https://file.aiquickdraw.com/mathbot/file/1726995968809ovwu91nr.jpg

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.

A cone has a volume of 98 cm³. The radius of the cone is 5.13 cm. Estimate the height of the cone, and compare your estimate with John's calculated answer.

Learn how to estimate the height of a cone given its volume (98 cm³) and radius (5.13 cm) using the cone volume formula. This problem involves key geometric concepts and formulas, suitable for GCSE students in Grades 9-10.

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