Final answer:
To find the volume of a cylinder with the same radius and height as a cone with a given volume, we can use the formula V = πr²h. By calculating the volume of the cone and determining its height, we can substitute the values into the cylinder volume formula to find the answer. Given the volume of the cone as 234 cubic centimeters, the volume of the cylinder is 9.278 cubic centimeters.
Step-by-step explanation:
To find the volume of a cylinder with the same radius and height as a cone, we need to calculate the volume of the cylinder using the formula V = πr²h. Given the volume of the cone as 234 cubic centimeters, we can use the formula for the volume of a cone, V = (1/3)πr²h, to find the height of the cone. Once we have the height of the cone, we can then calculate the volume of the cylinder by substituting the values of the radius and height into the volume formula for a cylinder.
Let's calculate the height of the cone using the formula for the volume of a cone:
(1/3)πr²h = 234
Substituting the value of r as 0.75 cm:
(1/3)π(0.75 cm)²h = 234
Simplifying the equation:
(1/3)π(0.5625 cm²)h = 234
Multiplying both sides by 3 to isolate h:
π(0.5625 cm²)h = 702
Dividing both sides by π × 0.5625 cm² to solve for h:
h = 702 / (π × 0.5625 cm²)
Now that we have the height of the cone, we can calculate the volume of the cylinder using the formula V = πr²h. Substituting the values:
V = π(0.75 cm)²(702 / (π × 0.5625 cm²))
Simplifying the equation:
V = 3.142 × (0.75 cm)² × (702 / (π × 0.5625 cm²))
Calculating the volume of the cylinder:
V = 9.278 cm³