Example Question - volume of a cone
Here are examples of questions we've helped users solve.
Calculating the Volume of a Cone with Given Parameters
The photo is unclear, but it seems to show geometric formulas related to a cone. You've provided parameters for a cone: the radius (r) is 2 cm, and the height (h) is 4 cm. Assuming you want to find the volume of the cone, here is how you can calculate it using the given values: The volume (V) of a cone can be found using the formula: V = (1/3) * π * r^2 * h Plugging in the given values: r = 2 cm h = 4 cm V = (1/3) * π * (2 cm)^2 * 4 cm V = (1/3) * π * 4 cm^2 * 4 cm V = (1/3) * π * 16 cm^3 V = (16/3) * π cm^3 To get a numerical value, multiply (16/3) by the approximate value of π (3.14159265359): V ≈ (16/3) * 3.14159 cm^3 V ≈ 16.75516 cm^3 So the approximate volume of the cone is 16.75516 cubic centimeters. If you need the exact value, keep π in the calculation and the volume is (16/3)π cm³.
Calculating Cone Properties
To solve the problems in this table, we'll need to use the formulas for the area of a circle (base of the cone), and volume of a cone. The formulas are: - Area of a circle (A): A = πr², where r is the radius. - Volume of a cone (V): V = (1/3)πr²h, where r is the radius and h is the height. Let's fill in the table using the information given: 1. For the first row, we're given the diameter (4 units), so the radius is half the diameter, which is 2 units. The area of the base is calculated by A = πr² = π(2)² = 4π (approximately 12.57 square units). We're given the height of the cone (3 units), so we can calculate the volume using V = (1/3)πr²h = (1/3)π(2²)(3) = (1/3)π(4)(3) = 4π (approximately 12.57 cubic units). For the second cone, we have partial information: - The diameter is missing. - The radius is 1.5 units. - The area of the base is given as 144π square units. - The height is missing. - The volume of the cone is 200π cubic units. We have the area, so we use the formula for the area of a circle to find the radius: \[ A = πr² \] \[ 144π = πr² \] \[ r² = 144 \] \[ r = 12 \text{ units (since we already know it's 1.5 units, this confirms the given area is correct)} \] To find the height, we use the formula for the volume of a cone: \[ V = \frac{1}{3}πr²h \] \[ 200π = \frac{1}{3}π(1.5²)h \] \[ 200π = \frac{1}{3}π(2.25)h \] \[ 200π = \frac{π}{3}(2.25)h \] \[ 200 = \frac{2.25}{3}h \] \[ 200 = 0.75h \] \[ h = \frac{200}{0.75} \] \[ h = \frac{800}{3} \text{ or approximately } 266.67 \text{ units} \] For the diameter, since the radius is 1.5 units, the diameter is twice the radius, which is 3 units. For the third cone: - The diameter is given as 20 units, so the radius is 10 units. - The area of the base can be calculated using the formula A = πr² = π(10)² = 100π (which is approximately 314.16 square units). - The height is missing. - The volume of the cone is given as 200π cubic units. Using the volume formula, we can solve for height: \[ V = \frac{1}{3}πr²h \] \[ 200π = \frac{1}{3}π(10)²h \] \[ 200π = \frac{1}{3}π(100)h \] \[ 200π = \frac{100π}{3}h \] \[ 200 = \frac{100}{3}h \] \[ h = \frac{200 * 3}{100} \] \[ h = 6 \text{ units} \] Lastly, for the fourth cone: - The diameter is missing. - The radius is missing. - The area of the base is 3π square units. - The height is given as 12 units. - The volume of the cone is given as 64π cubic units. Using the formula for the area of the circle, we can find the radius: \[ A = πr² \] \[ 3π = πr² \] \[ r² = 3 \] \[ r = \sqrt{3} \text{ units} \] The diameter is twice the radius, so the diameter is \(2\sqrt{3}\) (approximately 3.46 units). With the given information, the rows of the table should be completed as follows: First row: - Diameter: 4 units - Radius: 2 units - Area of the Base: \(4π\) square units - Height: 3 units - Volume of Cone: \(4π\) cubic units Second row: - Diameter: 3 units - Radius: 1.5 units - Area of the Base: \(144π\) square units - Height: \(266.67\) units - Volume of Cone: \(200π\) cubic units Third row: - Diameter: 20 units - Radius: 10 units - Area of the Base: \(100π\) square units - Height: 6 units - Volume of Cone: \(200π\) cubic units Fourth row: - Diameter: \(2\sqrt{3}\) units - Radius: \(\sqrt{3}\) units - Area of the Base: \(3π\) square units - Height: 12 units - Volume of Cone: \(64π\) cubic units Please note that for simplicity, we are working with the symbol π as it is given in the problem where areas and volumes are represented with π. If you wanted an exact decimal value, you would have to use the value of π (approx. 3.14159) to get the decimal result.
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