Question

a gardener has 85 feet of fencing to be used to enclose a rectangular garden that has a border 2 - foot wide surrounding it (see the figure). use this information to answer the following. (b) the width of the garden is 11.5 feet. (round to the nearest tenth as needed.) what is the area of the garden? area = 264.5 square feet (round to the nearest tenth as needed.) (c) if the length and width of the garden are to be the same, what would be the dimensions of the garden? the length of the garden is 17.3 feet. (round to the nearest tenth as needed.) the width of the garden is 17.3 feet. (round to the nearest tenth as needed.) (d) what would be the area of the square garden? area = (round to the nearest tenth as needed.)

Explanation:

Step1: Recall area formula for rectangle

The area formula for a rectangle is $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width.

Step2: Solve for part (b)

Given $w = 11.5$ feet and $A=264.5$ square - feet. We can find $l$ using the formula $l=\frac{A}{w}$. Substitute the values: $l=\frac{264.5}{11.5}=23$ feet.

Step3: Solve for part (c)

If $l = w$, and the perimeter of the rectangle (including the 2 - foot border) is related to the available fencing. Let the side length of the square - shaped garden be $x$. The perimeter of the outer rectangle (including the border) is $P = 2(x + 4)+2(x + 4)$ (since there is a 2 - foot border on each side). But we are not using the perimeter information here. Since $A = l\times w$ and $l = w$, then $A=x^{2}$. We know from part (b) the area formula. If $l = w$, and $A = 264.5$ square feet, then $x=\sqrt{264.5}\approx16.2$ feet.

Step4: Solve for part (d)

If the garden is square - shaped and we want to find the area, and we know from part (c) that when the length and width are equal and the area is based on the previous non - square rectangle's area concept, if we assume the side of the square is $s$, and we know that for a square $A=s^{2}$. Since we found in part (c) that the side length of the square - equivalent garden (in terms of area) is approximately $16.2$ feet, then $A=(16.2)^{2}=262.44\approx262.4$ square feet.

Answer:

(b) Length of the garden is 23.0 feet.
(c) Dimensions of the square - shaped garden are approximately 16.2 feet by 16.2 feet.
(d) Area of the square garden is approximately 262.4 square feet.