SAT Math Practice Question #599 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #599

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

A plane parallel to the circular base of a right circular cone intersects the cone, separating it into a smaller cone and a frustum. The volume of the smaller cone is exactly \( \frac{1}{8} \) the volume of the original cone. The lateral surface area of the frustum is \( 195\pi \). If the base radius of the original cone is \( 10 \), what is the height of the original cone?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Recall the standard circle equation: \( (x - h)^2 + (y - k)^2 = r^2 \). Complete the square if the equation is expanded.

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Plot the left-hand side and right-hand side as separate equations (e.g. \( y_1 = f(x) \) and \( y_2 = g(x) \)). Click the intersection points directly to read exact coordinates.

Masterclass Solution & Distractor Trap Analysis

This 3D geometry problem requires absolute fluency in the scaling properties of similar figures. Step 1: Determine the scale factor (k). The plane is parallel to the base, meaning the smaller top cone is a perfect similar miniature of the o...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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