Hey Bot I need your help?
The volume of a sphere is 48 m3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
m3
To find the volume of a cylinder, we need the formula for its volume, which is given by V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the radius of the cylinder is the same as the sphere's, and its height is equal to the sphere's diameter, we can use the following information:
- Radius of the sphere (r) = ?
- Height of the cylinder (h) = diameter of the sphere
The diameter of the sphere can be calculated by using the formula D = 2r, where D is the diameter.
Given that the volume of the sphere is 48 m^3, we need to find the radius of the sphere first. We can do this by rearranging the formula for the volume of a sphere, V = (4/3)πr^3, to solve for r:
V = (4/3)πr^3
48 = (4/3)πr^3
r^3 = 36/π
r ≈ 2.81 meters (rounded to two decimal places)
Now we know the radius of the sphere is approximately 2.81 meters, and the height of the cylinder is equal to the diameter of the sphere, which is 2 * 2.81 = 5.62 meters.
Finally, we can calculate the volume of the cylinder using V = πr^2h:
V = π * (2.81)^2 * 5.62
V ≈ 79.36 m^3 (rounded to two decimal places)
Therefore, the volume of the cylinder is approximately 79.36 m^3.
Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3 The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.
(4 points)
To find the total volume of ice cream in and on top of the cone, we need to consider the volume of the ice cream cone and the volume of the hemisphere.
Let's start with the volume of the cone. The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where V_cone is the volume of the cone, r is the radius of the cone, and h is the height of the cone.
Given that the diameter of the ice cream cone is equal to its height, we can say that r = h/2. Therefore, the formula for the volume of the cone becomes V_cone = (1/3)π(h/2)^2h.
Let's move on to the volume of the hemisphere. The volume of a hemisphere is given by the formula V_hemisphere = (2/3)πr^3, where V_hemisphere is the volume of the hemisphere, and r is the radius of the hemisphere.
Given that the diameter of the hemisphere is equal to the height of the cone, we can say that r = h/2. Therefore, the formula for the volume of the hemisphere becomes V_hemisphere = (2/3)π(h/2)^3.
Now, we are given that the volume of the hemisphere is 4 in^3. We can use this information to solve for the height of the cone. We'll substitute this height value into the formulas for the cone and hemisphere to find their volumes.
Let's substitute the height of the cone (h) into the formula for the volume of the hemisphere:
4 = (2/3)π(h/2)^3.
Simplifying this equation:
4 = (2/3)π(h^3/8).
12 = 2π(h^3/8).
6 = π(h^3/8).
48 = πh^3.
h^3 = 48/π.
h ≈ 3.67 inches (rounded to two decimal places).
Now that we have the height of the cone, we can calculate its volume using the formula V_cone = (1/3)π(h/2)^2h:
V_cone = (1/3)π(3.67/2)^2 * 3.67.
V_cone ≈ 13.08 in^3 (rounded to two decimal places).
Finally, to find the total volume of ice cream in and on top of the cone, we add the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere.
Total volume ≈ 13.08 + 4 = 17.08 in^3 (rounded to two decimal places).
Therefore, the total volume of ice cream in and on top of the cone is approximately 17.08 in^3.
Can you put this in an simple summary/ explanation
Certainly! In this problem, we are given an ice cream cone with a diameter and height that are equal to one another. A hemisphere of ice cream is placed on top of the cone. We are asked to find the total volume of ice cream in and on top of the cone.
To solve this, we use the formulas for the volume of a cone and the volume of a hemisphere. By knowing that the diameter of the cone is equal to its height, we can find the radius and height of the cone.
We are given the volume of the hemisphere, and by using that information, we can solve for the height of the cone. Once we know the height, we can calculate the volume of the cone using its formula.
Finally, we add the volume of the cone and the volume of the hemisphere to find the total volume of ice cream in and on top of the cone.
In summary, the total volume of ice cream in and on top of the cone is found by calculating the volumes of the cone and hemisphere separately and then adding them together.