Question

on a float trip, nate traveled 33 miles downstream in the same amount of time that it would take him to row 14 miles upstream. if the speed of the current is 2 mph, find his speed in still water. a) using the variable x to represent nate’s speed in still water, write an equation using the information as it is given above that can be solved the problem. equation: b) nate’s speed in still water is mph. (round your answer to 1 decimal place.)

Explanation:

Part A

Step1: Determine downstream and upstream speeds

Downstream speed = speed in still water + current speed = \( x + 2 \) mph.
Upstream speed = speed in still water - current speed = \( x - 2 \) mph.

Step2: Recall time formula (\( \text{time} = \frac{\text{distance}}{\text{speed}} \))

Time downstream: \( \frac{33}{x + 2} \) hours.
Time upstream: \( \frac{14}{x - 2} \) hours.

Step3: Set times equal (same time)

Since times are equal, the equation is \( \frac{33}{x + 2} = \frac{14}{x - 2} \).

Step1: Cross - multiply the equation

From \( \frac{33}{x + 2}=\frac{14}{x - 2} \), cross - multiply to get \( 33(x - 2)=14(x + 2) \).

Step2: Expand both sides

\( 33x-66 = 14x + 28 \).

Step3: Solve for \( x \)

Subtract \( 14x \) from both sides: \( 33x-14x-66=14x - 14x+28 \), which simplifies to \( 19x-66 = 28 \).
Add 66 to both sides: \( 19x-66 + 66=28 + 66 \), so \( 19x=94 \).
Divide both sides by 19: \( x=\frac{94}{19}\approx4.9 \).

Answer:

\( \frac{33}{x + 2} = \frac{14}{x - 2} \)

Part B