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)
Make it simple please
Ok great but can you just solve the questions and then explain why it is 16/3 in^3.
Make it simple please
Ok now put it into a paragraph and show the work as well.
explain in a short paragraph
yo make it shorter
Make it super short!
To solve this problem, we can use the relationship between the formulas for the volumes of cones and spheres.
The volume of a cone is given by the formula Vcone = (1/3)πr^2h, where r is the radius of the cone's base and h is the height of the cone.
The volume of a sphere is given by the formula Vsphere = (4/3)πr^3, where r is the radius of the sphere.
In this problem, the diameter of the cone is equal to its height. Let's assume the diameter and height of the cone are both d.
The radius of the cone's base (and also the height) is equal to d/2.
We know that the volume of the hemisphere on top of the cone is 4 in^3. The volume of a whole sphere with radius d/2 is (4/3)π(d/2)^3.
Since the volume of the hemisphere is half the volume of the whole sphere, we can write the equation (4/3)π(d/2)^3 / 2 = 4.
Simplifying this equation, we get (4π/3) * (d/2)^3 = 4.
Now, let's solve for d.
Divide both sides of the equation by (4π/3) to isolate (d/2)^3: (d/2)^3 = 4 / (4π/3).
The coefficient (4π/3) simplifies with 4: (d/2)^3 = 3 / π.
Take the cube root of both sides of the equation to solve for d/2: d/2 = (3 / π)^(1/3).
Multiply both sides of the equation by 2 to solve for d: d = 2 * (3 / π)^(1/3).
Now that we know the value of d, we can calculate the volume of the cone using the formula Vcone = (1/3)πr^2h.
The radius of the cone's base (and also the height) is d/2, so let's substitute the value into the formula: Vcone = (1/3)π((d/2)^2)(d/2).
Simplifying this equation, we get Vcone = (1/3)π(d^2/4)(d/2).
Multiplying the fractions, we have Vcone = (π/12)d^3.
Substituting the value of d, we get Vcone = (π/12)(2 * (3 / π)^(1/3))^3.
Simplifying further, we get Vcone = (π/12)(8 * (3 / π)).
Multiplying, we have Vcone = 2/3 * 8 = 16/3.
Therefore, the total volume of ice cream in and on top of the cone is 16/3 in^3.
Sure! Here's a simpler explanation:
- The volume of the hemisphere on top of the cone is 4 in^3.
- The diameter of the cone is equal to its height.
- Let's assume the diameter and height of the cone are both d.
- The radius of the cone's base (and also the height) is equal to d/2.
- The volume of a sphere with radius d/2 is (4/3)π(d/2)^3.
- Since the volume of the hemisphere is half the volume of the sphere, we can write the equation (4/3)π(d/2)^3 / 2 = 4.
- Simplifying the equation gives (d/2)^3 = 3 / π.
- Solve for d/2 by taking the cube root, which gives d/2 = (3 / π)^(1/3).
- Multiply both sides by 2 to solve for d: d = 2 * (3 / π)^(1/3).
- Use the formula for the volume of a cone, Vcone = (1/3)πr^2h, where r is the radius and h is the height.
- The volume of the cone is Vcone = (1/3)π((d/2)^2)(d/2).
- Simplifying this equation, we get Vcone = (1/3)π(d^2/4)(d/2).
- Substituting the value of d, we get Vcone = (π/12)(2 * (3 / π)^(1/3))^3.
- Simplifying further, we get Vcone = 16/3.
- Therefore, the total volume of ice cream in and on top of the cone is 16/3 in^3.
Sure! Here's a simplified explanation with the steps shown:
To find the total volume of ice cream in and on top of the cone, we start by knowing that the volume of the hemisphere on top of the cone is 4 in^3. Since the diameter of the cone is the same as its height, let's assume that the diameter and height of the cone are both d. The radius of the cone's base (and also its height) is equal to d/2.
To find the volume of the hemisphere, we use the formula for the volume of a sphere, which is (4/3)πr^3. Since the hemisphere is half of a sphere, the volume of the hemisphere is (4/3)π(d/2)^3.
Setting this volume equal to 4 in^3, we have (4/3)π(d/2)^3 = 4.
Simplifying this equation, we find (d/2)^3 = 3/π.
To solve for d/2, we take the cube root of both sides: d/2 = (3/π)^(1/3).
Multiplying both sides by 2 gives us d = 2 * (3/π)^(1/3).
Now that we know the value of d, we can find the volume of the cone using the formula for the volume of a cone, which is (1/3)πr^2h.
Substituting the radius of the cone's base (d/2) into the formula, we have Vcone = (1/3)π((d/2)^2)(d/2).
Simplifying this equation, we get Vcone = (1/3)π(d^2/4)(d/2).
Substituting the value of d, we have Vcone = (π/12)(2 * (3/π)^(1/3))^3.
Simplifying further, we get Vcone = 16/3.
Therefore, the total volume of ice cream in and on top of the cone is 16/3 in^3.
Certainly! Here's the solution and explanation:
To find the total volume of ice cream in and on top of the cone, we need to find the individual volumes of the cone and the hemisphere, and then add them together.
First, let's find the volume of the cone. Since the diameter of the cone is equal to its height, let's assume that the diameter and height of the cone are both 'd'. The radius of the cone's base (and also its height) is equal to d/2.
The formula for the volume of a cone is Vcone = (1/3)πr^2h, where r is the radius of the cone's base and h is the height of the cone.
Substituting the radius (d/2) and height (d) into the formula, we have:
Vcone = (1/3)π(d/2)^2(d) = (1/3)π(d^2/4)(d) = (π/12)d^3.
Next, let's find the volume of the hemisphere. Given that its volume is 4 in^3, we know that the volume of a whole sphere with radius d/2 would be (4/3)π(d/2)^3. However, we only have half of a sphere, so the volume of the hemisphere would be half of that.
Setting up the equation, we have:
(1/2)(4/3)π(d/2)^3 = 4 in^3.
(2/3)π(d/2)^3 = 4 in^3.
Now, let's solve for 'd'. We'll divide both sides by (2/3)π and take the cube root:
(d/2)^3 = (4 in^3) / [(2/3)π].
(d/2)^3 = (6/π) in^3.
Taking the cube root of both sides, we get:
d/2 = [(6/π)^(1/3)] in.
Multiplying both sides by 2, we have:
d = 2 * [(6/π)^(1/3)] in.
Now, let's substitute the value of 'd' into the volume of the cone equation:
Vcone = (π/12)d^3 = (π/12)[2 * (6/π)^(1/3)]^3.
Simplifying this further, we get:
Vcone = (π/12)(8 * (6/π)^(1/3))^3.
Vcone = (π/12)(8 * (6/π)^(1/3))(8 * (6/π)^(1/3))(8 * (6/π)^(1/3)).
Vcone = (8/3)(6/π)(6/π)(6/π) = (384/π^3) in^3.
Adding the volumes of the cone and hemisphere, we have:
Vtotal = Vcone + Vhemisphere = (384/π^3) + (4 in^3) = (384/π^3) + (4/1)(π^3/π^3) = (384 + 4π^3)/π^3 in^3.
Therefore, the total volume of ice cream in and on top of the cone is (384 + 4π^3)/π^3 in^3.