Tutor profile: Will B.

A triangle has one side with angle 45 degrees. The two adjacent sides have lengths of 2 and 4. Find the length of the third side.

We can solve this with the law of cosines, $$c=\sqrt{a^2+b^2-2abcos(\gamma)}$$, where a and b are the adjacent sides, c is the opposite side, and $$\gamma$$ is the angle. Plugging into this equation a=2, b=4, and $$\gamma$$=45 degrees, we find $$c=\sqrt{2^2+4^2-2*4cos(45)}=\sqrt{4+16-\frac{8}{\sqrt{2}}}$$, or approximately 5.07.

Solve the indefinite integral $$\int sin(x)cos(x)dx$$.

This problem can be simplified with u substitution. Let $$u=sin(x)$$. Then $$du=cos(x)dx$$. We can substitute this into the equation to get $$\int u du$$. Using the power rule, we simply fund that this integral is equal to $$\frac{1}{2} u^2+C$$ for some constant C. Plugging back in $$u=sin(x)$$, we find that the integral is equal to $$\frac{1}{2} sin(x)^2+C$$. Using the chain rule to differentiate this confirms that we have performed the integral correctly.

An stationary object explodes into three pieces of shrapnel of equal mass, A, B, and C. A travels directly east, B travels directly south, and C travels at 3 m/s at 30 degrees north of west. What are the velocities of A and B?

By conservation of momentum, we can break up the vertical and horizontal momentum of C to find the momentum (and therefore velocity) of A and B. All shrapnels have the same mass, so it cancels. A's momentum is equal to C's horizontal momentum, so the velocity of A is $$3 cos(30)$$ east. Similarly, B's velocity is $$3sin(30)$$ south.