All Questions
214
questions
6
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4
answers
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Definite integral: $\int^{4}_0 (16-x^2)^{\frac{3}{2}}\,dx$
The following integral can be computed using the substitution $x = 4\sin\theta~$ and then proceeding with $dx = 4\cos\theta~ d\theta~$, and evaluating the integral of $\cos^4\theta$ :
$$\int^{4}_0 (...
- 1,498
6
votes
5
answers
6k
views
Why is the distance between two circles/spheres that don't intersect minimised at points that are in the line formed by their centers?
From GRE 0568
From MathematicsGRE.Com:
I'm guessing the idea applies to circles also?
Is there a way to prove this besides the following non-elegant way?
Form a line between centers $C_1$ and $C_2$
...
- 13.7k
6
votes
3
answers
4k
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Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$
Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$ respectively,then find $2M+6m.$
Let $x=\cos\theta$ and $y=\sin\theta$,...
- 5,868
6
votes
4
answers
218
views
Tangent to $x^2+y^2-6x-6y=-13$ and $x^2+y^2+2x+2y=-1$
Considering the circles $\lambda: x^2+y^2-6x-6y=-13$ and $\theta: x^2+y^2+2x+2y=-1$ find the line simultaneously tangent to them.
I found the implicit derivative of those two,
$\lambda: y'=-\frac{x-3}{...
- 109
6
votes
5
answers
728
views
The hardest geometry question with "a triangle" and "a circle" - Circle intersecting triangle equally in 5 parts
I received this question long time ago from one of my old friends who is mathematician/physicist. He called it the hardest geometry question with "a triangle" and "a circle". I am ...
- 725
6
votes
1
answer
3k
views
Is *njwildberger* wrong about area and circumference of a circle?
In this video, njwildberger says that the area and circumference of a circle are proof-less theorems. But I heard that we can derive both the area and circumference of a circle using calculus? So are ...
6
votes
2
answers
170
views
How do I evaluate the following limit?
I understand that I need to somehow use that $PR=AP=AQ$ as the point $A \to P$. But beyond that, I am unable to use that information to find $OB$.
This problem is from the textbook "Calculus with ...
- 63
6
votes
1
answer
235
views
Triangle and Circle maximization problem
So I was playing around GeoGebra and found this thing out, I don't know if this problem has a name or something.
Triangle ABC is inscribed inside a circle, from point D which is located inside the ...
- 61
6
votes
2
answers
2k
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Shortest path between two points around an obstacle?
I'm trying to figure out a problem that goes like this:
A particle originally placed at the origin tries to reach the point $(12,16)$ whilst covering the shortest distance possible. But there is a ...
- 71
6
votes
0
answers
131
views
Is this a valid way of deriving the area of a circle?
On the Wikipedia article about deriving the area of a circle, it mentions that the formula
$$
\text{area} = \pi r^2
$$
can be derived by evaluating the integral
$$
2 \int_{-r}^{r} \sqrt{r^2-x^2} \, dx ...
- 20.7k
5
votes
2
answers
359
views
Geometric Identities involving $π^2$
Are there any known geometric identities that have $π^2$ in the formula?
- 1,844
5
votes
2
answers
23k
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Rotate a point on a circle with known radius and position
Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the ...
- 567
5
votes
2
answers
193
views
Question from applications of derivatives.
Prove that the least perimeter of an isoceles triangle in which a circle of radius $r$ can be inscribed is $6r\sqrt3$.
I have seen answer online on two sites. One is on meritnation but the problem is ...
- 847
5
votes
1
answer
731
views
Average distance to a non-central point in a circle
If I pick a point within the circle about the origin of radius $R$, say $(r,\theta) = (0.5 R, \frac{\pi}{2})$, what is the average distance of all other points to that point?
Things which are ...
- 61
4
votes
3
answers
582
views
Help with Calculus Optimization Problem!
We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and green is the seating area). For safety reasons, ...
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