Chapter 12.4, Problem 1C

To determine

To find: the probability that the other submarine is within $400$ m .

Answer to Problem 1C

The probability that the other submarine is within $400$ m is $0.0093$ .

Explanation of Solution

Given information:

Two research submarines are assigned to take any two positions in the ocean at a depth between $100$ m and $500$ m and within a square area with $6000$ m sides.

First submarine takes its position in the center of the square area at the $100$ m depth.

Calculation:

Probability that the other submarine is within $400$ m :

  $\begin{array}{l}P=\frac{\text{Volume of hemispherical area where submarine can be found}}{\text{Volume of cuboidal area where submarine can be found}}\\ =\frac{\frac{1}{2}\times \frac{4}{3}\pi \times r^3}{lbh}\end{array}$

Take $r=400$ m, $l=6000\text{ m},b=6000\text{ m},h=400\text{ m}$

  $\begin{array}{l}P=\frac{\frac{1}{2}\times \frac{4}{3}\pi \times {\left(400\right)}^3}{6000\times 6000\times 400}\\ =\frac{\frac{1}{2}\times \frac{4}{3}\pi \times 400\times 400\times 400}{6000\times 6000\times 400}\\ =\frac{\frac{2}{3}\pi \times 16}{3600}\\ =\frac{32\pi }{3\times 3600}\end{array}$

  $\begin{array}{l}=\frac{32\times 3.14}{10800}\\ =\frac{100.48}{10800}\\ =0.0093\end{array}$

Therefore, the probability that the other submarine is within $400$ m is $0.0093$ .