Answer:
[tex]169[/tex] bags of sand would be required.
Step-by-step explanation:
Refer to the diagram attached. When viewed from above, the garden and the path form two circles with the same center.
The question states that the diameter of the garden (the inner circle) is [tex]38\; \rm yd[/tex]. Thus, the radius of this circle would be [tex](38\; \rm yd) / 2 = 19\; \rm yd[/tex].
The outer circle corresponds to outer rim of the path. The question states that the width of this path is [tex]5\; \rm yd[/tex]. There would thus be a [tex]5\; \rm yd\![/tex] gap between the two circles.
Hence, the radius of the outer circle would be [tex]5\; \rm yd \!\![/tex] larger than the radius (not diameter) of the inner circle for a total of [tex](5 \; {\rm yd} + 19\; {\rm yd}) = 24\; \rm yd[/tex].
The area of a circle of radius [tex]r[/tex] is [tex]\pi \, r^{2}[/tex]. The area of the outer circle (the garden and the path) would be [tex](24\; \rm yd)^{2}\, \pi[/tex], whereas the radius of the inner circle (the garden) would be [tex](19\; {\rm yd})^{2}\, \pi[/tex].
The area of the inner circle (the garden) plus the area of the ring between the two circles (the path) would give the area of the outer circle. That is:
[tex]\begin{aligned}& \text{Area of Garden} + \text{Area of Path} \\ &= \text{Area of Garden and Path}\end{aligned}[/tex].
Rearrange to get an expression for the area of the path. Evaluate to obtain:
[tex]\begin{aligned} & \text{Area of Path} \\ =\; & \text{Area of Garden and Path} \\ &-\text{Area of Garden} \\ =\; & \text{Area of Outer Circle} \\ &- \text{Area of Inner Circle} \\ =\; & (24\; {\rm yd})^{2}\, \pi - (19\; {\rm yd})^{2}\, \pi \\ \approx\; & 675\; {\rm yd^{2}}\end{aligned}[/tex].
The number of bags of sand required would be approximately [tex]169[/tex] (round up the result to the next integer.)
