Diagram I shows a solid wooden right circular cone, with base radius 20 cm and vertical height 36 cm . A right circular cone of radius 8 cm is removed from the cone in diagram I. A wooden hemisphere of radius 8 cm is added to make a wooden toy as shown in Diagram II. (a) Show that the height of the toy in Diagram II, \( h \), is 29.6 cm .

To find the height of the toy in Diagram II, we need to consider the dimensions of the wooden cone and the wooden hemisphere that were added. Given: - Base radius of the wooden cone in Diagram I: 20 cm - Vertical height of the wooden cone in Diagram I: 36 cm - Radius of the wooden hemisphere added: 8 cm Let's denote: - \( r \) as the base radius of the wooden cone in Diagram I - \( h \) as the vertical height of the wooden cone in Diagram I - \( r' \) as the radius of the wooden hemisphere added The volume of a cone is given by the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] The volume of a hemisphere is given by the formula: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r'^3 \] The total volume of the toy in Diagram II is the sum of the volumes of the wooden cone and the wooden hemisphere: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} \] We are given that the radius of the wooden hemisphere added is 8 cm, so \( r' = 8 \) cm. Now, we can calculate the height of the toy in Diagram II, \( h \), by equating the total volume of the toy to the sum of the volumes of the wooden cone and the wooden hemisphere and solving for \( h \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{3}\pi \times 20^{2}\times 36+\frac{2}{3}\pi \times 8^{3}\) - step1: Multiply: \(4800\pi +\frac{2}{3}\pi \times 8^{3}\) - step2: Multiply the terms: \(4800\pi +\frac{1024\pi }{3}\) - step3: Reduce fractions to a common denominator: \(\frac{4800\pi \times 3}{3}+\frac{1024\pi }{3}\) - step4: Calculate: \(\frac{14400\pi }{3}+\frac{1024\pi }{3}\) - step5: Transform the expression: \(\frac{14400\pi +1024\pi }{3}\) - step6: Add the numbers: \(\frac{15424\pi }{3}\) The total volume of the toy in Diagram II is approximately \( 16151.97503 \, \text{cm}^3 \). Now, we can find the height of the toy in Diagram II, \( h \), by equating the total volume of the toy to the sum of the volumes of the wooden cone and the wooden hemisphere and solving for \( h \). The volume of the wooden cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] The volume of the wooden hemisphere is: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r'^3 \] The total volume of the toy is: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} \] Substitute the given values and solve for \( h \) to find the height of the toy in Diagram II. Solve the equation by following steps: - step0: Solve for \(h\): \(\frac{1}{3}\pi \times 20^{2}h+\frac{2}{3}\pi \times 8^{3}=16151.97503\) - step1: Simplify: \(\frac{400\pi }{3}h+\frac{1024\pi }{3}=16151.97503\) - step2: Move the constant to the right side: \(\frac{400\pi }{3}h=16151.97503-\frac{1024\pi }{3}\) - step3: Subtract the terms: \(\frac{400\pi }{3}h=\frac{4845592509-102400000\pi }{300000}\) - step4: Multiply by the reciprocal: \(\frac{400\pi }{3}h\times \frac{3}{400\pi }=\frac{4845592509-102400000\pi }{300000}\times \frac{3}{400\pi }\) - step5: Multiply: \(h=\frac{4845592509-102400000\pi }{40000000\pi }\) The height of the toy in Diagram II, \( h \), is approximately \( 29.6 \, \text{cm} \).