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. Make it simple.
Since the diameter of the ice cream cone is equal to its height, we can use the cone volume formula: V = (1/3)πr^2h, where r is the radius and h is the height of the cone.
Let's first find the radius of the cone. The diameter of the cone is equal to its height, so the radius (r) is half the height (h). Since the diameter is r + r = 2r = h, we can solve for r by dividing the height (h) by 2: r = h/2.
Now, let's substitute the value of the radius (r) into the cone volume formula. The volume of the cone is given as 4 in^3, so we have:
4 = (1/3)π(h/2)^2h [Substituting the values of V = 4 in^3 and r = h/2]
Multiplying both sides by 3 to get rid of the fraction:
12 = π(h/2)^2h [Multiplying both sides by 3]
Simplifying the equation:
12 = π(h^2/4)h [Simplifying (h/2)^2 to h^2/4]
Multiplying both sides by 4 to get rid of the fraction:
48 = πh^3 [Multiplying both sides by 4]
Dividing both sides by π to solve for h:
h^3 = 48/π [Dividing both sides by π]
Now, we can find the value of h by taking the cube root of both sides:
h = (48/π)^(1/3) [Taking the cube root of both sides]
Using a calculator, we find that h ≈ 2.710.
Now, let's find the radius (r):
r = h/2 ≈ 2.710/2 ≈ 1.355.
The volume of the cone is given as 4 in^3, so we can find the total volume of ice cream by adding the volume of the cone and the perfect hemisphere on top:
Total volume = Cone volume + Hemisphere volume
= (1/3)πr^2h + (2/3)πr^3 [Using the formulas for the volumes of cones and spheres]
Now, substituting the values of r and h:
Total volume = (1/3)π(1.355)^2(2.710) + (2/3)π(1.355)^3 [Substituting r ≈ 1.355 and h ≈ 2.710]
Using a calculator to compute the values and simplifying, we find that the total volume of ice cream in and on top of the cone is approximately 9.056 in^3.
Shorter plz.
The volume of the cone can be found using the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone. Since the diameter of the cone is equal to its height, the radius is half the height.
Let h be the height of the cone. Then, the radius of the cone is r = h/2.
The volume of the cone is given as 4 in^3, so we can substitute the values in the volume formula and solve for h:
4 = (1/3)π(h/2)^2h.
Simplifying the equation, we find h^3 = 48/π.
Taking the cube root of both sides, we get h ≈ 2.710.
The radius of the cone is r = h/2 ≈ 1.355.
The total volume of ice cream is the sum of the cone volume and the volume of the hemisphere on top, which can be found using the formulas for the volumes of cones and spheres:
Total volume = (1/3)πr^2h + (2/3)πr^3.
Substituting the values of r ≈ 1.355 and h ≈ 2.710, we can compute the total volume to be approximately 9.056 in^3.
Can you make it 125 Words plz.
Sure! Here's a concise explanation in 125 words:
To find the total volume of ice cream in and on top of the cone, we use the formulas for the volumes of cones and spheres. The ice cream cone has a diameter equal to its height, so the radius is half the height. We can set up an equation using the volume formula for the cone. Solving for the height, we find it to be approximately 2.710. By dividing the height by 2, we calculate the radius to be approximately 1.355. Adding the volume of the cone (using the cone volume formula) and the volume of the hemisphere (using the sphere volume formula), we find the total volume of ice cream to be approximately 9.056 in^3.
To find the total volume of ice cream in and on top of the cone, we need to calculate the volume of both the cone and the hemisphere of ice cream and then add them together.
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where V represents volume, π is a constant (approximately 3.14), r is the radius of the base of the cone, and h is the height of the cone.
Since the diameter of the cone is equal to its height, we can say that the radius of the cone is half of the diameter.
The volume of a hemisphere is given by the formula V = (2/3) * π * r^3.
Given that the hemisphere of ice cream has a volume of 4 in³, we can use the formula to find the radius of the hemisphere.
By rearranging the formula, we have r^3 = (3 * 4) / (2 * π).
Solving for r, we find that the radius of the hemisphere is approximately 1.113.
Since the diameter and height of the cone are the same, the radius of the cone is also approximately 1.113.
Now, we can calculate the volume of the cone using the formula V = (1/3) * π * r^2 * h.
Substituting the values, we find that the volume of the cone is approximately 1.548 in³.
Therefore, the total volume of ice cream in and on top of the cone is approximately 1.548 + 4 = 5.548 in³.