### Math Notes

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#### Algebra Solutions

##### Topics || Problems

A motorboat takes 4 hours to travel 30 miles downstream and then 18 miles upstream on a river whose current flows at the rate of 3 miles per hour. How fast can the boat travel in still water?

Let $$V_d$$, $$V_u$$ and $$V_s$$ be the the velocity of the motorboat downstream, upstream and in still water respectively.

Let $$t, ~ t_d, ~and ~t_u$$ be the total time travel, time moving downstream and time moving upstream respectively. Thus $$4 = t_u + t_d$$

$$V = \frac{distance}{time}$$

$$V_d = V_s + 3$$

$$\frac{30}{4-t_u} = V_s + 3$$

$$V_u = V_s - 3$$

$$\frac{18}{t_u} = V_s - 3$$

$$\frac{30}{4-t_u} - \frac{18}{t_u}= V_s +3 -(V_s - 3)$$

$$\frac{30}{4-t_u} - \frac{18}{t_u}= 6$$

$$30t_u - 18 (4-t_u) = 6 (t_u)(4 - t_u)$$

$$30t_u - 72 +18t_u) = 24t_u -6t_{u}^{2})$$

$$6t_{u}^2 + 24 t_u -72 = 0$$

$$t_u = 2$$

$$V_s = \frac{18}{2} +3$$

$$V_s = 12 ~mph$$