A boater finds herself 4 miles fromm the nearest point on a straight shore line whech is 6 miles from a shoreside motel. She plans to row on shore and then walk to a motel. If she can walk at 3 mph, at what speed must she be able to row so that the quickest way to to the motel is to row directly?
Well, to calculate that, we should take into account both the rowing distance and the walking distance. Let's call the speed of rowing "R":
The rowing distance can be calculated using the Pythagorean theorem: sqrt(4^2 + 6^2) = 7.21 miles (approximately).
Now we can compare the time it takes to row directly versus the time it takes to row to shore and then walk to the motel.
For rowing, it will take her 7.21/R hours, and for walking, it will take her 6/3 = 2 hours.
So, the total time it takes to row and then walk is 7.21/R + 2.
If she wants the quickest way to the motel, that means the total time rowing and walking needs to be minimized. To do that, we can differentiate the time with respect to R:
d(7.21/R + 2)/dR = -7.21/R^2.
Setting this equal to zero and solving for R, we get R = 7.21/0, which is undefined.
Well, apparently, our math has taken a detour just like this boater's plans. It seems there's a bit of a funny situation here. Maybe the quickest way to the motel is to just embrace the absurdity and stay afloat with laughter! 😄
To determine the speed at which the boater must row so that the quickest way to the motel is to row directly, we need to compare the time it takes to row directly to the time it takes to row to the nearest point on the shoreline and then walk to the motel.
Let's start by calculating the time it takes to row directly to the motel.
The distance from the boater to the motel is given as 6 miles. Let's assume she rows at a speed of R mph.
Time taken to row directly to the motel = Distance / Speed
Time taken = 6 miles / R mph = 6/R hours
Now, let's calculate the time it takes to row to the nearest point on the shoreline and then walk to the motel.
The boater is initially 4 miles away from the nearest point on the shoreline. Let's assume she rows at a speed of R mph.
Time taken to row to the shoreline = Distance / Speed
Time taken to row = 4 miles / R mph = 4/R hours
Once she reaches the shoreline, she needs to walk from there to the motel. The distance she needs to walk is 6 miles, and her walking speed is 3 mph.
Time taken to walk = Distance / Speed
Time taken to walk = 6 miles / 3 mph = 2 hours
Total time taken to row to the shoreline and then walk to the motel = Time taken to row + Time taken to walk
Total time taken = (4/R) hours + 2 hours = 4/R + 2 hours
To find the quickest way, the rowing time directly to the motel should be less than the time taken to row to the shoreline and then walk to the motel.
Therefore, 6/R < 4/R + 2
To isolate R, we can subtract 4/R from both sides of the inequality:
6/R - 4/R < 2
Combining the fractions on the left side of the inequality:
(6 - 4)/R < 2
Simplifying further:
2/R < 2
Cross multiplying:
2 < 2R
Divide both sides by 2:
1 < R
Therefore, the boater must be able to row at a speed greater than 1 mph (R > 1 mph) to make rowing directly to the motel the quickest way.