The dimensions of a rectangular photograph are even consecutive integers. The area of the photograph is 224square inches. Find the lengths of each side.
x (x + 2) = 224 ... x^2 + 2 x - 224 = 0
solve for x
the dimensions of a rectangular photograph are even consecutive integers. the area of the photograph is
168
square
inches
.
find the lengths of each side.
Let's assume the consecutive even integers representing the dimensions of the photograph are n and (n + 2).
The area of the photograph is given as 224 square inches.
We can express the area as the product of the dimensions:
n * (n + 2) = 224
Expanding the equation:
n^2 + 2n = 224
Rearranging the equation to form a quadratic equation:
n^2 + 2n - 224 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Since we have -224 as the constant term, let's try factoring.
Factoring the quadratic equation:
(n + 16)(n - 14) = 0
Setting each factor equal to zero:
n + 16 = 0 or n - 14 = 0
Solving for n in each case:
n = -16 or n = 14
Since the dimensions cannot be negative, we discard n = -16.
Thus, the value of n is 14.
Therefore, the consecutive even integers representing the dimensions of the photograph are 14 and (14 + 2), which are 14 and 16, respectively.
Hence, the lengths of each side of the photograph are 14 inches and 16 inches.
To find the lengths of each side of the rectangular photograph, we can use the given information that the area of the photograph is 224 square inches.
Let's assume the smaller dimension is x. Since the dimensions are even consecutive integers, the larger dimension will be (x + 2).
The area of a rectangle is calculated by multiplying the length and width. In this case, the area is 224 square inches. So, we can set up the equation:
x * (x + 2) = 224
Simplifying the equation, we have:
x^2 + 2x - 224 = 0
Now, we need to solve this quadratic equation. We can either factor it or use the quadratic formula:
Factoring:
(x + 16)(x - 14) = 0
From this, we can see that either (x + 16) = 0 or (x - 14) = 0.
However, the dimensions of a rectangle cannot be negative, so we exclude the solution (x - 14) = 0.
Therefore, x + 16 = 0, which gives us x = -16.
Again, we discard this solution since the dimensions cannot be negative.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 2, and c = -224.
Plugging in these values, we get:
x = (-(2) ± √((2)^2 - 4(1)(-224))) / (2(1))
x = (-2 ± √(4 + 896)) / 2
x = (-2 ± √900) / 2
x = (-2 ± 30) / 2
Now, we have two possible values for x:
x = (-2 + 30) / 2 = 28 / 2 = 14
x = (-2 - 30) / 2 = -32 / 2 = -16
Again, we discard the negative solution, so the smaller dimension is x = 14.
The larger dimension will be (14 + 2) = 16.
Therefore, the lengths of each side of the rectangular photograph are 14 inches and 16 inches.