If H, C and V are respectively the height, curved surface area and volume of a cone, then what is 3πVH 3+ 9V 2equal to?
C 2H 2
Understanding Cone Formulas and Properties
The question asks us to find the value of the expression \(3\pi VH^3 + 9V^2\), where H is the height, C is the curved surface area, and V is the volume of a cone. To solve this, we need to use the standard formulas for the volume and curved surface area of a cone in terms of its height (H) and base radius (r).
Let's list the relevant formulas for a cone:
- Volume (V): \(V = \frac{1}{3}\pi r^2 H\)
- Curved Surface Area (C): \(C = \pi r l\), where \(l\) is the slant height.
- Slant height (\(l\)): The relationship between height (H), radius (r), and slant height (l) is given by the Pythagorean theorem: \(l = \sqrt{r^2 + H^2}\).
Evaluating the Given Expression \(3\pi VH^3 + 9V^2\)
Our goal is to simplify the expression \(3\pi VH^3 + 9V^2\) and express it in terms of C and H. We will substitute the formula for the volume, \(V = \frac{1}{3}\pi r^2 H\), into the expression.
The expression is:
\[ 3\pi VH^3 + 9V^2 \]Substitute \(V = \frac{1}{3}\pi r^2 H\) into the expression:
\[ 3\pi \left(\frac{1}{3}\pi r^2 H\right) H^3 + 9\left(\frac{1}{3}\pi r^2 H\right)^2 \]Now, let's simplify each term:
The first term is:
\[ 3\pi \left(\frac{1}{3}\pi r^2 H\right) H^3 = \pi^2 r^2 H \cdot H^3 = \pi^2 r^2 H^{1+3} = \pi^2 r^2 H^4 \]The second term is:
\[ 9\left(\frac{1}{3}\pi r^2 H\right)^2 = 9 \left(\frac{1}{3}\right)^2 (\pi r^2 H)^2 = 9 \cdot \frac{1}{9} \cdot \pi^2 (r^2)^2 H^2 = \pi^2 r^4 H^2 \]So, the expression becomes the sum of these two simplified terms:
\[ \pi^2 r^2 H^4 + \pi^2 r^4 H^2 \]We can factor out common terms from this expression. Both terms have \(\pi^2\), \(r^2\), and \(H^2\).
\[ \pi^2 r^2 H^2 (H^2 + r^2) \]This is the simplified form of the given expression in terms of \(r\) and \(H\).
Relating the Expression to \(C\) and \(H\)
Now let's look at the curved surface area (C) and see how \(C^2 H^2\) relates to our simplified expression.
The formula for curved surface area is \(C = \pi r l\), where \(l = \sqrt{r^2 + H^2}\).
Substitute \(l\) into the formula for C:
\[ C = \pi r \sqrt{r^2 + H^2} \]Now, let's square C:
\[ C^2 = (\pi r \sqrt{r^2 + H^2})^2 = \pi^2 r^2 (r^2 + H^2) \]Finally, let's calculate \(C^2 H^2\):
\[ C^2 H^2 = [\pi^2 r^2 (r^2 + H^2)] H^2 = \pi^2 r^2 H^2 (r^2 + H^2) \]Comparing the Results
We found that the simplified expression \(3\pi VH^3 + 9V^2\) is equal to \(\pi^2 r^2 H^2 (H^2 + r^2)\).
We also found that \(C^2 H^2\) is equal to \(\pi^2 r^2 H^2 (r^2 + H^2)\).
Comparing these two results, we see that:
\[ 3\pi VH^3 + 9V^2 = C^2 H^2 \]Thus, the expression \(3\pi VH^3 + 9V^2\) is equal to \(C^2 H^2\). This matches option 1.
Revision Table: Key Cone Formulas
| Property | Symbol | Formula (using radius r, height H, slant height l) |
|---|---|---|
| Volume | V | \(\frac{1}{3}\pi r^2 H\) |
| Curved Surface Area | C | \(\pi r l\) or \(\pi r \sqrt{r^2 + H^2}\) |
| Base Area | Abase | \(\pi r^2\) |
| Total Surface Area | Atotal | \(C + A_{base} = \pi r l + \pi r^2 = \pi r (l+r)\) |
| Slant Height relation | \(l^2\) | \(l^2 = r^2 + H^2\) |
Additional Information on Cone Properties
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. The height (H) is the perpendicular distance from the apex to the center of the base. The radius (r) is the radius of the circular base. The slant height (l) is the distance from any point on the circumference of the base to the apex, measured along the surface of the cone. The volume (V) measures the space occupied by the cone, and the curved surface area (C) measures the area of the sloping surface excluding the base.
The relationship between the height, radius, and slant height (\(l^2 = r^2 + H^2\)) is a direct application of the Pythagorean theorem, forming a right-angled triangle with the height, radius, and slant height as its sides. Understanding these fundamental definitions and formulas is crucial for solving problems involving cones, like the one discussed here which relates volume and curved surface area to the cone's dimensions.
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