If two point charges \(+q\) and \(−q\) are separated by distance \(d,\) then according to Coulomb's law, the attractive force between them is
\[F_e=k_e\frac{q^2}{d^2},\]
where \(k_e\) is Coulomb's constant. But what if the charges were distributed on spheres instead of at points? Two charged conducting spheres with charges \(+q\) and \(−q\) are separated by center-to-center distance \(d.\)
The attractive force between them is \(\text{__________}.\)
Are you sure you want to view the solution?
\(n\) | \(\hspace{10mm} 2^n\) | Concatenation of the powered numbers | Divisibility checked |
0 | \(\hspace{5mm} 2^0=1\) | \(\hspace{25mm} 1\) | \(\hspace{8mm} 1\, \big|\, 1\) |
1 | \(\hspace{5mm} 2^1=2\) | \(\hspace{25mm} 12\) | \(\hspace{8mm} 2\, \big|\, 12\) |
2 | \(\hspace{5mm} 2^2=4\) | \(\hspace{25mm} 124\) | \(\hspace{8mm} 4\, \big|\, 124\) |
3 | \(\hspace{5mm} 2^3=8\) | \(\hspace{25mm} 1248\) | \(\hspace{8mm} 8\, \big|\, 1248\) |
4 | \(\hspace{5mm} 2^4=16\) | \(\hspace{25mm} 124816\) | \(\hspace{8mm} 16\, \big|\, 124816\) |
As we get greater and greater numbers in column 3 of the table by concatenation (i.e. 12481632, 1248163264, ...) for \(n>4,\) will the divisibility in the last column still hold?
Are you sure you want to view the solution?
It's common knowledge that the graph of \(x^2 + y^2 \leq 1\) is a unit disk, and the area of the region is \(\pi.\)
However, it's not common for someone to know the shape of the graph \[ \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1,\] where \(\lfloor \cdot \rfloor\) is the floor function. To three decimal places, what is the area of the region on the coordinate plane that satisfies the equation \( \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1?\)
\(\)
Note: Try to draw a hand sketch of this curve and provide analysis instead of using any software!
Are you sure you want to view the solution?
Does there exist a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for every two distinct real numbers \(a\) and \(b,\) \(f(a)\) and \(f(b)\) differ by at least \(1?\)
Are you sure you want to view the solution?
Three identical 30-60-90 right triangles are arranged as shown in the figure.
What is the ratio of the \(\color{red}\text{red segment's}\) length to the \(\color{blue}\text{blue segment's}\) length?
Are you sure you want to view the solution?
Problem Loading...
Note Loading...
Set Loading...