QQuestionMathematics
QuestionMathematics
Calculate the total number of gumballs that need to fit in the machine if each gumball has a diameter of 1.25 inches.
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Answer
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Step 1:: First, we need to find the volume of a single gumball.
r = \frac{1.25 \text{ inches}}{2} = 0.625 \text{ inches}
Since a gumball is a sphere, we can use the formula for the volume of a sphere: Since the diameter is 1.25 inches, the radius is half of that:
Step 2:: Now, substitute the value of the radius into the formula for the volume of a sphere:
V_{gumball} = \frac{4}{3} \pi (0.625 \text{ inches})^{3}
Step 3:: Calculate the volume of the gumball:
V_{gumball} \approx \frac{4}{3} \pi (0.263 \text{ inches}^3) \approx 0.348 \text{ inches}^3
Step 4:: The question asks for the total number of gumballs that fit in the machine, so we need to find the volume of the gumball machine's gumball compartment.
r = \frac{10 \text{ inches}}{2} = 5 \text{ inches}
Let's assume the gumball machine has a cylindrical gumball compartment with a height of 30 inches and a diameter of 10 inches. First, calculate the radius of the cylinder:
Step 5:: Now, calculate the volume of the cylindrical gumball compartment:
V_{compartment} = \pi r^{2} h = \pi (5 \text{ inches})^{2} (30 \text{ inches})
Step 6:: Calculate the volume of the gumball compartment:
V_{compartment} = \pi (25 \text{ inches}^2) (30 \text{ inches}) = 2250 \pi \text{ inches}^3
Step 7:: Divide the volume of the gumball compartment by the volume of a single gumball to find out how many gumballs fit in the machine:
Number \ of \ gumballs = \frac{V_{compartment}}{V_{gumball}} = \frac{2250 \pi \text{ inches}^3}{0.348 \text{ inches}^3} \approx 20496.86
Final Answer
Approximately 20497 gumballs can fit in the gumball machine, assuming the machine has a cylindrical gumball compartment with a height of 30 inches and a diameter of 10 inches.
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