The radius and height of a right circular cone are in the ratio 3 : 7. If the volume of the cone is 528 cm 3, then what is the height of the cone? \(\left( {{\rm{Take}}\,\,{\rm{\pi }}\,{\rm{ = }}\frac{{22}}{7}} \right)\)
14.0 cm
Calculating the Height of a Cone from Volume and Radius-Height Ratio
This problem involves finding the height of a right circular cone given its volume and the ratio of its radius to its height. We are provided with the volume of the cone, the ratio of its radius and height, and the value of \(\pi\) to use.
Let the radius of the cone be \(r\) and the height be \(h\). The given ratio of the radius to the height is 3 : 7. This can be written as:
\(\frac{r}{h} = \frac{3}{7}\)
We can express \(r\) and \(h\) in terms of a common variable, say \(x\). So, we can write:
\(r = 3x\)
\(h = 7x\)
The volume \(V\) of a right circular cone is given by the formula:
\(V = \frac{1}{3}\pi r^2 h\)
We are given that the volume \(V = 528 \text{ cm}^3\) and \(\pi = \frac{22}{7}\). Substitute these values and the expressions for \(r\) and \(h\) into the volume formula:
\(528 = \frac{1}{3} \times \frac{22}{7} \times (3x)^2 \times (7x)\)
Now, let's simplify and solve for \(x\):
\(528 = \frac{1}{3} \times \frac{22}{7} \times (9x^2) \times (7x)\)
\(528 = \frac{1}{3} \times \frac{22}{7} \times 9x^2 \times 7x\)
Cancel out the 7 in the numerator and denominator, and the 3 in the denominator with the 9 in the numerator:
\(528 = \frac{1}{\cancel{3}} \times 22 \times \cancel{9}^3 x^2 \times \cancel{7} x\)
\(528 = 22 \times 3x^2 \times x\)
\(528 = 66x^3\)
Now, isolate \(x^3\):
\(x^3 = \frac{528}{66}\)
Let's perform the division:
\(528 \div 66\). We can see that \(66 \times 8 = (60+6) \times 8 = 480 + 48 = 528\).
So,
\(x^3 = 8\)
To find \(x\), we take the cube root of 8:
\(x = \sqrt[3]{8}\)
\(x = 2\)
We are asked to find the height of the cone, which is \(h = 7x\). Substitute the value of \(x = 2\):
\(h = 7 \times 2\)
\(h = 14\)
The height of the cone is 14 cm.
Final Answer
The calculated height of the cone is 14.0 cm.
Revision Table: Cone Volume Calculation
| Concept | Formula/Value | Notes |
|---|---|---|
| Radius : Height Ratio | \(r : h = 3 : 7\) | Implies \(r=3x, h=7x\) |
| Volume of Cone | \(V = \frac{1}{3}\pi r^2 h\) | Standard formula |
| Given Volume | \(V = 528 \text{ cm}^3\) | Used in equation |
| Given \(\pi\) | \(\pi = \frac{22}{7}\) | Used in equation |
| Calculated \(x\) | \(x = 2\) | From solving equation |
| Height \(h\) | \(h = 7x\) | Calculated using \(x\) |
Additional Information: Properties of Right Circular Cones
A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat base (which is a circle) to a point called the apex or vertex. The apex is directly above the center of the circular base, forming a right angle with the base radius. Key properties include:
- Base: A circular base.
- Apex: The point opposite the base.
- Height (h): The perpendicular distance from the apex to the center of the base.
- Radius (r): The radius of the circular base.
- Slant Height (l): The distance from any point on the circumference of the base to the apex. It is related to the radius and height by the Pythagorean theorem: \(l^2 = r^2 + h^2\).
- Volume (V): The space occupied by the cone, given by \(V = \frac{1}{3}\pi r^2 h\). This formula is \(\frac{1}{3}\) times the volume of a cylinder with the same base radius and height.
- Surface Area: Includes the area of the base (\(\pi r^2\)) and the lateral surface area (\(\pi r l\)). Total surface area \(A = \pi r^2 + \pi r l\).
Understanding these properties is crucial for solving various problems related to cones, including those involving volume, surface area, and dimensions like radius, height, and slant height.
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