Eli is making a model castle out of clay. One of the roof peaks is in the shape of a cone with a diameter of 14 inches and a slant height of 20 inches
What is the surface area of the cone peak? Round your answer to the nearest hundredth. Use 3.14 for pi. (1 point)
The surface area of a cone is given by the formula:
Surface Area = πr(r + l)
where r is the radius of the base of the cone and l is the slant height.
Given the diameter (d) of the base of the cone is 14 inches, we can calculate the radius (r) by dividing the diameter by 2:
r = d/2 = 14/2 = 7 inches
Plugging in the values of r = 7 inches and l = 20 inches into the formula, we get:
Surface Area = π(7)(7 + 20) = 3.14 * 7 * 27
Calculating this, we get:
Surface Area = 3.14 * 7 * 27 = 592.86
Rounding this to the nearest hundredth, the surface area of the cone peak is approximately 592.86 square inches.
Find the volume of the cylinder, in cubic millimeters. Round your answer to the nearest hundredth the height is 13mm the radius is 8mm
The formula for the volume of a cylinder is given by:
Volume = πr^2h
where r is the radius of the circular base, and h is the height of the cylinder.
Given that the radius (r) is 8 mm and the height (h) is 13 mm, we can calculate the volume as follows:
Volume = π(8^2)(13) = π(64)(13) = 832π
To find the numerical value of the volume, we substitute the value of π as 3.14:
Volume ≈ (3.14)(832) = 2619.68
Rounding this to the nearest hundredth, the volume of the cylinder is approximately 2619.68 cubic millimeters.
A water bottle has a height of 16 inches and a radius of 4 inches. What is the volume, in cubic inches, of the water bottle? Use 3.14 for pi. (1 point)
The formula for the volume of a cylinder is given by:
Volume = π * r^2 * h
where r is the radius of the circular base, and h is the height of the cylinder.
Given that the radius (r) is 4 inches and the height (h) is 16 inches, we can calculate the volume as follows:
Volume = 3.14 * (4^2) * 16
= 3.14 * 16 * 16
= 803.84 cubic inches
Therefore, the volume of the water bottle is 803.84 cubic inches.
A cone-shaped container on a farm is used for storing animal feed. The container has a radius of 4 feet and a height of 12 feet. What volume of feed, in cubic feet, can the container store? Round your answer to the nearest hundredth. Use 3.14 for . (1 point)
The formula for the volume of a cone is given by:
Volume = (1/3) * π * r^2 * h
where r is the radius of the base of the cone, and h is the height of the cone.
Given that the radius (r) is 4 feet and the height (h) is 12 feet, we can calculate the volume as follows:
Volume = (1/3) * 3.14 * (4^2) * 12
= (1/3) * 3.14 * 16 * 12
= (1/3) * 3.14 * 192
= 201.06 cubic feet
Therefore, the container can store approximately 201.06 cubic feet of feed, rounded to the nearest hundredth.
The diameter of a sphere (1 point)
The diameter of a sphere is a straight line that passes through the center of the sphere and touches two points on its surface. It is twice the value of the radius of the sphere.
Find the surface area of a hemisphere with a.radius 10m. Leave your answer in terms of pi. (1 point)
The formula for the surface area of a hemisphere is given by:
Surface Area = 2πr^2
where r is the radius of the hemisphere.
Given that the radius (r) is 10m, we can calculate the surface area as follows:
Surface Area = 2π(10^2)
= 2π(100)
= 200π
Therefore, the surface area of the hemisphere is 200π square meters.
The surface area of a sphere is 7.84 ft.^2 What is the radius of this sphere? (1 point)
The formula for the surface area of a sphere is given by:
Surface Area = 4πr^2
where r is the radius of the sphere.
Given that the surface area is 7.84 ft², we can set up the equation:
7.84 = 4πr^2
Dividing both sides of the equation by 4π:
1.96 = r^2
Taking the square root of both sides:
r ≈ √1.96
r ≈ 1.4
Therefore, the radius of the sphere is approximately 1.4 feet.
How much ice cream is in a spherical scoop of ice cream with a radius of 3 cm? Use 3.14 as an approximation for pi. Round your answer to the nearest whole number. (1 point)
The formula for the volume of a sphere is given by:
Volume = (4/3) * π * r^3
where r is the radius of the sphere.
Given that the radius (r) is 3 cm, we can calculate the volume as follows:
Volume = (4/3) * 3.14 * (3^3)
≈ (4/3) * 3.14 * 27
≈ 113.04 cm^3
Rounding this to the nearest whole number, the scoop of ice cream contains approximately 113 cm^3.
Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone? (1 point)
The relationship between the volume formulas for the sphere, cylinder, and cone can be determined by comparing their respective formulas.
The formula for the volume of a sphere is given by:
Volume_sphere = (4/3) * π * r^3
The formula for the volume of a cone is given by:
Volume_cone = (1/3) * π * r^2 * h
The formula for the volume of a cylinder is given by:
Volume_cylinder = π * r^2 * h
From the given information, we can infer that the height of the cone and cylinder is twice the radius: h = 2r.
Comparing the formulas, we can observe that the volumetric formulas for both the cone and the cylinder involve the factor of 'h', which represents the height, whereas the volume of the sphere does not include any height term.
Therefore, the true relationship about the volume formulas is that the volume formula for a sphere is distinct from the formulas for both the cylinder and cone, as it does not depend on the height of the shape.
• sphere - cone = cylinder
© sphere = cylinder + cone
O 1 cylinder = 2 cones + 1 sphere
The correct relationship between the volume formulas for the sphere, cone, and cylinder would be:
1 sphere = 1 cylinder - 2 cones
The volume of a sphere is 48 m^3 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)
The formula for the volume of a sphere is given by:
Volume_sphere = (4/3) * π * r^3
Given that the volume of the sphere is 48 m^3, we can set up the equation:
48 = (4/3) * π * r^3
To find the radius (r), we need to solve for r by rearranging the equation:
r^3 = (3/4) * (48 / π)
r^3 = 36 / π
Taking the cube root of both sides:
r ≈ (36/π)^(1/3)
Now, we know that the height of the cylinder is equal to the diameter of the sphere, so it is twice the radius of the sphere. Therefore, the height (h) of the cylinder is:
h ≈ 2 * r ≈ 2 * (36/π)^(1/3)
Using these values, we can now find the volume of the cylinder. The formula for the volume of a cylinder is given by:
Volume_cylinder = π * r^2 * h
Volume_cylinder ≈ π * [(36/π)^(1/3)]^2 * [2 * (36/π)^(1/3)]
Volume_cylinder ≈ π * (36/π)^(2/3) * 2 * (36/π)^(1/3)
Volume_cylinder ≈ 2 * 36 * (36/π)^(2/3) * (36/π)^(1/3)
Volume_cylinder ≈ 2 * 36 * (36/π)^(2/3 + 1/3)
Volume_cylinder ≈ 72 * (36/π)^(1)
Therefore, the volume of the cylinder with the given specifications is approximately 72 * (36/π) cubic meters.
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.
Let's denote the height and radius of the ice cream cone as h and r, respectively. Since the diameter of the cone is equal to its height, we have d = h.
The volume of the cone is given by the formula:
Volume_cone = (1/3) * π * r^2 * h
Given that the diameter (d) is equal to the height (h), we can substitute:
h = d
So the volume of the cone can be expressed as:
Volume_cone = (1/3) * π * r^2 * d
Now, we know that the hemisphere placed on top of the cone has a volume of 4 in.^3, which we can denote as Volume_hemisphere.
The relationship between the volume of a cone and the volume of a hemisphere is given by:
Volume_cone = (1/2) * Volume_hemisphere
Substituting in the formulas for the volumes, we have:
(1/3) * π * r^2 * d = (1/2) * 4
Simplifying:
π * r^2 * d = 6
Since we know that the diameter (d) is equal to the height (h), we can substitute:
π * r^2 * h = 6
Now, we are given that the volume of the hemisphere is 4 in.^3. The formula for the volume of a hemisphere is given by:
Volume_hemisphere = (2/3) * π * r^3
Substituting the given value:
4 = (2/3) * π * r^3
Simplifying:
r^3 = (3/2) * (4/π)
r^3 = 6/π
To find the radius (r), we take the cube root of both sides:
r = (6/π)^(1/3)
Now, to find the total volume of the ice cream in and on top of the cone, we add the volume of the cone and the volume of the hemisphere:
Total volume = Volume_cone + Volume_hemisphere
Total volume = (1/3) * π * r^2 * d + Volume_hemisphere
Total volume = (1/3) * π * ((6/π)^(1/3))^2 * ((6/π)^(1/3)) + Volume_hemisphere
Total volume = (1/3) * π * (6/π)^(2/3) * (6/π)^(1/3) + 4
Therefore, the total volume of ice cream in and on top of the cone can be calculated using the given information and the relationships between the formulas for cone and hemisphere volumes.