suppose you want to build a rectangular sandbox where the width is 4 ft less than the length and the diagonal is 4 ft longer than the length. what are the dimensions
(x+4)^2 = x^2 + (x-4)^2
x^2 + 8 x + 16 = x^2 + x^2 - 8 x + 16
x^2 -16 x = 0
x = 0 or x = 16
so x = 16
12, 16, 20
Suppose you want to build a rectangular picture frame where the width is 2 centimeters less than the length and the diagonal is 2 centimeters longer than the length. What are the dimensions of the picture frame?
To find the dimensions of the rectangular sandbox, let's assume the length of the sandbox is represented by 'L' feet.
According to the given information, the width is 4 feet less than the length. So, the width can be expressed as (L - 4) feet.
The diagonal of a rectangle can be found using the Pythagorean theorem, which states that the square of the length plus the square of the width equals the square of the diagonal. In this case, the diagonal is 4 feet longer than the length. Thus, the diagonal can be expressed as (L + 4) feet.
Using the Pythagorean theorem, we can set up the equation:
(L^2) + (L - 4)^2 = (L + 4)^2
Expanding the equation:
L^2 + (L^2 - 8L + 16) = L^2 + 8L + 16
Simplifying the equation:
2L^2 - 8L + 16 = L^2 + 8L + 16
Moving all terms to one side:
2L^2 - L^2 - 8L + 8L = 0
Combining like terms:
L^2 = 0
Taking the square root of both sides:
L = 0 or L = 0
Since length cannot be zero, we discard the L = 0 solution.
Therefore, the length of the sandbox is undetermined based on the information provided.
To find the dimensions of the sandbox, let's assume the length of the sandbox is represented by "L" in feet.
According to the given information, the width of the sandbox is 4 feet less than the length, so we can express the width as: L - 4.
The diagonal of the rectangle is 4 feet longer than the length. We can use the Pythagorean theorem to relate the length, width, and diagonal of a rectangle. The theorem states that the square of the diagonal is equal to the sum of the squares of the other two sides.
In this case, the length and width are two sides of the rectangle, and the diagonal is the hypotenuse. So, we have the equation:
(L^2) + (L - 4)^2 = (L + 4)^2
Expanding and simplifying the equation:
L^2 + L^2 - 8L + 16 = L^2 + 8L + 16
Rearranging the terms:
2L^2 - 16L + 16 = L^2 + 8L + 16
Combining like terms:
L^2 - 24L = 0
Factoring out L:
L(L - 24) = 0
Setting each factor equal to zero:
L = 0 or L - 24 = 0
Since a length of zero wouldn't make sense in this context, we discard that solution.
Therefore, L - 24 = 0, so L = 24.
Now we have determined that the length of the sandbox is 24 feet. To find the width, we substitute this value back into L - 4:
Width = L - 4 = 24 - 4 = 20 feet.
Hence, the dimensions of the sandbox are 24 feet by 20 feet.