# michaelcai

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### The lateral surface area of the frustum of a solid right cone is the product of one-half the slant height ($L$) and the sum of the circumfer

michaelcai  10 hours ago
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### A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has a lower ba

michaelcai  10 hours ago
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### A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer a

michaelcai  11 hours ago
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### The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of me

michaelcai  11 hours ago
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### If the sum of three real numbers is $0$ and their product is $17$, then what is the sum of their cubes?

michaelcai  Oct 12, 2017
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### Suppose that $a$, $b$, and $c$ are real numbers for which ​ and $a michaelcai Oct 12, 2017 0 40 3 +298 ### What real number is equal to the expression ​, where the$1$s and the$2$s alternate? michaelcai Oct 11, 2017 0 54 2 +298 ### The sequence$x_1$,$x_2$,$x_3$, . . ., has the property that$x_n = x_{n - 1} + x_{n - 2}$for all$n \ge 3$. If$x_{11} - x_1 = 99$, then michaelcai Oct 11, 2017 0 45 1 +298 ### Points$A$,$B$,$C$, and$T$are in space such that each of$\overline{TA}$,$\overline{TB}$, and$\overline{TC}$is perpendicular to the o michaelcai Oct 11, 2017 0 42 1 +298 ### Rectangle$ABCD$is the base of pyramid$PABCD$. If$AB = 3$,$BC = 2$,$\overline{PA}\perp \overline{AD}$,$\overline{PA}\perp \overline{AB

michaelcai  Oct 9, 2017
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### Each of the eight edges of a regular square pyramid has length 6. Find the volume of the pyramid.

michaelcai  Oct 9, 2017
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### Find the sum of the real values of $x$ such that the infinite geometric series $x+\frac{1}{2}x^3+\frac{1}{4}x^5+\frac{1}{8}x^7+\dots$ is equ

michaelcai  Oct 6, 2017
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### Compute the sum $$\frac{2}{1 \cdot 2 \cdot 3} + \frac{2}{2 \cdot 3 \cdot 4} + \frac{2}{3 \cdot 4 \cdot 5} + \cdots$$

michaelcai  Oct 5, 2017
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### The $n^{\text{th}}$ term of a certain geometric series is given by $a\cdot r^{n-1}$, where $a$ and $r$ are positive integers and $r$ is

michaelcai  Oct 5, 2017
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### What is the value of the sum $\frac{2}{3}+\frac{2^2}{3^2}+\frac{2^3}{3^3}+ \ldots +\frac{2^{10}}{3^{10}}$? Express your answer as a common f

michaelcai  Oct 5, 2017
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### Let $$f(x) = \frac{x^2}{x^2 - 1}.$$Find the largest integer $n$ so that $f(2) \cdot f(3) \cdot f(4) \cdots f(n-1) \cdot f(n) < 1.98.$

michaelcai  Oct 5, 2017
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### Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 60^\circ$, and $AC=10$. Find the radius of the incircle of $\triangle ABC$

michaelcai  Oct 2, 2017
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### The set $\{2, 4, 6, \dots, n\}$ contains the positive consecutive even integers from 2 through $n$. When one of the integers from the set is

michaelcai  Sep 29, 2017