An observatory is shaped like a cylinder standing on one of its bases with a dome on top. The diameter of the floor of the observatory is 64 feet, as shown in the diagram. 64 ft Which measurement is closest to the circumference of the base of the observatory in feet? F 200.96 ft G 3,215.36 ft H 100.48 ft J 401.92 ft
The Correct Answer and Explanation is:
To find the circumference of the base of the observatory, use the formula for the circumference of a circle:Circumference=π×diameter\text{Circumference} = \pi \times \text{diameter}Circumference=π×diameter
The diameter of the base is given as 64 feet. Substituting into the formula:Circumference=π×64≈3.1416×64=201.06176 feet\text{Circumference} = \pi \times 64 \approx 3.1416 \times 64 = 201.06176 \text{ feet}Circumference=π×64≈3.1416×64=201.06176 feet
Rounding to two decimal places gives approximately 200.96 feet.
Among the choices provided:
- F) 200.96 ft
- G) 3,215.36 ft
- H) 100.48 ft
- J) 401.92 ft
The value 200.96 feet is closest to the actual circumference based on the calculation. Therefore, the correct answer is:
F) 200.96 ft
Explanation:
The observatory is modeled as a cylinder, and the question focuses only on the circular base. The key to solving this lies in knowing that the circumference of a circle is directly proportional to its diameter. Multiplying the diameter by the mathematical constant π (approximately 3.1416) provides the circumference.
This principle arises from the definition of π as the ratio of a circle’s circumference to its diameter. Once the diameter is provided, it becomes a straightforward application of this geometric rule.
The other answer choices are not plausible when compared to the scale of the structure. A circumference of 3,215.36 feet would suggest a diameter over 1,000 feet, which contradicts the given diameter of only 64 feet. Similarly, 100.48 feet corresponds to a much smaller diameter, and 401.92 feet is roughly double the correct value, suggesting an error such as using radius instead of diameter in the formula.
Thus, the value 200.96 feet accurately reflects the correct mathematical relationship.
