Question
Tim is an elementary school art teacher. His students are sculpting a replica of a shark out of clay. Tim has given them one block of clay to make 20 conical shark teeth for the sculpture. The block contains 81 cm3 of clay. If each tooth is solid and has a 2.5 cm base diameter, what is the maximum height each tooth can be? Assume that the students use all the clay.
Answers
GPT 3.5
First, let's calculate the volume of one conical tooth.
The volume of a cone can be calculated using the formula: V = 1/3 * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
Given that the base diameter is 2.5 cm, the radius (r) is half of the diameter, so r = 2.5/2 = 1.25 cm.
Since we know the volume of the clay block is 81 cm^3 and there are 20 teeth to be made, we can divide to find the volume allotted to one tooth: 81 cm^3 / 20 teeth = 4.05 cm^3 per tooth.
Now, let's solve for the height (h) of the cone:
4.05 cm^3 = 1/3 * π * (1.25 cm)^2 * h
4.05 cm^3 = (1/3) * π * 1.5625 cm^2 * h
4.05 cm^3 = 0.5236 cm^2 * h
h = 4.05 cm^3 / 0.5236 cm^2
h ≈ 7.73 cm
Therefore, the maximum height each tooth can be is approximately 7.73 cm.
The volume of a cone can be calculated using the formula: V = 1/3 * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
Given that the base diameter is 2.5 cm, the radius (r) is half of the diameter, so r = 2.5/2 = 1.25 cm.
Since we know the volume of the clay block is 81 cm^3 and there are 20 teeth to be made, we can divide to find the volume allotted to one tooth: 81 cm^3 / 20 teeth = 4.05 cm^3 per tooth.
Now, let's solve for the height (h) of the cone:
4.05 cm^3 = 1/3 * π * (1.25 cm)^2 * h
4.05 cm^3 = (1/3) * π * 1.5625 cm^2 * h
4.05 cm^3 = 0.5236 cm^2 * h
h = 4.05 cm^3 / 0.5236 cm^2
h ≈ 7.73 cm
Therefore, the maximum height each tooth can be is approximately 7.73 cm.