A scoop of yogurt is a perfect sphere with a radius of 5 cm and fits exactly so that its equator fits in the top circle of a sugar cone.  Suppose the yogurt melts and the cone itself does not absorb any of the yogurt.  

a)  If the cone is 10 cm tall, will it hold the melted yogurt?  Explain.

b) What is the minimum height for the cone in order for it to hold exactly all the melted yogurt? Explain.
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1 Answer

If r=radius, volume of sphere=4πr³/3; volume of cone=πr²h/3, where h is the height of the cone. If the melted sphere is to fit into the cone, 

4πr³/3≤πr²h/3, 4r≤h. So h≥4r, the height of the cone must be at least 4 times the radius of the sphere/cone. If r=5cm, h must be at least 20cm. So if the height is 10cm, it will not contain all the melted yoghurt.

a) No

b) 20cm

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