Question - Calculating Cone Properties
Solution:
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 unitsSecond row:- Diameter: 3 units- Radius: 1.5 units- Area of the Base: $$144π$$ square units- Height: $$266.67$$ units- Volume of Cone: $$200π$$ cubic unitsThird row:- Diameter: 20 units- Radius: 10 units- Area of the Base: $$100π$$ square units- Height: 6 units- Volume of Cone: $$200π$$ cubic unitsFourth 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 unitsPlease 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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