Question

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14. Find the distance between $P_{1}(7,\frac{217}{180}\pi)$ and $P_2(5,-\frac{23}{36}\pi)$ on the polar plane.
16. Makya was conducting a physics experiment. He rolled a ball down a ramp and calculated the distance covered by the ball at different times. The ball rolled a distance of 1 foot during the first second, 3 feet during the next second, and so on. If the distances the ball rolled down the ramp each second form an arithmetic sequence, determine the distance the ball rolled down during the fifteenth second.

Answer 1

14. Convert the points to Cartesian coordinates.

$P_1$ becomes

(7 cos(217/180 π), 7 sin(217/180 π)) ≈ (-5.59, -4.21)

and $P_2$ is

(5 cos(-23/36 π), 5 sin(-23/36 π)) ≈ (-2.11, -4.53)

Then the distance between the two points is

$\sqrt{(-5.59+2.11)^2+(-4.21+4.53)^2}\approx\boxed{3.49}$

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 $d_n$ where

$\begin{cases}d_1=1\\d_n=d_{n-1}+2&\text{for }n>1\end{cases}$

We can solve for the n-th distance $d_n$ in terms of the first $d_1$ using the recursive rule above.

$d_2=d_1+2$

$d_3=d_2+2=(d_1+2)+2=d_1+4$

$d_4=d_3+2=(d_1+4)+2=d_1+6$

and so on, up to

$d_{15}=d_1+2(15-1)=\boxed{29}$