Two people take identical jet skis across a river, traveling at the same speed relative to the water. Jet ski A heads directly across the river and is carried downstream by the current before reaching the opposite shore. Jet ski B travels in a direction that is 35° upstream, and arrives at the opposite shore directly across from the starting point.

(a)Find the ratio of the time it takes for the two jet skis to cross the river.

Ah, jet skis and river crossings. Now we're talking! Let's dive into this river-crossing conundrum, shall we?

Since we're talking about ratios here, let's compare the time it takes for each jet ski to cross the river. We'll call the time it takes for jet ski A to cross as "tA" and the time it takes for jet ski B to cross as "tB".

Now, jet ski A goes straight across the river, while jet ski B takes a bit of a detour upstream before reaching the other side. So we need to find the ratio between these two times.

The key here is that the distance traveled by both jet skis is the same - they both end up on the opposite shore, right? So we can set up a little equation using the distances traveled.

Let's say the distance across the river is "d". Jet ski A covers this distance directly, while jet ski B has to cover the same distance but at an angle of 35° upstream. Don't worry, math is funnier than it sounds!

Now, the distance covered by jet ski A can be calculated as "d", since it goes straight across. But for jet ski B, we need to break it down into two components: the distance traveled upstream and the distance traveled horizontally.

The distance traveled upstream can be calculated using trigonometry (yup, long-forgotten high school math is back for a fun visit!). The formula is "d * sin(35°)".

Now, we combine the distance traveled upstream with the horizontal distance traveled, which is the same as jet ski A's distance, "d".

So, for jet ski B, the total distance covered is "d + d * sin(35°)".

Now we can set up our equation: tA / tB = d / (d + d * sin(35°)).

Dividing both sides by "d", we get: tA / tB = 1 / (1 + sin(35°)).

So, the ratio of the time it takes for jet ski A to cross to the time it takes for jet ski B to cross is 1 : (1 + sin(35°)).

Now, I could give you the numerical answer, but why ruin a good joke? Let's just say it involves some sines, cosines, and a dash of laughter. Enjoy cracking that mathematical code, my friend!

To find the ratio of the time it takes for the two jet skis to cross the river, we can use the concept of vector addition.

Let's assume the speed of the jet skis relative to the water is V and the speed of the river's current is C.

For Jet ski A, it travels directly across the river. The resultant velocity can be found by adding the velocities of the jet ski and the current. Since the current is carrying the jet ski downstream, the resultant velocity will have a component in the downstream direction and a component perpendicular to the river.

For Jet ski B, it travels at an angle of 35° upstream. This means it is heading against the current. Again, the resultant velocity can be found by adding the velocities of the jet ski and the current.

Now, let's break down the velocities into their components:

For Jet ski A:
- The downstream component of the velocity (Vd) is V + C
- The perpendicular component of the velocity (Vp) is zero since it travels directly across the river.

For Jet ski B:
- The upstream component of the velocity (Vu) is V - C
- The perpendicular component of the velocity (Vp) is zero since it reaches directly across the river.

Since the distances traveled by both jet skis are the same (as they reach directly across the river), we can compare their speeds using the time taken.

Let's assume the width of the river is D, and the time taken for Jet ski A to cross the river is TA.

For Jet ski A:
Distance = D (width of the river)
Speed = Vd
Time = D / (V + C) = TA

For Jet ski B:
Distance = D (width of the river)
Speed = Vu (upstream component)
Time = D / (V - C) = TB

Now, let's find the ratio of the two times:

TA / TB = (D / (V + C)) / (D / (V - C))
TA / TB = (V - C) / (V + C)

So, the ratio of the time it takes for the two jet skis to cross the river is (V - C) / (V + C).

To find the ratio of the time it takes for the two jet skis to cross the river, we first need to determine the velocities of each jet ski.

Let's consider the velocity of the river current as Vc and the speed of the jet skis as Vj.

For Jet ski A:
Since it goes directly across the river, its velocity with respect to the ground is the vector sum of its speed and the velocity of the river current. Let's call this Vja.

For Jet ski B:
It travels at an angle of 35° upstream, which means its velocity with respect to the ground is the vector difference of its speed and the velocity of the river current. Let's call this Vjb.

Now, let's break down the velocities into their horizontal and vertical components using trigonometry.

For Jet ski A:
The horizontal component of Vja is given by Vj * cos(0°) since it's traveling directly across the river. Therefore, the horizontal component of Vja is Vj.

The vertical component of Vja is given by Vc since Jet ski A is being carried downstream by the river current.

For Jet ski B:
The horizontal component of Vjb is given by Vj * cos(35°) since it's traveling at an angle of 35° upstream. Therefore, the horizontal component of Vjb is Vj * cos(35°).

The vertical component of Vjb is given by -Vj * sin(35°) since it's traveling upstream against the river current. The negative sign indicates the opposite direction of the river current.

Now, we can calculate the time it takes for each jet ski to cross the river using the formula:

Time = Distance / Speed

For Jet ski A, the distance to cross the river is the width of the river, and its speed is Vja.

For Jet ski B, since it reaches the opposite shore directly across from the starting point, the distance it travels horizontally is also the width of the river, and its speed is Vjb.

To find the ratio of the time it takes for Jet ski A and Jet ski B to cross the river, we can divide the time for Jet ski A by the time for Jet ski B:

Ratio = Time for Jet ski A / Time for Jet ski B

By substituting the expressions for time, distance, and speed for each jet ski, we can find the ratio.