A right circular cone has height 45 centimetres. It is cut by a plane parallel to the base at a height of 15 centimetres above the base, forming a smaller cone at the top. If the volume of this smaller cone is 18480 cubic centimetres, what is the volume, in cubic centimetres, of the original cone?

Introduction / Context:
This problem involves a cone that is cut by a plane parallel to its base, creating a smaller similar cone at the top and a frustum below. We are given the height of the original cone and the height from the base where the cut is made, as well as the volume of the smaller cone. Using similarity of cones and the cubic relationship between linear dimensions and volume, we can find the volume of the original cone without needing the actual radius values.

Given Data / Assumptions:

• Height of the original cone H = 45 cm.
• A plane parallel to the base cuts the cone at a height 15 cm above the base.
• Thus, the distance from the apex to the cutting plane is H − 15 = 45 − 15 = 30 cm.
• The smaller cone at the top has height h_small = 30 cm.
• Volume of the smaller cone V_small = 18480 cm^3.
• We must find the volume of the original cone V_original.
• The two cones are similar, so all linear dimensions scale in the same ratio.

Concept / Approach:
When two cones are similar, the ratio of their corresponding linear dimensions (such as heights and radii) is the same. Volumes of similar solids scale as the cube of the linear scale factor. Here, the height of the smaller cone is (H − 15) and the height of the original cone is H. Therefore, the volume ratio V_small / V_original equals (h_small / H)^3. We compute this ratio, then solve for V_original using the given smaller volume.

Step-by-Step Solution:
Step 1: Height of original cone H = 45 cm. Step 2: Height of smaller top cone h_small = H − 15 = 30 cm. Step 3: Because the cut is parallel to the base, the smaller cone is similar to the original cone with linear scale factor k = h_small / H = 30 / 45 = 2 / 3. Step 4: Volumes of similar solids scale as the cube of the linear factor, so V_small / V_original = k^3 = (2 / 3)^3 = 8 / 27. Step 5: Let V_original be the volume of the original cone. Then V_small = (8 / 27) * V_original. Step 6: Substitute V_small = 18480 cm^3 to get 18480 = (8 / 27) * V_original. Step 7: Solve for V_original: V_original = 18480 * (27 / 8). Step 8: Compute 18480 / 8 = 2310, then V_original = 2310 * 27. Step 9: Calculate 2310 * 27 = 2310 * 20 + 2310 * 7 = 46200 + 16170 = 62370 cm^3.

Verification / Alternative check:
As a quick check, compute the ratio 18480 / 62370 and simplify. Dividing numerator and denominator by 10 gives 1848 / 6237. This ratio should equal 8 / 27. Multiplying 8 / 27 by 62370 indeed returns 18480, confirming that the volume scale factor 8 / 27 is correct. Since we used only similarity ratios and no approximate constants, the value 62370 cm^3 is exact and consistent with the given smaller volume.

Why Other Options Are Wrong:
Other volumes such as 34650, 61600, 36960 and 54000 do not maintain the correct ratio between V_small and V_original equal to (2 / 3)^3. For instance, if 34650 were the correct original volume, then V_small would be 34650 * 8 / 27, which does not equal 18480. Only 62370 satisfies the exact similarity relationship derived from the geometry of similar cones.

Common Pitfalls:
A common mistake is to incorrectly use a linear or square ratio instead of the cube when relating volumes. Some students treat the height ratio 2 / 3 as if it directly applies to volume, giving V_small = (2 / 3) * V_original, which is incorrect. Others may misinterpret the phrase “cut at a height of 15 cm from its base” and take the smaller cone height as 15 cm instead of 30 cm. Carefully visualising the cone and remembering that volume scales with the cube of the linear factor avoids these errors.

Final Answer:
The volume of the original cone is 62370 cm^3.