Estimation Acone has a volume of \( 98 \mathrm{~cm}^{3} \). The radius of the cone is 5.13 cm . Work out an estimate for the height of the cone. John uses a calculator to work out the height of the cone to 2 decimal places. Willyour estimate be more than John's answer or less than John's answer? Give reason for your answer.

To find the height of the cone, we can use the formula for the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. We can rearrange this formula to solve for the height \( h \): \[ h = \frac{3V}{\pi r^2} \] Given: - Volume \( V = 98 \, \text{cm}^3 \) - Radius \( r = 5.13 \, \text{cm} \) Now, we can substitute the values into the formula: 1. Calculate \( r^2 \): \[ r^2 = (5.13)^2 \approx 26.3769 \, \text{cm}^2 \] 2. Calculate \( \pi r^2 \): \[ \pi r^2 \approx 3.14 \times 26.3769 \approx 82.83 \, \text{cm}^2 \] 3. Now, substitute \( V \) and \( \pi r^2 \) into the height formula: \[ h = \frac{3 \times 98}{82.83} \approx \frac{294}{82.83} \approx 3.55 \, \text{cm} \] Now, we need to consider whether our estimate will be more or less than John's answer, which is calculated to two decimal places. ### Reasoning: 1. The value of \( \pi \) we used (3.14) is slightly less than the actual value of \( \pi \) (approximately 3.14159). This means that our calculation for \( \pi r^2 \) is slightly less than it should be. 2. Since we used a smaller value for \( \pi r^2 \), the height \( h \) we calculated will be slightly larger than the actual height calculated using a more precise value of \( \pi \). Thus, our estimate of \( h \approx 3.55 \, \text{cm} \) will be **more than John's answer**, because John's answer will be based on a more accurate calculation that uses a more precise value of \( \pi \).