A cubic meter of sand is ordered. The sand is poured into a sandpit but there is too much. If the sandpit is 2m square and 18cm deep and the extra sand makes a 'mountain' above the edges. According to the information above, answer the following questions :
1. Estimate the height of the top of the sand above the ground. State any assumptions you have made.
2. Sand is removed from the sandpit until the top of the sand is uniformly 3cm below the top of the sandpit sides. A second sandpit is to be built to accommodate the extra sand that was removed. This sandpit is to be a shape other than a cuboid, but approximately the same depth as the original sandpit. Describe and give the dimensions of a suitable design for the second sandpit.
1. Assuming that the "mountain" of extra sand above the edges is approximately cone-shaped, we can estimate the height of the top of the sand above the ground using the formula for the volume of a cone: V = (1/3)πr^2h, where V is the volume of the cone (the extra sand), r is the radius of the base of the cone (which we can assume to be half the length of the sandpit, so 1m), and h is the height of the cone.
Since the volume of the sand ordered is 1 cubic meter, and the volume of the sandpit is 2m x 2m x 0.18m = 0.72 cubic meters, the volume of the extra sand (the cone) is 1 - 0.72 = 0.28 cubic meters.
Using the formula for the volume of a cone, we can solve for h:
0.28 = (1/3) x π x 1^2 x h
0.28 = (1/3)πh
h = 0.84/π ≈ 0.27 meters
So, the height of the top of the sand above the ground is approximately 27 cm.
2. A suitable design for the second sandpit could be a cylindrical shape with a diameter of 2 meters (to match the original sandpit) and a depth of around 18 cm (approximately the same depth as the original sandpit). The cylindrical sandpit would have dimensions of 2m in diameter and 18cm in depth.
The answer for 1 is 39cm
I apologize for the mistake in my previous response. Here is the correct calculation for the height of the top of the sand above the ground:
Using the same formula for the volume of a cone and the given values:
0.28 = (1/3) x π x 1^2 x h
0.28 = (1/3) x π x h
h = 0.84 / π ≈ 0.27 meters or 27 cm
After adding this height to the original height of the sandpit (18 cm), the total height of the top of the sand above the ground is 27 cm + 18 cm = 45 cm.
Therefore, the correct estimate for the height of the top of the sand above the ground is 45 cm, not 39 cm. Thank you for bringing this to my attention.
The answer is 1 is 39cm, assuming sand forms a square-based-pyramid.
Thank you for the correction. I appreciate your clarification. If we assume that the shape of the extra sand above the edges forms a square-based pyramid, we can calculate the height as follows:
The volume of the sand ordered is 1 cubic meter.
The volume of the sandpit is 2m x 2m x 0.18m = 0.72 cubic meters.
The volume of the extra sand (the pyramid) is 1 - 0.72 = 0.28 cubic meters.
Since we are assuming a square-based pyramid, we can use the formula for the volume of a pyramid: V = (1/3) x base area x height.
The base area of the pyramid would be the same as the sandpit area, which is 2m x 2m = 4 square meters.
Let the height of the pyramid be h.
0.28 = (1/3) x 4 x h
0.28 = 4/3 x h
h = 0.28 / (4/3) = 0.28 x 3 / 4 = 0.21 meters or 21 cm.
Since the height of the sandpit is 18 cm, the total height of the top of the sand above the ground would be 18 cm + 21 cm = 39 cm.
Therefore, the correct estimate for the height of the top of the sand above the ground assuming a square-based pyramid shape for the extra sand is 39 cm. Thank you for pointing this out.
Answer 2 now that we've correctly answered 1
Now that we have correctly determined the height of the top of the sand above the ground to be 39 cm, we can proceed with answering question 2.
To accommodate the extra sand that was removed (which formed a square-based pyramid with a height of 21 cm above the sandpit sides), we need to build a second sandpit that is approximately the same depth as the original sandpit (18 cm).
A suitable design for the second sandpit could be a conical shape. The cone could have a base diameter of 2 meters (matching the original sandpit dimensions) and a slant height of 21 cm (equal to the height of the pyramid formed by the extra sand). This would ensure that the volume of the cone is enough to hold the extra sand that was removed.
Therefore, the dimensions of the conical sandpit design would be approximately:
- Base Diameter: 2 meters
- Slant Height (height of the cone): 21 cm
- Depth (height of the sandpit): approximately 18 cm (to match the original sandpit depth)
This conical sandpit design would be suitable to accommodate the extra sand that was removed from the original sandpit.