Question

The width of a rectangle is 9 less than twice its length. If the area of the rectangle is 61 cm. what is the length of the diagonal?

Answer 1

l(2l - 9) = 61

2l² - 9l - 61 = 0

l = 9 ± √((-9)² - 4(2)(-61)))/(2(2))

= (9 ± √(81 + 488))/4

= (9 + √569)/4

w = 2((9 + √569)/4) - 9

= (1/2)(9 + √569) - 9

= (-9 + √569)/2

l² = ((9 + √569)/4)²

= (81 + 18√569 + 569)/16

= (650 + 18√569)/16

= (325 + 9√569)/8

w² = ((-9 + √569)/2)

= (81 - 18√569 + 569)/4

= (650 - 18√569)/4

= (325 - 9√569)/2

l² + w²

= (325 + 9√569 + 4(325 - 9√569))/8

= (5(325) - 27√569)/8

= (1,625 - 27√569)/8

= (3,250 - 54√569)/16

Length of diagonal

= (1/4)√(3,250 - 54√569)) = about 11.07 cm

Answer 2

The length of the diagonal of the rectangle is approximately 8.60 cm.

Step-by-step explanation:

To find the length of the diagonal of a rectangle, we can use the Pythagorean theorem, which states that in a right 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 (the length and width of the rectangle).

Let's assume the length of the rectangle is x. According to the given information, the width is 9 less than twice the length, so the width can be represented as 2x - 9.

The area of the rectangle is given as 61 cm², so we can set up the equation: x(2x - 9) = 61.

Simplifying the equation, we get 2x² - 9x - 61 = 0.

Using factoring or the quadratic formula, we find that the solutions for x are x = 7 and x = -4. Since a negative length is not possible, we discard -4 and conclude that the length of the rectangle is 7 cm.

Now, we can use the length and width to find the diagonal. Using the Pythagorean theorem, we have: diagonal² = length² + width² = 7² + (2(7) - 9)² = 49 + 5² = 74.

Taking the square root of both sides, we find that the length of the diagonal is √74 cm, which is approximately 8.60 cm.