A 3 by 5 inch picture is put into frame with a mat that has width x. Draw picture inside the frame, write the expression of the width and length of the frame AND the picture together. If the total area of the frame with the picture is 35 in ^2, find the width and length of the frame.
frame = (3+2x) by (5+2x)
(3+2x)(5+2x)= 35
15 + 16x + 4x^2 = 35
4x^2 + 16x - 20 = 0
x^2 + 4x - 5 = 0
(x-1)(x+5) = 0
x = 1 or x = -5 , the last one is not possible
the frame is 5 cm by 7 cm
To draw the picture inside the frame, we can represent the frame with a rectangle and the picture with a smaller rectangle inside it. Let's assume the width of the frame to be x inches.
The width of the picture is 3 inches, and the length of the picture is 5 inches.
So, the width of the frame, including the picture, is x + 2(3) = x + 6 inches. (Since the picture is placed on both sides of the frame)
The length of the frame, including the picture, is x + 2(5) = x + 10 inches. (Since the picture is placed on both sides of the frame)
The area of the frame with the picture is the product of the width and length:
Area = (x + 6) * (x + 10)
Given that the total area of the frame with the picture is 35 in^2, we can set up the equation:
(x + 6) * (x + 10) = 35
Now, we can solve this equation to find the values of x, which will give us the width and length of the frame.
To solve this problem, let's start by drawing a diagram to visualize the situation.
Since the picture is 3 by 5 inches, we can represent it as a rectangle with those dimensions. Let's assume the width of the frame is x, as given in the problem.
To find the width and length of the frame, we need to add the width of the frame to both the picture's width and length.
So, the width of the frame with the picture will be the sum of:
- The width of the frame on the left side of the picture
- The width of the picture
- The width of the frame on the right side of the picture
The length of the frame with the picture will be the sum of:
- The width of the frame on the bottom side of the picture
- The length of the picture
- The width of the frame on the top side of the picture
Let's use algebra to express this:
Width of frame with picture = x (left frame) + 3 (picture width) + x (right frame)
Length of frame with picture = x (bottom frame) + 5 (picture length) + x (top frame)
Now, we can find the total area of the frame with the picture, which is given as 35 in^2.
Area of frame with picture = Width of frame with picture * Length of frame with picture
Substituting the expressions we derived earlier:
35 = (x + 3 + x) * (x + 5 + x)
Simplifying this equation gives us:
35 = (2x + 3) * (2x + 5)
To solve this quadratic equation, we can expand the expression and set it equal to zero:
4x^2 + 10x + 6x + 15 = 35
4x^2 + 16x - 20 = 0
To further simplify the equation, we can divide every term by 4:
x^2 + 4x - 5 = 0
Now, we can factor or use the quadratic formula to solve for x. Factoring this quadratic equation, we have:
(x + 5)(x - 1) = 0
Setting each factor equal to zero, we find two possible values for x:
x + 5 = 0 --> x = -5 (discard this solution since we're dealing with measurements)
x - 1 = 0 --> x = 1
Therefore, the width of the frame is 1 inch.
To find the length of the frame, we can substitute the value of x into the expression we derived earlier:
Length of frame with picture = x (bottom frame) + 5 (picture length) + x (top frame)
= 1 (bottom frame) + 5 + 1 (top frame)
= 6 + bottom frame + top frame
Since the frame is a rectangle, the top and bottom frames have the same width:
Length of frame with picture = 6 + 2 (same width)
= 6 + 2x
= 6 + 2(1)
= 6 + 2
= 8
Therefore, the length of the frame is 8 inches.
To summarize:
- The width of the frame is 1 inch
- The length of the frame is 8 inches.