A rectangular picture is 4 inches longer than it is wide. The picture is surrounded by a 3 inch wide frame. The area of the picture and frame is 144 square inches more than the area of the picture. What is the width of the picture?
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the outside of a picture frame has a length which is 4 cm more than width. The area enclosed by the outside of the picture frame is 165 square cm. Find the width of the outside of the picture frame.
Let's break down the problem step by step:
Step 1: Assign variables
Let's denote the width of the picture as "W".
Step 2: Express the length of the picture
According to the problem statement, the rectangular picture is 4 inches longer than its width. Therefore, the length of the picture can be expressed as "W + 4".
Step 3: Calculate the area of the picture
The area of a rectangle is given by the formula: Area = length × width.
So, the area of the picture can be expressed as W × (W + 4).
Step 4: Calculate the total area (picture + frame)
The total area is given as 144 square inches more than the area of the picture. Therefore:
Total Area = Picture Area + 144
Step 5: Express the total area
The total area can be calculated by adding the areas of the picture and the frame. Since the frame surrounds the picture on all sides, the total width is increased by 2 times the frame width (3 inches). Similarly, the total length is increased by 2 times the frame width (3 inches). Therefore:
Total Area = (W + 2(3))(W + 4 + 2(3))
Step 6: Set up the equation
Since the total area is 144 square inches more than the area of the picture, we can set up the equation:
Total Area = Picture Area + 144
Substituting the expressions for the total and picture areas:
(W + 2(3))(W + 4 + 2(3)) = W × (W + 4) + 144
Step 7: Solve the equation
Expanding both sides of the equation:
(W + 6)(W + 10) = W² + 4W + 144
W² + 10W + 6W + 60 = W² + 4W + 144
W² + 16W + 60 = W² + 4W + 144
Simplifying the equation:
W² - W² + 16W - 4W = 144 - 60
12W = 84
Step 8: Solve for the width
Dividing both sides of the equation by 12:
W = 84 / 12
W = 7
Therefore, the width of the picture is 7 inches.
To solve this problem, let's denote the width of the picture as "x" inches.
According to the given information, the length of the picture is 4 inches longer than its width, so the length would be (x + 4) inches.
The frame is 3 inches wide on all sides of the picture. Therefore, the overall width of the picture and frame would be (x + 2 * 3) = (x + 6) inches, and the overall length would be ((x + 4) + 2 * 3) = (x + 10) inches.
The area of the picture is simply the width multiplied by the length, which would be x * (x + 4) square inches.
The area of the picture and frame is given as 144 square inches more than the area of the picture. So, the equation for this can be set up as:
(x * (x + 4)) + 144 = ((x + 6) * (x + 10))
To solve this equation, we can simplify it:
x^2 + 4x + 144 = x^2 + 16x + 60
Next, we cancel out the common terms:
4x - 16x = 60 - 144
Simplifying further:
-12x = -84
Now, dividing both sides of the equation by -12:
x = -84 / -12
x = 7
Therefore, the width of the picture is 7 inches.