The area of a rectangle is 2160 square units and the diagonal is 78 units long. What are the length and width of the rectangle?

I know the width and length are 30 and 72, but I am having trouble with the actual work.

Mr. Robredo is building a fence around a rectangular region with 36 feet of fencing. The area of the rectangular region is 65 square feet. What are the length and the width of the rectangular region? Enter your answers in the boxes.

Whats the answer?????

5896

To find the length and width of the rectangle, we can use the formula for the area of a rectangle and the Pythagorean theorem to solve for the unknown sides.

Let's start by setting up the formula for the area of a rectangle:

Area = length × width

Given that the area is 2160 square units, we have:

2160 = length × width [Equation 1]

Next, we can use the Pythagorean theorem to relate the length, width, and diagonal of the rectangle. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the other two sides (in this case, the length and width).

Let's represent the length as 'l', the width as 'w', and the diagonal as 'd'. Using the Pythagorean theorem, we have:

l² + w² = d² [Equation 2]

Given that the diagonal is 78 units long, we have:

l² + w² = 78² [Equation 3]

Now we have a system of equations consisting of Equation 1 and Equation 3.

To solve this system of equations, we can substitute Equation 1 into Equation 3:

(l × w) + w² = 6084 [Replacing l² in Equation 3 with (l × w) from Equation 1]

Now, we can solve this quadratic equation. Rearranging the equation, we get:

w² + w × l - 6084 = 0

To find the values of l and w, we can use factoring or the quadratic formula. Factoring this quadratic equation might be a bit more complicated, so in this case, let's use the quadratic formula:

w = [-b ± √(b² - 4ac)] / (2a)

Plugging in the values into the quadratic formula, we have:

w = [-(1) ± √((1)² - 4(1)(-6084))] / (2(1))

Simplifying further, we get:

w = [-1 ± √(1 + 24336)] / 2

w = [-1 ± √24437] / 2

Now, calculating the value under the square root:

√24437 ≈ 156.263

Using this value, we can calculate two possible values for the width:

w₁ = (-1 + 156.263) / 2 ≈ 77.632

w₂ = (-1 - 156.263) / 2 ≈ -78.632

Since width cannot be negative in this context, we discard the negative value.

Now that we have the width, we can substitute it back into Equation 1 to solve for the length:

2160 = length × 77.632

length ≈ 2160 / 77.632 ≈ 27.831

Therefore, the approximate length and width of the rectangle are 27.831 units and 77.632 units, respectively.