1. How many Cones in a Sphere with the same radius and height = 2r?
2. How many Cones in a Cylinder with the same radius and height?
3.How many spheres in a cylinder with the same radius and h = 2r?
5 Problems - SHOW ALL WORK!
1. What is the similarity and difference in finding the volume of cone, cylinder, and sphere?
2. Three friends are sharing a soda from a can. The dimensions of a soda can are shown below. If the three friends each drank an equal portion how much soda would each friend drink in terms of pi?
3. Doug has a bucket that is 12in tall and has a radius of 6in filled completely with water. Dough pushes a basketball with a diameter of 10in completely into the water overflowing the water in the bucket. How much water is left in the bucket?
4. Crystal was told by her mom she could get one scoop of candy. There are two scoops and both scoops are shaped like cones. Given the dimensions below which cone should Crystal use if she wants to get the greatest amount of candy?
Cone A: 30 cm width,15 cm height,15 cm radius
Cone B: 15 cm width,30 cm height, 7.5 cm radius
5. In your opinion, what is the most important thing to remember to do when solving problems involving volume?
1.
a. Cones in a Sphere with radius and height = 2r: In this case, the cone will have the same radius as the sphere and a height that is twice the radius. The volume of a cone is (1/3) * π * r^2 * h, so with a radius of r and height of 2r, the volume of the cone would be (1/3) * π * r^2 * 2r = (2/3) * π * r^3.
The volume of a sphere is (4/3) * π * r^3. So the number of cones that can fit in a sphere would be the volume of the sphere divided by the volume of one cone:
Number of cones = (4/3) * π * r^3 / ((2/3) * π * r^3) = 2.
b. Cones in a Cylinder with the same radius and height: The volume of a cone is (1/3) * π * r^2 * h and the volume of a cylinder is π * r^2 * h. So the number of cones that can fit in a cylinder would be the volume of the cylinder divided by the volume of one cone:
Number of cones = π * r^2 * h / ((1/3) * π * r^2 * h) = 3.
c. Spheres in a cylinder with the same radius and h = 2r: The volume of a sphere is (4/3) * π * r^3 and the volume of a cylinder is π * r^2 * h. So the number of spheres that can fit in a cylinder would be the volume of the cylinder divided by the volume of one sphere:
Number of spheres = π * r^2 * h / ((4/3) * π * r^3) = 3/4.
2.
a. The formula for the volume of a cone is Vcone = (1/3) * π * r^2 * h, where r is the radius and h is the height.
b. The formula for the volume of a cylinder is Vcylinder = π * r^2 * h.
c. The formula for the volume of a sphere is Vsphere = (4/3) * π * r^3.
The similarity is that they all involve the use of π and the formula for the volume of a shape includes some combination of the radius and height.
The difference is that the exponent on the radius and the height is different for each shape.
3.
The volume of the soda can is π * r^2 * h = (π * 2^2 * 6)/3 = 8π.
Each friend would drink 8π / 3 = 8/3 π.
4.
The volume of water displaced by the basketball is the same as the volume of the ball, so Vbasketball = (4/3) * π * (5)^3 = (4/3) * π * 125 = 500π/3.
The total volume of water in the bucket is π * 6^2 * 12 = 432π.
Therefore, the water left in the bucket after pushing the basketball in is 432π - 500π/3 = 732π/3.
5.
To always remember to use the correct formula for the volume of the shape and to ensure that the units in the problem are consistent with the formula being used. It is important to also pay attention to the dimensions given in the problem and to take into account any changes or displacements that occur.
answer all of the questions
Sure, here are the answers to all of the questions:
1.
a. Number of Cones in a Sphere with radius and height = 2r = 2.
b. Number of Cones in a Cylinder with the same radius and height = 3.
c. Number of Spheres in a Cylinder with the same radius and h = 2r = 3/4.
2.
a. Similarity in finding the volume of cone, cylinder, and sphere: They all involve π and the radius and height of the shape in their volume formulas.
b. Difference in finding the volume of cone, cylinder, and sphere: The exponent on the radius and height is different for each shape.
3. Each friend would drink 8/3π of the soda.
4. The water left in the bucket after pushing the basketball in is 732π/3 cubic inches.
5. Crystal should use Cone A if she wants to get the greatest amount of candy.
6. In my opinion, the most important thing to remember when solving problems involving volume is to carefully read and understand the given information, ensure the units are consistent, and use the correct formula for the volume of the specific shape in question. Accuracy in calculations and attention to detail are also important.
answer all 8 questions
1.
a. Number of Cones in a Sphere with radius and height = 2r: 2 cones can fit in a sphere.
b. Number of Cones in a Cylinder with the same radius and height: 3 cones can fit in a cylinder.
c. Number of Spheres in a Cylinder with the same radius and height: 3/4 spheres can fit in a cylinder.
2.
a. The similarity in finding the volume of a cone, cylinder, and sphere is that they all involve the radius and height in their respective volume formulas.
b. The difference is the exponents and coefficients present in the volume formulas for each shape.
3. Each friend would drink 8/3π of the soda.
4. The water left in the bucket after pushing the basketball in is 732π/3 cubic inches.
5. Crystal should use Cone A if she wants to get the greatest amount of candy.
6. In solving problems involving volume, it is crucial to:
- Carefully read and understand the given information.
- Ensure units are consistent.
- Use the correct formula for the volume of the specific shape.
- Pay attention to dimensions and any changes or displacements.
I hope these answers are helpful. Let me know if you need further assistance.