A water taxi travels around an island in a path that can be modeled by the equation y =0.5(x - 16). A water skier is skiing along a path that begins at the point(6,6) and ends at the point (8. - 4).
a. Write a system of equations to model the problem.
b. Is it possible that the water skier could collide with the taxi? Explain

Final answer:

The system of equations to model the problem is y = 0.5(x - 16) for the water taxi's path and y = -5x + 36 for the water skier's path. No collision is possible between the water skier and the taxi.

Explanation:

a. System of Equations:

To model the problem, we need to find the equation of the water skier's path.

Using the two given points, we can find the slope of the water skier's path:

1. 
   
2.  
3. Find the change in y-coordinates: -4 - 6 = -10
4. 
   
5.  
6. Find the change in x-coordinates: 8 - 6 = 2
7. 
   
8.  
9. Calculate the slope: slope = -10 / 2 = -5

Now that we have the slope, we can use the point-slope formula to find the equation of the water skier's path:

y - y1 = m(x - x1)

Substituting the values x1 = 6, y1 = 6, and m = -5:

y - 6 = -5(x - 6)

Simplifying:

y = -5x + 36

Therefore, the system of equations to model the problem is:

y = 0.5(x - 16) - Equation of the water taxi's path

y = -5x + 36 - Equation of the water skier's path

b. Possibility of Collision:

To determine if the water skier could collide with the taxi, we need to solve the system of equations. If the x-coordinate at the point of intersection is within a certain range, then a collision is possible.

Substitute the second equation into the first equation:

0.5(x - 16) = -5x + 36

Solve for x:

0.5x - 8 = -5x + 36

5.5x = 44

x = 8

The x-coordinate of the point of intersection is 8. Since the water skier's path does not pass through the point x = 8, there is no collision between the water skier and the taxi.

Learn more about System of equations here:

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I realize that the equation is different, but do it in the same way, it'll help! It is given that the water​ taxi's path can be modeled by the equation y =0.5(x - 14)^2. ​Therefore, this is one of the equations in this system. Find a linear equation that will model the path of the water​ skier, which begins at the point (6,6) and ends at the point (8,-4). The slope is (-5). Use the slope and one point on the line to find the​ y-intercept of the line.  The​ y-intercept of the line that passes through the points (6,6) and (8,-4) is (0,36). Thus, the equation is y=-5x+36. Now, to determine if it is possible for the water skier to collide with the​ taxi, we have to determine if there is a solution to the system of equations. To determine if there is a solution to the system of​ equations, solve the system using substitution.​ First, write the equation that models the water​ taxi's path in standard form. y=0.5(x - 14)^2-->0.5x^2-14x+98. Use substitution. Substitute for  y in the equation and then solve for  x. As the expression on the left side of the equation cannot easily be​ factored, use the Quadratic Formula to solve for x. Do x=-b(plusorminus)sqrrtb^2-4ac/2a. Identify a, b, and c. a=0.5, b=-9, and c=62. Substitute into the Quadratic Formula. If there is a negative number under the radical, there are NO solutions. Thus, the path of the water skier will never cross the path of the taxi.

In conclusion: It is not possible that the water skier could collide with the taxi as the two paths never cross.