Area, Surface Area and Volume: Question 3

Worked solution

Step 1: Choose the correct formula

The solid is a cone, so use the volume of a cone:

$V = \frac{1}{3}\pi r^2 h$

The key features are the factor $\tfrac{1}{3}$ (a cone fills one third of the enclosing cylinder) and that the radius is squared, not the height.

Step 2: Substitute the values

Here $r = 6\text{ cm}$ and $h = 14\text{ cm}$:

$V = \frac{1}{3}\,\pi \,(6)^2(14)$

Work out the numbers inside step by step:

$(6)^2 = 36, \qquad 36 \times 14 = 504$

So

$V = \frac{1}{3}\,\pi \times 504 = 168\,\pi.$

Step 3: Evaluate

Using $\pi = 3.142$:

$V = 168 \times 3.142 = 527.856 \approx 528\text{ cm}^3 \ (3\text{ s.f.})$

(Using the more precise $\pi$ on a calculator gives $527.79\ldots$, which also rounds to $528\text{ cm}^3$.)

Why the other options are wrong

• $1580\text{ cm}^3$ comes from forgetting the $\tfrac{1}{3}$ and computing the full cylinder $\pi r^2 h = 504\pi$.
• $905\text{ cm}^3$ comes from using the sphere formula $\tfrac{4}{3}\pi r^3 = 288\pi$, which wrongly ignores the height.
• $2110\text{ cm}^3$ comes from using the diameter $12$ in place of the radius: $\tfrac{1}{3}\pi(12)^2(14) = 672\pi$.

Final answer

$\boxed{V \approx 528\text{ cm}^3}$

So the correct option is A.