a picture 8 in. by 12 in. is placed in a frame which has a uniform width. if the area of the frame is equal to the area of the picture, find the dimension of the frame
(if you would,just give me the formula and i'll take it from there)
thank you
let the width of the frame be x cm
width of whole thing = 8+2x
length of the whole thing = 12 + 2x
whole thing - actual picture = actual picture
(8+2x)(12+2x) - 96 = 96
expand and simplify, take everything to the left and you have a quadratic equation, which you said you can handle.
(hint: you should get x=2)
Double a number is equal to the number decreased by five what is the number
Well, well, well! Looks like we've got ourselves a picture-perfect problem, my friend! Let's get cracking.
To find the dimensions of the frame, we'll use a bit of mathematical magic. Here's the formula you need:
Area of a rectangle = Length × Width
Now, let's break it down, clown-style!
The picture measures 8 inches by 12 inches, so its area is 8 × 12 = 96 square inches.
The frame adds extra width to both sides and top and bottom of the picture. Let's call this extra width "x" (I'm feeling a bit eXtra today).
The total dimensions (including the frame) will be:
Length: 8 + 2x
Width: 12 + 2x
The area of the frame will be:
(Frame Length) × (Frame Width) = Total Area - Picture Area
(8 + 2x) × (12 + 2x) = (8 + 2x)(12 + 2x) - 96
Now, it's time for you to be the hero! Solve the equation to find the value of "x" (that cheeky frame width), and plug it into the formulas for the Length and Width of the frame.
Remember, laughter is the best frame of mind! Have fun with the calculations!
To find the dimensions of the frame, we can use the formula for area of a rectangle.
Let's denote the width of the frame as "x".
The overall dimensions of the frame including the picture will be:
Width: 8 in + 2x
Length: 12 in + 2x
To find the area of the picture, we calculate 8 in by 12 in:
Area of the picture = Length × Width = 8 in × 12 in = 96 in²
The area of the frame will be equal to the area of the picture, so we can set up the equation:
Area of the frame = (8 in + 2x) × (12 in + 2x)
Since the areas are equal, we can solve for "x" by setting up the equation:
96 in² = (8 in + 2x) × (12 in + 2x)
To solve this equation, we can distribute and simplify:
96 in² = 96 in² + 32x + 24x + 4x²
Combine like terms:
0 = 4x² + 56x
Now we have a quadratic equation. Set it equal to zero:
4x² + 56x = 0
Divide both sides by 4 to simplify the equation:
x² + 14x = 0
Factor out an "x" in order to solve the equation:
x(x + 14) = 0
Now we have two solutions:
x = 0 or x + 14 = 0
Since the width of the frame cannot be zero, we ignore the x = 0 solution.
Solve for x + 14 = 0:
x = -14
Since we are dealing with dimensions, we take the positive value:
x = 14
Therefore, the width of the frame is 14 inches.