At a lunar research site, Nova and Luna planned an observation shelter made from a cylindrical base topped by a hemispherical dome. A vertical cross-section showed a rectangle beneath a semicircle, helping them connect the drawing to the three-dimensional structure. They compared two designs. Design A had radius 3 meters and cylinder height 4 meters, while Design B had radius 2 meters and cylinder height 7 meters. Excluding the floor and the hidden circle where the parts joined, Design A required 42π square meters of exterior material and enclosed 54π cubic meters. Design B required only 36π square meters but enclosed about 33.3π cubic meters. The shelter needed at least 50π cubic meters of interior space, and 45π square meters of construction material was available. Although Design B used less material, it was too small. Nova and Luna selected Design A, which met both limits, and construction of the lunar observation dome began.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
A valid design must satisfy both the material limit and the minimum interior-space requirement.
A student says Design B should be chosen because it requires only 36π square meters of material, less than Design A's 42π square meters. Why is this reasoning incomplete?
Design B uses too little material, so it cannot be constructed.Design A and Design B actually require the same amount of material.Design B meets the material limit but encloses only about 33.3π cubic meters, which is below the required 50π cubic meters.Design B exceeds the available 45π square meters of material.
Correct. Design B saves material but does not provide enough interior space.
Check Design B against both requirements, not just the amount of material it uses.
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Hint
Subtract the amount Design A requires from the amount available.
Design A uses 42π square meters of the available 45π square meters of construction material. How many square meters, as a multiple of π, remain unused? Enter only the coefficient of π.
Correct. There are 3π square meters of material remaining.
Compare the available 45π square meters with the 42π square meters required by Design A.
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Hint
Add the curved surface area of the cylinder, 2πrh, to the curved surface area of the hemisphere, 2πr^2.
Nova checks Design A from its dimensions instead of relying on the recorded exterior area. The cylinder has radius 3 meters and height 4 meters, and the hemisphere has the same radius. Excluding the floor and the hidden circle where the parts join, which calculation verifies the exterior area of 42π square meters?
2π(3)(4) + 2π(3)^2 = 42ππ(3)^2 + 2π(3)(4) = 33π2π(3)(4) + 4π(3)^2 = 60ππ(3)(4) + 2π(3)^2 = 30π
Correct. The exposed cylinder contributes 24π and the hemisphere contributes 18π, for a total of 42π square meters.
Do not include the floor or the hidden circular joining surface. Add only the cylinder's curved surface and the hemisphere's curved surface.
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Ay Gözlem Kubbesini İnşa Etmek
TR
ARC_G10_GM_02_STORY_03
Ay Gözlem Kubbesini İnşa Etmek
Building the Lunar Observation Dome
EN
ARC_G10_GM_02_STORY_03
Building the Lunar Observation Dome