Question

The length of a rectangle is 3 m less than the diagonal and the width is 8 m less than the diagonal. If the area is 74 m^2, how long is the diagonal in meters? Round your answer to the nearest tenth.​

Answer 1

Explanation:

The diagram of the rectangle, ABCD is shown in the attached photo. The diagonal of the rectangle forms a triangle, ABC

Applying Pythagoras theorem,

d^2 = (d - 8)^2 + (d - 3 )^2

d^2 = d^2 - 16d + 64 + d^2 - 6d + 9

d^2 = 2d^2 - 22d + 73

d^2 - 22d + 73 = 0

d^2 = 22d - 73 - - - - - - 1

If the area is 74 m^2, it means that

(d- 8)(d- 3) = 74

d^2 - 11d + 24 = 74

d^2 = 74 - 24 + 11d

d^2 = 50 + 11d - - - - - - - -2

Equating equation 1 and 2, it becomes

22d - 73 = 50 + 11d

22d - 11d = 50 + 73

11d = 123

d = 123/11 = 11.182

diagonal = 11.2 m to the nearest tenth.

Answer 2

14.5 m

Explanation:

Let x represent the length of the diagonal. Then the length of the rectangle is (x-3) and its width is (x-8). The area is the product of these, so is ...

(x -3)(x -8) = 74

x^2 -11x +24 = 74 . . . . eliminate parentheses

x^2 -11x = 50 . . . . . . . .subtract 24

x^2 -11x +30.25 = 80.25 . . . . add 30.25 to complete the square

(x -5.5)^2 = 80.25 . . . . . . write as square

x - 5.5 = √80.25 ≈ 8.958 . . . . take the square root

x = 8.958 + 5.5 = 14.458 . . . . .add 5.5

The length of the diagonal is about 14.5 meters.