How do you set up this problem?
A boater travels 16 miles per hour on the water on a still day. During one particular windy day, he finds that he travels 48 miles with the wind behind him in the same amount of time that he travels 16 miles into the wind. Find the rate of the wind.
To set up this problem, we can use the concept of speed, time, and distance. Let's denote the rate of the boater's speed in still water as "b" and the rate of the wind as "w."
Given that the boater travels 16 miles into the wind, it means that his apparent speed is reduced by the speed of the wind. So, the effective speed against the wind would be "b - w."
Similarly, when the boater is traveling with the wind, his effective speed would be increased by the speed of the wind, i.e., "b + w."
We are also given that the boater travels 48 miles with the wind behind him in the same amount of time that he travels 16 miles into the wind. Mathematically, we can express this as:
Time taken with the wind = Time taken against the wind
Distance traveled with the wind / Effective speed with the wind = Distance traveled against the wind / Effective speed against the wind
48 / (b + w) = 16 / (b - w)
Now, we can solve this equation to find the value of "w" (the rate of the wind).
To set up this problem, we need to identify the variables involved and establish equations based on the given information.
Let's denote the rate of the boat in still water as "b" and the rate of the wind as "w".
We know that the boater can travel 16 mph in still water. However, we need to consider the effect of the wind, which can either help or hinder the boat's progress.
When the boater is traveling with the wind, their effective speed is increased by the rate of the wind, so we can set up the equation:
b + w = 48/𝑡, where 𝑡 represents the time in hours to travel 48 miles.
Conversely, when the boater is traveling against the wind, the effective speed is reduced by the rate of the wind, so we have:
b - w = 16/𝑡, where 𝑡 represents the time in hours to travel 16 miles.
Now, we have two equations to solve simultaneously. By solving these equations, we can find the value of "w" (the rate of the wind) that satisfies both equations.
this one is similar to the previous two questions you asked and were helped by bobpursley.
Remember Time = Distance/Rate, so..
Time for first case = 48/(16+x)
time for second case = 16/(16-x)
but the times were the same
48/(16+x) = 16/(16-x)
cross-multiply and solve for x