A garden measuring 12m by 16m is to have a pedestrian pathway installed around it, increasing the total area to 285m^2.

Question

What will be the new width and length of the pathway? Include a diagram please.

Answer 1

Because the width of the pathway is uniform (the same) all around the length would be 16 plus 2x this dimension and the width would be 12 plus 2x this dimension.

Let x = width of pathway

16 + 2x = length of pathway

12 + 2x = width of pathway

A = area of pathway and garden = 285

(16 + 2x)(12 + 2x) = 285

192 + 32x + 24x +4x² = 285

4x² + 32x + 24x + 192 - 285 = 0

4x² + 56x - 93 = 0

Using the Quadratic Formula we can solve for x:

x = (-b ± √b² - 4ac)/2a

x = (-56 ± √(56² -4(4)(-93))/2(4)

x = (-56 ± √(3136 + 1488))/8

x = (-56 ± √4624)/8

x = (-56 ± 68)/8

x = -124/8, x = 12/8

Because x cannot be negative, the answer for x is 12/8 = 3/2 or 1.5. Therefore, the new length would be 16 + 2x = 16 + 2(1.5) = 16 + 3 =19m and the new width would be 12 + 2x = 12 + 2(1.5) = 12 + 3 = 15m. As a check 19 x 15 = 285--->285 = 285 CORRECT!

Hope this helps!

Answer 2

The area of the garden is 192.

the area of the walkway is then 93

(16+2x)(12+2x) = 285

192 + 32x + 24x + 4x^2 = 285

192 + 56x + 4x^2 = 285

4x^2 + 56x - 93 = 0

(2x + 31)(2x - 3 )=0

x = 3/2