a rectangular picture frame measures 8 inches by 4 inches. You want to triple the frame's area by adding the same distance, x, to the length and width.
Find the value of x and find the new dimensions of the frame.
How do you do this?? Please show work
L = Lenght = 8 in
W = Width = 4 in
A = Area = L*W = 8*4 = 32 in^2
3A = 3*32 = 96 in^2
(L+x)*(W+x) = 3A
(8+x)*(4+x) = 96
4*8 + 4*x + x*8 + x*x = 96
32 + 4x + 8x + x^2 = 96
x^2 + 12x + 32 - 96 = 0
x^2 + 12x - 64 = 0
The exact solutions of this equation are:
x = 4
and
x = -16
Distance can't be negative so:
x=4
(8+4) * (4+4) = 12 * 8 = 96 in^2 = 3A
The new dimensions of the frame are:
Lenght = 8+4 = 12 in
Width = 4+4 = 8 in
To triple the frame's area, we need to find the value of x and determine the new dimensions of the frame.
Let's begin by finding the current area of the frame:
Area = length × width
Area = 8 inches × 4 inches
Area = 32 square inches
To triple the area, we need to multiply it by 3:
New Area = 3 × 32 square inches
New Area = 96 square inches
Now, let's consider adding the same distance, x, to the length and width. The new length would be 8 inches + x, and the new width would be 4 inches + x.
Multiplying the new length and new width, we should get the new area of 96 square inches:
(New length) × (New width) = New Area
(8 inches + x) × (4 inches + x) = 96 square inches
Expanding the equation:
(8x + 32 + 4x + x^2) = 96
Rearranging the equation:
x^2 + 12x + 32 = 96
Subtracting 96 from both sides:
x^2 + 12x - 64 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. In this case, factoring is an efficient method.
(x + 16)(x - 4) = 0
Setting each factor equal to zero:
x + 16 = 0 or x - 4 = 0
Solving for x:
x = -16 or x = 4
Since the length and width cannot be negative, the value of x must be positive. Therefore, x = 4.
Substituting x = 4 into the expressions for the new length and width:
New length = 8 inches + 4 inches = 12 inches
New width = 4 inches + 4 inches = 8 inches
So, the new dimensions of the frame are 12 inches by 8 inches.
To find the value of x and the new dimensions of the frame, we can use the given information and some algebraic equations.
Let's start by finding the area of the original frame. The formula to calculate the area of a rectangle is A = length × width.
Given:
Length of the original frame = 8 inches
Width of the original frame = 4 inches
Original area = 8 inches × 4 inches = 32 square inches
Now we want to triple the frame's area by adding the same distance x to the length and width.
The new length of the frame will be: 8 inches + x
The new width of the frame will be: 4 inches + x
So the area of the new frame is given by: (8 inches + x) × (4 inches + x)
We know that the new area needs to be three times the original area, which gives us the equation: (8 inches + x) × (4 inches + x) = 3 × 32 square inches
Now let's solve this equation to find the value of x.
Expanding the equation:
(8 inches + x) × (4 inches + x) = 96 square inches
Using the distributive property of multiplication, we have:
32 inches^2 + 8x inches + 4x inches + x^2 inches^2 = 96 square inches
Combining like terms, we get:
x^2 + 12x + 32 = 96
Rearranging the equation to make it a quadratic equation equal to zero, we have:
x^2 + 12x + 32 - 96 = 0
Simplifying:
x^2 + 12x - 64 = 0
Now we can use the quadratic formula to solve for x. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 12, and c = -64. Plugging these values into the formula, we have:
x = (-(12) ± √((12)^2 - 4(1)(-64))) / (2(1))
Simplifying further:
x = (-12 ± √(144 + 256)) / 2
x = (-12 ± √(400)) / 2
x = (-12 ± 20) / 2
This gives us two possible solutions for x:
x1 = (-12 + 20) / 2 = 8/2 = 4 inches
x2 = (-12 - 20) / 2 = -32/2 = -16 inches
Since the distance cannot be negative, we discard x2 = -16 inches as a valid solution.
Therefore, the value of x is 4 inches.
To find the new dimensions of the frame, we substitute x = 4 inches into the expressions we derived earlier:
New length = 8 inches + 4 inches = 12 inches
New width = 4 inches + 4 inches = 8 inches
Therefore, the new dimensions of the frame are 12 inches by 8 inches.