The square of a binomial expands as follows:

Expanding all the binomials, we have

Sum like terms:

Simplify terms appearing on both sides:

Move all terms involving k to the left and all numbers to the right:

Divide both sides by -20:

10xt=190
T represents the amount of trips and 10 represents the amount of money she spends each trip. Since she has $190 already it will be on the other side of the equation after the equal sign
Standard deviation (s): 2.7386127875258
Simplify the integrands by polynomial division.


Now computing the integrals is trivial.
5.

where we use the power rule,

and a substitution to integrate the last term,

8.

using the same approach as above.