14. Convert the points to Cartesian coordinates.
[tex]P_1[/tex] becomes
(7 cos(217/180 π), 7 sin(217/180 π)) ≈ (-5.59, -4.21)
and [tex]P_2[/tex] is
(5 cos(-23/36 π), 5 sin(-23/36 π)) ≈ (-2.11, -4.53)
Then the distance between the two points is
[tex]\sqrt{(-5.59+2.11)^2+(-4.21+4.53)^2}\approx\boxed{3.49}[/tex]
16. Between the first two seconds, the ball rolls 2 ft. If the distance traveled by the ball between each second forms an arithmetic sequence, then the distances themselves are
1, 3, 5, 7, 9, ...
That is, the distances form a sequence [tex]d_n[/tex] where
[tex]\begin{cases}d_1=1\\d_n=d_{n-1}+2&\text{for }n>1\end{cases}[/tex]
We can solve for the n-th distance [tex]d_n[/tex] in terms of the first [tex]d_1[/tex] using the recursive rule above.
[tex]d_2=d_1+2[/tex]
[tex]d_3=d_2+2=(d_1+2)+2=d_1+4[/tex]
[tex]d_4=d_3+2=(d_1+4)+2=d_1+6[/tex]
and so on, up to
[tex]d_{15}=d_1+2(15-1)=\boxed{29}[/tex]