Solve this plz
The area of a rectangle is 1680msq.if it's diagonal is 58 m long find the length & width of the rectangle
L = 1680/W
L^2 + W^2 = 58^2
Substitute 1680/W for L in the second equation and solve for W. Insert that value into the first equation to solve for L. Check by putting both values into the second equation.
To solve this problem, we can use the properties of a rectangle and apply the Pythagorean theorem.
Let's start by assigning variables to the length and width of the rectangle. Let's say the length is L and the width is W.
The area of a rectangle is given by the formula: Area = Length × Width.
So, we have the equation: L × W = 1680.
The diagonal of a rectangle creates a right triangle with the length, width, and diagonal as the legs and hypotenuse, respectively. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (length and width).
Using this information, we can write the equation: L^2 + W^2 = diagonal^2.
Now let's substitute the given values into the equations.
Since we know that the diagonal of the rectangle is 58 meters long, we have:
L × W = 1680,
L^2 + W^2 = 58^2.
We can now solve this system of equations to find the length and width of the rectangle.
To solve it algebraically, we can use either substitution or elimination method. Let's use the substitution method.
From the first equation, we can express L in terms of W: L = 1680/W.
Substituting this into the second equation, we get:
(1680/W)^2 + W^2 = 58^2.
Simplifying the equation:
(1680^2)/W^2 + W^2 = 58^2.
Now, we can solve this equation for W. We multiply both sides by W^2 to eliminate the denominator:
1680^2 + W^4 = 58^2 * W^2.
Rearranging the equation:
W^4 - (58^2) * W^2 + (1680^2) = 0.
We now have a quadratic equation in terms of W^2. Let's solve this equation using the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a.
In our case, the coefficient of W^4 is 1, the coefficient of W^2 is -(58^2), and the constant term is (1680^2).
Plugging these values into the quadratic formula, we get:
W^2 = [-(58^2) ± sqrt((58^2)^2 - 4(1)(1680^2))] / 2(1).
Simplifying further:
W^2 = [-(58^2) ± sqrt(3364 - 4 * 1680^2)] / 2.
At this point, you can use a calculator to evaluate the square root and simplify the expression inside the square root.
Once you have the values for W^2, you can take the square root of those values to find the possible widths of the rectangle. Then, substitute those values back into the equation L × W = 1680 to find the corresponding lengths.
After following these steps, you will have the length and width of the rectangle.