Let the original length and width be $l$ and $w$. The original area is $lw=72$. After both dimensions increase by $3$, the new area is $(l+3)(w+3)=117$. Expanding gives $lw+3l+3w+9=117$. Since $lw=72$, substitute to get $72+3l+3w+9=117$, so $3l+3w=36$, and therefore $l+w=12$. The original perimeter is $2(l+w)=2(12)=24$. Now simplify choice B: $2\left(\frac{117-72}{3}\right)-12=2(15)-12=30-12=24$. Therefore, choice B represents the original perimeter.

Use the area formula for a circle, $A=\pi r^2$, for each garden.

Garden A: $\pi (3{)}^2=9\pi$

Garden B: $\pi (5{)}^2=25\pi$

Garden C: $\pi (8{)}^2=64\pi$

Total area of the three gardens:

The company will cover 40% of this total area, so multiply by $0.4$:

Therefore, the total area covered with mulch is $\boxed{39.2\pi }$ square meters.

The circle with center at the origin and radius $10$ has equation $x^2+y^2=100$. Since the chord lies on the line $y=6$, substitute $y=6$ into the equation:

So the line intersects the circle at $(-8,6)$ and $(8,6)$. The chord length is the horizontal distance between these points:

Therefore, the chord has length $16$.

Let the width of the flower bed be $w$ feet. Then its length is $w+4$ feet. Since the gravel border is 2 feet wide on each side, the outer dimensions are $(w+4)$ feet wider total and 4 feet longer total than the flower bed dimensions. So the outer width is $w+4$ and the outer length is $(w+4)+4=w+8$. The problem gives two area facts:

Flower bed area:

Outer total area:

Use the first equation:

Factor:

So $w=12$ or $w=-16$. A width must be positive, so $w=12$.

Check with the total area: the outer dimensions would be $16$ by $20$, and

which matches. Therefore, the width of the flower bed is $12$ feet.

First find the common area of the flower beds using the triangle area formula, $A=\frac{1}{2}bh$. From Bed 1,

square feet. This matches the others: Bed 2 gives $\frac{1}{2}(18)(10)=90$, and Bed 3 gives $\frac{1}{2}(20)(9)=90$.

So Bed 4 must also have area $90$ square feet. Let its height be $h$. Using its base of 24 feet,

so

The question asks for the amount of edging material, which is the perimeter, not the area. Bed 4 has base 24 feet and two equal slanted sides of 13 feet each, so its perimeter is

Therefore, the flower bed needs $50$ feet of edging material.