Let's solve the question step by step.

Part (a) - Estimating the height of the cone

We are given:

• Volume of the cone $V = 98 \, \text{cm}^3$
• Radius of the cone $r = 5.13 \, \text{cm}$

The formula for the volume of a cone is:

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

We need to find the height $h$. Rearranging the formula to solve for $h$:

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

Now, substituting the given values:

$h = \frac{3 \times 98}{\pi \times (5.13)^2}$

Let's calculate this step by step.

Step 1: Square the radius

$5.13^2 = 26.3169$

Step 2: Multiply by $\pi$

$\pi \times 26.3169 \approx 82.685$

Step 3: Multiply the volume by 3

$3 \times 98 = 294$

Step 4: Divide by $\pi r^2$

$h = \frac{294}{82.685} \approx 3.56 \, \text{cm}$

So, an estimate for the height of the cone is 3.56 cm.

Part (b) - Comparison with John's answer

John uses a calculator to work out the height to 2 decimal places. Since the question asks for an estimate and we used rounded values for $\pi$, John's answer may be slightly different. In general, our estimate may be slightly less accurate than John's since John's calculator likely uses a more precise value for $\pi$ and does more precise arithmetic. Therefore, our estimate could be either more or less than John's answer, but it will be very close.

Would you like to go into more detail on any of these steps?

Here are five related questions to practice:

1. If the volume of the cone were 150 cm³, what would be the height of the cone?
2. How does changing the radius affect the height of the cone for a fixed volume?
3. What happens to the height if the volume of the cone is doubled while keeping the radius constant?
4. If John had rounded the radius to 5 cm, how would that change his answer?
5. What is the formula for the surface area of a cone, and how can it be derived from the height?

Tip: When estimating with formulas involving $\pi$, it's helpful to use at least two decimal places for $\pi$ (3.14 or more) for better accuracy.