Browsing category ask uncle colin

Ask Uncle Colin: A Partition Enigma

Dear Uncle Colin, In reading Sir Dermot Turing’s XY&Z, he states that the number of species of cycle is 101 - and after a bit of thought, I figured out that that’s the number of partitions of the number 13. However, I couldn’t work out how to get 101! Can

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Ask Uncle Colin: Family

Dear Uncle Colin, I have a probability question that involves a weird place where every family has three children, and every child is either a girl or a boy. 50-50. Independent of each other. If I take a random family, and choose at random one of the children, I have

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Ask Uncle Colin: The Timeless SUVAT Equation

Dear Uncle Colin, I can derive the timeless SUVAT equation $v^2 = u^2 + 2as$, but I can’t intuitively see where it comes from. Any clues? - Everyone Needs Explanations, Really Get Yours Hi, ENERGY, and thanks for your message! This is one that I never really picked up on

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Ask Uncle Colin: A Parametric Integration

Dear Uncle Colin, I have the parametric equations $x = (t+1)^2$ and $y = \frac{1}{2}t^3 + 3$ and the lines $y = 16 - x$ and $x=1$. I need to find the area enclosed by the curve, these two lines and the $x$-axis, but my answer doesn’t agree with the

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Ask Uncle Colin: Multiple solutions

Dear Uncle Colin, I just solved $2\cos^2(4x)=1$ between 0 and $2\pi$ and found four solutions: $\frac{1}{16}\pi$, $\frac{3}{16}\pi$, $\frac{5}{16}\pi$ and $\frac{7}{16}\pi$. The answer scheme says there are sixteen solutions! Where have I gone wrong? Have You Perhaps A Trig Identity Answer? Hi, HYPATIA, and thanks for your message! Looking at your

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Ask Uncle Colin: A Pair Of Birds

Dear Uncle Colin, I have a little problem. You see, there’s this bird, A, in its nest at time $t=0$ - the nest is at $(20, -17)$ - and it travels with a velocity of $-6\bi + 7\bj$ (in the appropriate units). But there’s another bird, B, whose nest is

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Ask Uncle Colin: Meeting Graphs

Dear Uncle Colin, I’m told that the graphs of the functions $f(x) = x^3 + (a+b)x^2 + 3x -4$ and $g(x) = (x-3)^3 + 1$ touch - and I need to express $a$ in terms of $b$. Can you help? - Can’t Understand Basic Introductory Calculus Hi, CUBIC, and thanks

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Ask Uncle Colin: An Area To Find

Dear Uncle Colin, I need to find the area between the curves $y=16x$, $y= \frac{4}{x}$ and $y=\frac{1}{4}x$, as shown. How would you go about that? Awkward Regions, Exhibit A Hi, AREA, and thanks for your message! As usual, there are several possible approaches here, but I’m going to write up

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Ask Uncle Colin: A Missing Digit

Dear Uncle Colin, A seven-digit integer has 870,720 as its last six digits. It is the product of six consecutive even integers. What is the missing first digit? Please Reveal Our Digit! Underlying Calculation Too Hi, PRODUCT, and thanks for your message! There are several approaches to this (as usual)

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Ask Uncle Colin: An Infinite Product

Dear Uncle Colin, I have to show that $\Pi_1^\infty \frac{(2n+1)^2 - 1}{(2n+1)^2} > \frac{3}{4}$. How would you do that? Partial Results Obtained Don’t Undeniably Create Truth Hi, PRODUCT, and thanks for your message! That’s a messy one. I can see two reasonable approaches: one is to take the whole thing

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