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1. Challenge Find the ratio of the shaded area to the area of the largest semi-circle

Consider the following diagram: What fraction of the area of the largest semicircle is the shaded area?
2. Practice Describe reciprocal function

Consider the graph of a rational function, we'll call it $$f(x)$$: The asymptotes and zeroes are shown. Now, consider the function: g(x)=\frac{1}{f(x)} Where will $$g(x)$$ have its asymptotes, and zeroes?
3. Ant on a rubber rope

I saw an interesting problem on another site, and I wanted to post a more general solution. Suppose an ant is going to crawl along a rubber rope, at a constant rate of $$v$$ units per second (relative to the rope). Initially the rope has length $$\ell$$. At the moment the any begins crawling...
4. Why people mistake rejection of theism as a hatred for their god

-j8ZMMuu7MU I hope people will watch this, and actually make some comments.
5. Lesson Another Cool Analytic Geometry Problem

Consider the following diagram: In blue, is a parabola of the form: y=ax^2 where $$0<x$$. Let us define a normal line as a line that is orthogonal to the parabola in the first quadrant. Five normal lines are shown in the figures above. The significance of the solid line in black will be...
6. Lesson A Bernoulli Equation

Suppose we are given to solve: \d{y}{x}+\frac{y}{x}=y^2 A first order ODE that can be written in the form: \frac{dy}{dx}+P(x)y=Q(x)y^n where $$P(x)$$ and $$Q(x)$$ are continuous on an interval $$(a,b)$$ and $$n$$ is a real number, is called a Bernoulli equation. This equation was proposed...
7. Lesson Neat Analytic Geometry Problem

Let $$f(x)$$ be an unknown function defined on $$[0,\infty)$$ with $$f(0)=0$$ and $$f(x)\le x^2$$ for all $$x$$. For each $$0\le t$$, let $$A_t$$ be the area of the region bounded by $$y=x^2$$, $$y=ax^2$$ (where $$1<a$$) and $$y=t^2$$. Let $$B_t$$ be the area of the region bounded by $$y=x^2$$...
8. Lesson Related Rates: Lighthouse Beam

A lighthouse is fixed $$y$$ units from a straight shoreline. A spotlight in the lighthouse revolves at a rate of $$R$$ revolutions per minute, shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along the shoreline $$x$$ units from the shoreline point...
9. Riemann Hypothesis Proved?

Let’s get straight to the point: Sir Michael Atiyah announced that he has a proof of the Riemann Hypothesis (RH) and will present it in Heidelberg this Monday 24 of September. This is already worthy of headlines, but if it happens that the proof is correct we are talking about news of seismic...
10. Lesson Related Rates: A Sand Pile

Sand is being dumped from a conveyor belt at a constant rate, forming a conical pile. The ratio of the base radius of this pile to its height is constant. Find the height of the pile as a function of time. To begin this problem, let's define some constants: $$a$$ = the rate at which sand is...
11. Lesson Work Required To Pump Fluid From a Hemispherical Tank

First, let's let: $$R$$ = the radius of the hemispherical tank. $$\rho$$ = the weight density of the fluid. $$g$$ = the acceleration due to gravity. Now, let's imagine slicing the contents of the tank horizontally into circular sheets. The radius of each sheet will be a function of its...
12. Challenge Summation of Series

Begin by showing: \frac{r}{(r+1)!}=\frac{1}{r!}-\frac{1}{(r+1)!} i) Hence or otherwise , evaluate: S_n=\sum_{r=1}^n\left(\frac{r}{(r+1)!}\right) ii) Evaluate: S_{\infty}=\sum_{r=1}^{\infty}\left(\frac{r+2}{(r+1)!} \right)
13. Practice Remainder Theorem

When a polynomial $$f(x)$$ is divided by $$x-1$$ and $$x-2$$, the remainders are $$0$$ and $$-4$$ respectively. Find the remainder when $$f(x)$$ is divided by $$(x-1)(x-2)$$.
14. Lesson Maximizing Area of Norman Window

A Norman window has the shape of a semicircle atop a rectangle. Suppose such a window has a given perimeter $$P$$. What are the dimensions of the window having the greatest area? Consider the following diagram: One approach is to use Lagrange multipliers. We have the objective function...
15. The Greater Insult

ttevamkS6gw The author of this video wrote this to go along with the video: ---------- Even though I’m a nonbeliever, would I be scared to learn that God really does exist? No. Far from it. The belief that God desires praise, worship, and violent retribution, comes from a lack of...
16. Challenge Maximize Circular Arc

The following very interesting problem was posted on another forum: Find the radius r of a circle c whose center is on a fixed circle C of radius R such that the arc length of the part of c within C is a maximum. I created the following diagram to help the student who posted the problem...
17. Challenge Minimize Volume of Cone Circumscibing Sphere

A cone circumscribes a sphere as follows: The problem: Find the minimum volume of such a cone. Okay, our objective function is the volume of the cone: V(h,r)=\frac{1}{3}\pi r^2h To determine the constraint, I used similarity to write: \frac{h-R}{R}=\frac{\sqrt{r^2+h^2}}{r} After some...
18. Practice Rooting Around

We are given: a^{-\Large\frac{1}{4}}-a^{\Large\frac{1}{4}}=14 Compute the value of: a^{-\Large\frac{1}{6}}+a^{\Large\frac{1}{6}}
19. Practice Don't Go Off On A Tangent!

Or, perhaps going off on a tangent is exactly what we need here. :) Let's begin... Consider the following diagram: Show that: y=x\left(\tan(\alpha+\beta)-\tan(\beta)\right)
20. Lesson Perpendicular Lines Tangent To A Parabola

This is a problem I just made up which incorporates some of the topics we've recently discussed... Consider a parabola whose axis of symmetry is the $$y$$-axis, and whose vertex is at $$(0,b)$$ where $$0<b$$. Consider also two lines passing through the origin which are both tangent to the...