A pyramid and a cone are both 10 centimeters tall and have the same volume. What statement must be true about the two solids?A.The cross-sections of the pyramid and cone are the same shape.B.The area of the cross-sections of the pyramid and cone are multiples of each other.C.The vertical cross-sections of the pyramid and cone at the same width must have the same area.D.The horizontal cross-sections of the pyramid and cone at the same height must have the same area.
Question
A pyramid and a cone are both 10 centimeters tall and have the same volume. What statement must be true about the two solids?A.The cross-sections of the pyramid and cone are the same shape.B.The area of the cross-sections of the pyramid and cone are multiples of each other.C.The vertical cross-sections of the pyramid and cone at the same width must have the same area.D.The horizontal cross-sections of the pyramid and cone at the same height must have the same area.
Solution
The correct statement is D. The horizontal cross-sections of the pyramid and cone at the same height must have the same area.
Here's why:
A. The cross-sections of the pyramid and cone are not necessarily the same shape. A pyramid's cross-section would be a triangle or a polygon, while a cone's cross-section would be a circle.
B. The area of the cross-sections of the pyramid and cone are not necessarily multiples of each other. This would depend on the specific dimensions of the base and height of each figure.
C. The vertical cross-sections of the pyramid and cone at the same width do not necessarily have the same area. This would also depend on the specific dimensions of the base and height of each figure.
D. The horizontal cross-sections of the pyramid and cone at the same height must have the same area. This is because the volume of a solid is found by multiplying the area of the base (or cross-section) by the height. If the volumes of the pyramid and cone are equal and their heights are equal, then their cross-sectional areas must also be equal.
Similar Questions
Cone W has a radius of 10 cm and a height of 5 cm. Square pyramid X has the same base area and height as cone W.Paul and Manuel disagree on how the volumes of cone W and square pyramid X are related. Examine their arguments. Which statement explains whose argument is correct and why?Paul ManuelThe volume of square pyramid X is three times the volume of cone W. This can be proven by finding the base area and volume of cone W, along with the volume of square pyramid X. The base area of cone W is π(r2) = π(102) = 314 cm2. The volume of cone W is (area of base)(h) = (314)(5) = 523.33 cm3. The volume of square pyramid X is (area of base)(h) = (314)(5) = 1,570 cm3. The volume of square pyramid X is equal to the volume of cone W. This can be proven by finding the base area and volume of cone W, along with the volume of square pyramid X. The base area of cone W is π(r2) = π(102) = 314 cm2. The volume of cone W is (area of base)(h) = (314)(5) = 523.33 cm3. The volume of square pyramid X is (area of base)(h) = (314)(5) = 523.33 cm3.Group of answer choicesPaul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid X.Paul's argument is correct; Manuel used the incorrect base area to find the volume of square pyramid X.Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid X.Manuel's argument is correct; Paul used the incorrect base area to find the volume of square pyramid X.
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