A rectangular garden has a dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built ar

Question

A rectangular garden has a dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet?

Answer 1

area of path ow width w is

long sides: 2*18w

short sides: 2*14w

4 corners: 4*w²

add them up

516 = 2*18w + 2*14w + 4*w²

divide by 4

129 = 9w + 7w + w²

w² + 16w - 129 = 0

quadratic equation:

x = [-b ±√(b²-4ac)] / 2a

w = [-16 ±√(16²+4*129)] / 2

w = [-16 ±√(256+516)] / 2

w = [-16 ± 27.6)] / 2

skipping the negative answer

w = 5.8 feet.

.

Answer 2

I would say the path can be up to 12ft wide with 516sqft of gravel.

its common sense the area of the garden is 234sqft

so if it was 6ft like the other answer the outside dimensions would be 24x19 which is 456sq ft. so even if you were to fill the whole area in, it still wouldnt be 516sqft

If you were to use all 516sq ft of gravel, the outside dimensions around the path would be 25x30 wihch is 750sq ft

750-234 is 516

so you have your answer 12 ft wide!

Answer 3

8.3225inches wide

Answer 4

From the garden's size we know the perimeter, or length of the path:

P = 2*(18 + 13) = 62ft

Now to figure out the width of the path we have to do some geometry.  There are 3 basic shapes we will use for the path, they are the 13 ft long rectangle, the 18 ft long rectangle, and the squares formed at the corners. These squares' areas are equal to the width squared, whereas the rectangles are only equal to the length times the width.  Our total area then becomes:

A = 2*(13*w) + 2*(18*w) + 4*(w*w)

A = 4w^2 + 62w

And since we know the available area is 516 ft^2, putting this in for A and solving the quadratic equation we can get the width:

516 = 4w^2 + 62w

0 = 4w^2 + 62w - 516

Using the quadratic formula: (w = -b2+/-SQRT(b^2 - 4ac)/2a, where a = 4, b = 62, and c = -516)

w = 6, -21.5

Since we obviously cannot have a negative width, 6 must be our answer.  Therefore the path can be 6 feet wide!

Edit:  We can check this answer by doing the following.  Please note that by having a 6ft wide path you would actually be adding 12 ft to both dimensions, since you add 6 feet of length to both sides.  Your total area would then become:

18+(2*6) x 13+(2*6) = 30x25 = 750 ft^2

Subtracting the area of the garden gives you the are of the gravel required:

750ft^2 - (18*13)ft^2 = 516ft^2

Therefore a width of 6 feet is correct.