated by multiplying its length by its width. Using this information, we can set up the equation:
(w + 12)(w + 2) = (w + 10)(w) + 40
Next, let's solve this equation step-by-step.
w^2 + 14w + 24 = w^2 + 10w + 40
To simplify the equation, we can cancel out the common terms on both sides:
14w + 24 = 10w + 40
Now, let's isolate the variable by subtracting 10w from both sides of the equation:
14w - 10w + 24 = 10w - 10w + 40
4w + 24 = 40
Next, let's isolate the variable by subtracting 24 from both sides of the equation:
4w + 24 - 24 = 40 - 24
4w = 16
Finally, let's solve for w by dividing both sides of the equation by 4:
4w/4 = 16/4
w = 4
Therefore, the original width of the painting is 4 inches.
Since the problem states that the original painting was 10 inches longer than it was wide, the length can be calculated as:
w + 10 = 4 + 10 = 14
So, the length of the original painting is 14 inches.
I hope this helps you solve the problem! Let me know if you have any further questions.
Hello! I'd be happy to help you with this geometry word problem.
Let's break down the problem and try to find a solution step by step.
1. Let's represent the original width of the painting as "x" inches.
2. We know that the length of the painting is 10 inches longer than its width, so the length can be represented as "x + 10" inches.
3. The area of a rectangle is calculated by multiplying the length by the width. Therefore, the area of the original painting without its frame is given by "x * (x + 10)" square inches.
Now, let's move on to the second part of the problem.
4. When the painting is framed, the outside dimensions (length and width) increase by 2 inches each. So the length of the framed painting becomes "x + 10 + 2" inches, and the width becomes "x + 2" inches.
5. The area of the framed painting is given by multiplying the length and the width: "(x + 10 + 2) * (x + 2)" square inches.
Now, let's use the information provided in the problem to set up an equation.
6. The problem states that the area of the framed painting is 40 square inches more than the area of the original painting. Therefore, we can set up the equation:
"(x + 10 + 2) * (x + 2) = x * (x + 10) + 40"
Now we can solve the equation to find the value of "x" and determine the length of the original painting.
I hope this explanation helps! If you have any further questions or need assistance with solving the equation, please feel free to ask.