Question

Maja manages a community garden which has rectangular garden plots that are 15 ft by 10 ft. She wants to add a uniform path around the garden plot so that the combined area of the path is 350 square feet. How wide, in feet, should the path be?

Answer 1

The path should be 6.38 feet wide

Let x be the width of the path.

Since the rectangular plot has an area with dimensions 15 ft by 10 ft, we add x to each dimension to give the dimensions of the rectangular plot plus uniform path.

So, the dimensions are 15 + x by 10 + x respectively.

So, the area of the rectangular plot plus uniform path is A

A = (15 + x)(10 + x)

Since the combined area of the path is 350 square feet, A = 350 ft²

So, (15 + x)(10 + x) = 350

Expanding the brackets, we have

150 + 15x + 10x + x² = 350

x² + 25x + 150 = 350

x² + 25x + 150 - 350 = 0

x² + 25x - 200 = 0

Using the quadratic formula, we find x, the width of the path

So,

$x = \frac{-25 +/-\sqrt{25^{2} - 4X1 X (-200)} }{2 X1 } \\x = \frac{-25 +/-\sqrt{625 + 800)} }{2}\\x = \frac{-25 +/-\sqrt{1425} }{2}\\x = \frac{-25 +/-37.75 }{2}\\x = \frac{-25 + 37.75 }{2} or x = \frac{-25 - 37.75 }{2}\\x = \frac{12.75 }{2} or x = \frac{-62.75 }{2}\\x = 6.375 or x = -31.375$

Since x cannot be negative, we choose the positive answer.

x = 6.375 ft

x ≅ 6.38 ft

So, the path should be 6.38 feet wide

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