A cone has a volume of 300 units \( ^{3} \), and a cylinder as a volume of 100 units \( ^{3} \). Is it possil that the cylinder and cone have the same height and the same radius?

To determine if it is possible for the cylinder and cone to have the same height and the same radius, we need to consider the formulas for the volume of a cone and a cylinder. The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] The volume of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] Given that the volume of the cone is 300 units \( ^{3} \) and the volume of the cylinder is 100 units \( ^{3} \), we can set up the following equations: \[ \frac{1}{3} \pi r^2 h = 300 \] \[ \pi r^2 h = 100 \] We can solve these equations to find the relationship between the radius and height of the cone and cylinder. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{1}{3}\pi r^{2}h=300\\\pi r^{2}h=100\end{array}\right.\) - step1: Calculate: \(\left\{ \begin{array}{l}\frac{\pi }{3}r^{2}h=300\\\pi r^{2}h=100\end{array}\right.\) - step2: Solve the equation: \(\left\{ \begin{array}{l}h=\frac{900}{\pi r^{2}}\\\pi r^{2}h=100\end{array}\right.\) - step3: Substitute the value of \(h:\) \(\pi r^{2}\times \frac{900}{\pi r^{2}}=100\) - step4: Simplify: \(900=100\) - step5: Calculate: \(\left\{ \begin{array}{l}h \in \varnothing \\r \in \varnothing \end{array}\right.\) - step6: Rewrite: \((h, r) \in \varnothing\) The system of equations does not have any solution, which means it is not possible for the cylinder and cone to have the same height and the same radius.