The length of a rectangular sign is 3ft longer than the widt

The length of a rectangular sign is 3ft longer than the width. If the sign\'s area is 54sqft, find its length and width.

Use the pythagorean theorem and the square root property to solve, simplified radical form.

Solution

Let the width of the rectangular sign be x ft. Then its length is x + 3 ft. Since the sign\'s area is 54 sq. ft., we have x (x+3) = 54 or, x² + 3x - 54 = 0 or, x² + 9x - 6x - 54 = 0 or, x ( x + 9) - 6 (x + 9 ) = 0 or, ( x - 6) ( x + 9) = 0. Therefore, either x = 6 or x = -9. Now, since the width cannot be negative, x = 6 is the only acceptable solution. Therefore, the width and the length of the rectangular sign are 6 ft. and 9 ft. respectively.

Alternatively, using the square root property, we can proceed as under:

The area or the rectangular sign is x (x + 3) = 54 or, x² + 3x = 54. or, x² + 2* (3/2 ) x + (3/2)² = 54 + (3/2)² or,

(x + 3/2)² = 54 + 9/4 = 225/4 = (15/2)². Therefore, x + 3/2 = + 15/2 or - 15/2 . Therefore, x = 15/2 - 3/2 = 12/2 = 6 or, x = -15/2 - 3/2 = - 9. Since the width of the sign cannot be negative, the only acceptable value of x is 6. Therefore, the width and the length of the rectangular sign are 6 ft. and 9 ft. respectively.