To create a fence around a yard, you generally need fence posts and fence panels (either ones you make or pre-made ones). One recommendation is to dig a hole one third the length of the fence post deep into the ground. Another recommendation is to dig the hole so that the diameter of the hole is three times the with of the post. Another recommendation is to use concrete to fill the hole.
1.
A person is digging a hole for the fence post. The post is 8 feet
long. The recommendation is that the post hole be one third the length
of the post deep. How deep should the post hole be?
We are asked to calculate \(\frac{1}{3}\)rd of \(8\).
\[\frac{1}{3} \cdot 8 = \frac{8}{3} = 2 \frac{2}{3} \textbf{ feet}\]
2.
A person is using fence posts that are square bases. Each side of
the base is 3.5 inches. They are advised to make the diameter of the
whole to be three times the width of the post. What should be the
diameter of the hole?
We are asked to calculate three times the side of the base.
\[3.5 \cdot 3 = 10.5 \textbf{ inches}\]
3.
50 pounds of fast setting concrete makes 0.375 cubic feet when
mixed with water.
A fence post hole is suggested to be 3 times the width of the post. The hole should be one third of the length of the fence post.
A person has fence posts that are 3.5 inches in diameter and 8 feet tall. How many 50 pound bags of concrete do they need to purchase to fill 20 fence post holes that meet this criteria?
From our calculations above, we know the hole should be \(2 \frac{2}{3} \textbf{ feet}\) deep and \(10.5 \textbf{ inches}\) in diameter. The hole can be thought of as a right cylinder. The volume \(v\) of a right cylinder is
\[v = \pi r^2 h\]
We are told that one bag of cement makes \(0.375 \textbf{ ft}^3\). We can calculate the volume of the hole and divide that volume by the volume filled per bag to calculate the number of bags. As our units are in feet, we should convert \(10.5 \textbf{ inches}\) to feet first.
\[\frac{10.5 \textbf{ inches}}{12 \textbf{ inches per foot}} = 0.875 \textbf{ feet}\]
This is the diameter of the hole. The formula for volume needs the radius. Divide the diameter by 2 to get the radius.
\[r = \frac{0.875}{2} \textbf{ feet} = 0.4375 \textbf{ feet}\]
Substitute the radius and the height to calculate the volume.
\[v = \pi r^2 h\]
\[v = \pi (0.4375)^2 (2 \frac{2}{3})\]
\[v = \pi (0.19140625) (2 \frac{2}{3})\]
\[v \approx \pi (0.19140625) (2.6666667)\]
\[v \approx 1.60352 \textbf{ ft}^3\]
The hole, without the fence post, needs \(1.60352 \textbf{ ft}^3\) of cement each. However, we’re putting in a fence post that has a square base. That will take up some of the volume. We need to calculate the volume of the fence post and subtract this from the volume needed to fill the hole.
The fence post has a square base of 3.5 inches. That’s \(\frac{3.5 \textbf{ inches}}{12 \textbf{ inches}} = 0.2917\) feet. The height is 2 and 2/3rds feet. That means the volume of the fence post is
\[0.2917 \textbf{ feet} \cdot 0.2917 \textbf{ feet} \cdot 2 \frac{2}{3} \textbf{ feet} = 0.2269 ft^3\]
Subtract the volume of the fence post from the volume of the hole.
\[1.60352 \textbf{ ft}^3 - 0.2269 ft^3 = 1.3766 ft^3\]
We are told that one bag fills \(0.375 \textbf{ ft}^3\), or \(\frac{1\textbf{ bag}}{0.375 \textbf{ ft}^3}\). Multiply this to the volume of our hole so that the cubed feet units cancel.
\[1.3766 \textbf{ ft}^3 \cdot \frac{1\textbf{ bag}}{0.375 \textbf{ ft}^3} \approx 3.67 \textbf{ bags}\]
We need a little over three and one-half bags per hole. If there are 20 fence posts, we need
\[3.67 \textbf{ bags} \cdot 20 = 73.4 \textbf{ bags}\]
1. A person is digging a hole for the fence post. The post is 10 feet long. The recommendation is that the post hole be one third the length of the post deep. How deep should the post hole be?
2. A person is using fence posts that are square bases. Each side of the base is 5.5 inches. They are advised to make the diameter of the whole to be three times the width of the post. What should be the diameter of the hole?
3. 50 pounds of fast setting concrete makes 0.375 cubic feet when mixed with water.
A fence post hole is suggested to be 3 times the width of the post. The hole should be one third of the length of the fence post.
A person has fence posts that are 5.5 inches in diameter and 10 feet tall. How many 50 pound bags of concrete do they need to purchase to fill 4 fence post holes that meet this criteria?
1. The best depth of the fence post depends on the climate. In general, should a fence post hole be deeper in the northern half or in the southern half of the United States? Why?
2. What other considerations should be taken into account with the calculation of the volume of the hole?