To determine whether Bill's scoop of frozen yogurt would fill up the cone or not, we need to calculate the volumes of both the yogurt (modeled as a sphere) and the cone.
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Volume of the Sphere (Yogurt): The formula for the volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. Since the diameter of the scoop is 5 cm, the radius \( r \) is: \[ r = \frac{5}{2} = 2.5 \text{ cm} \] Now calculating the volume of the yogurt: \[ V = \frac{4}{3} \pi (2.5)^3 = \frac{4}{3} \pi (15.625) \approx 65.45 \text{ cm}^3 \]
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Volume of the Cone: The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] We need the radius \( r \) of the base of the cone and the height \( h \) of the cone to calculate its volume. Since the problem doesn’t specify the size of the cone, we'll assume it has a radius and height such that its volume is greater than the scoop of yogurt.
Let's analyze the responses based on the assumption that the cone has a reasonable size:
- False: volume of the cone is more than the volume of the yogurt
- True: volume of the yogurt is less than the volume of the cone
- False: volume of the cone is less than the volume of the yogurt
- True: volume of the yogurt is more than the volume of the cone
Given that we know the volume of the sphere (the yogurt) is approximately 65.45 cm³, it is reasonable to conclude that a typical cone used for serving ice cream or yogurt would have a larger volume than this.
Thus, the correct response is:
True: volume of the yogurt is less than the volume of the cone.