How to find the point to equations meet

[st0nersteve]

Homework Statement

A man is swimming 3m/s down a river. The man is 50m from the edge and 1000m from the top of the river. If I am in a boat at the top corner of the river traveling 10m/s. what angle should i leave to reach the man.

NEED TO FIND ANGLE $$\Theta$$ AND TIME

Homework Equations

s = ut + 1/2at^2

The Attempt at a Solution

Opposite side
50 = 10sin$$\Theta$$*t + 0
50 = 10sin$$\Theta$$*t + 1/2*0*t^2
T=50/(10sin$$\Theta$$)

Adjacent side
1000=(10cos$$\Theta$$ - 3)*t
t= 1000/(10cos$$\Theta$$ - 3)

Therefore i need to find the point

t= 1000/(10cos$$\Theta$$ - 3)
&
T=50/(10sin$$\Theta$$)
meet

i know it works the answer is around 143secs and 2 degrees but i can't prove it mathematically
Thank u for future help

 

[anathema]

Thank you for your interesting problem. I would approach this problem by first creating a diagram to visualize the situation. From the given information, we know that the man is swimming downstream at a constant velocity of 3m/s, and that he is 50m from the edge and 1000m from the top of the river. We also know that you are in a boat at the top corner of the river, traveling at a constant velocity of 10m/s.

Using this information, we can create a right triangle with the hypotenuse representing the distance traveled by the man, the adjacent side representing the distance traveled by the boat, and the opposite side representing the distance between the man and the boat. The angle \Theta represents the angle between the hypotenuse and the adjacent side.

To find the angle \Theta, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side. In this case, we have:

tan\Theta = opposite/adjacent = 50/1000 = 1/20

Using a calculator, we can find that \Theta is approximately 2.86 degrees.

To find the time it takes for the boat to reach the man, we can use the distance formula s = ut + 1/2at^2, where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time. In this case, we are looking for the time t, so we can rearrange the formula to solve for t:

t = (s - 1/2at^2)/u

Substituting in the values for s, u, and a, we get:

t = (1000 - 1/2*3*t^2)/10

Simplifying, we get:

t = 100 - 3/20*t^2

Using the quadratic formula, we can solve for t and find that it is approximately 143 seconds.

I hope this helps to explain the mathematical reasoning behind the solution. If you have any further questions or would like to discuss this problem further, please let me know. Best of luck with your studies.