Forgot password? New user? Sign up
Existing user? Log in
A monkey is typing a string of random letters, with all letters equally (and independently) likely to be typed.
Which word is more likely to appear first in the string, heart or earth?
Are you sure you want to view the solution?
Take a point $P$ on the graph of $y=2x^2,$ and draw a line going through $P$ parallel to the $y$-axis. Then call the red area $A:$ the area bounded by this line, the graph of $y=x^2,$ and the graph of $y=2x^2.$
Now, draw a line going through $P$ parallel to the $x$-axis. Then call the blue area $B:$ the area bounded by this line, the graph of $y=2x^2,$ and the graph of $y=f(x).$
This function $f(x)$ is continuous and displays the unique property that for every point $P$ on $y = 2x^2,$ the two areas $A$ and $B$ are equal.
Find $f(x)$ and evaluate $f(12).$
Are you sure you want to view the solution?
An infinitely thin, $k\text{ cm}$-long squeegee begins to slide down from the upper-left corner of a $k\text{ cm} \times k\text{ cm}$ square window. Its other end simultaneously slides toward the lower-right corner of the window, with the upper end kept in contact with the left side of the window.
If $k=\sqrt{\dfrac{2016}{\pi }}$, what is the area of the window $\big($in $\text{cm}^2\big)$ cleaned by the squeegee?
Note: you may end up with an integral expression that's difficult to evaluate analytically. Feel free to finish the job using the code environment below:
import math
Are you sure you want to view the solution?
Evaluate the following integral:
$\large \int_{ - \infty}^{\infty} \frac{\cos x}{1+x^2} dx.$
Are you sure you want to view the solution?
In the country of Brilliantia, there are $n > 60$ airports built recently such that the distances between any two airports are all different from one another.
A plane will always fly from an airport to the nearest airport. Let an origin of an airport $\text{A}$ be another airport that flies planes to $\text{A}$.
What is the maximum number of origins an airport can have?
Are you sure you want to view the solution?
Problem Loading...
Note Loading...
Set Loading...