Two skaters are practicing at the same time on the same rink. A coordinate grid is superimposed on the ice. One skater follows the path y=-4x+18, while the other skater follows the curve y=-3^2+17x .Find all the points where they might collide if they are not careful.
First, I believe there was an error in your equation. It seems like you wanted to write "y=-3x^2 + 17x". With this correction, to find the points where the two skaters could possibly collide, we need to set the two expressions for y equal to each other and solve for x:
-4x + 18 = -3x^2 + 17x
Rearranging the terms, we obtain:
3x^2 + 21x - 18 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0. We can solve for x using the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
In this equation, a = 3, b = 21, and c = -18. So we have:
x = [-21 ± sqrt((21)^2 - 4*3*(-18))] / (2*3)
x = [-21 ± sqrt(441 + 216)] / 6
x = [-21 ± sqrt(657)] / 6
x = [-21 ± sqrt(3*7*31)] / 6
x = [-21 ± sqrt(3)* sqrt(7)* sqrt(31)] / 6
As the square root of 657 is not a rational number, we can leave the solutions in this form for generality:
x1 = [-21 + sqrt(3)* sqrt(7)* sqrt(31)] / 6
x2 = [-21 - sqrt(3)* sqrt(7)* sqrt(31)] / 6
Substitute these x-values into either the linear equation y=-4x+18 or the quadratic equation y=-3x^2+17x to find the corresponding y-values.
Therefore, the points of possible collision are:
([-21 + sqrt(3)* sqrt(7)* sqrt(31)] / 6 , y1) and
([-21 - sqrt(3)* sqrt(7)* sqrt(31)] / 6 , y2)
where y1 and y2 are the y-values obtained from substituting the x-values into one of the original equations.
To find the points where the two skaters might collide, we need to solve the system of equations formed by their paths.
The paths of the two skaters are described by the equations:
Skater 1: y = -4x + 18
Skater 2: y = -3x^2 + 17x
To find the possible points of intersection, we'll set these two equations equal to each other and solve for x.
-4x + 18 = -3x^2 + 17x
Rearranging the equation and setting it equal to zero:
-3x^2 + 17x + 4x - 18 = 0
Combining like terms:
-3x^2 + 21x - 18 = 0
Next, we can try to factor the equation. However, it does not appear to easily factor. So, let's use the quadratic formula to find the values of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -3, b = 21, and c = -18.
x = (-21 ± √(21^2 - 4(-3)(-18))) / (2(-3))
Simplifying further:
x = (-21 ± √(441 - 216)) / (-6)
x = (-21 ± √225) / (-6)
The square root of 225 is 15:
x = (-21 ± 15) / (-6)
Now, we have two possible values for x:
x1 = (-21 + 15) / (-6) = -6/(-6) = 1
x2 = (-21 - 15) / (-6) = -36/(-6) = 6
Now that we have the x-values, we can substitute them back into the equations to find the corresponding y-values.
For x1 = 1:
Skater 1: y = -4(1) + 18 = -4 + 18 = 14
Skater 2: y = -3(1)^2 + 17(1) = -3 + 17 = 14
For x2 = 6:
Skater 1: y = -4(6) + 18 = -24 + 18 = -6
Skater 2: y = -3(6)^2 + 17(6) = -108 + 102 = -6
Therefore, the two skaters might collide at the points (1, 14) and (6, -6).
To find the points where the two skaters might collide, we need to find the coordinates (x, y) that satisfy both of their equations simultaneously.
Let's start by equating the two equations of the skaters:
-4x + 18 = -3^2 + 17x
Simplifying the equation, we get:
-4x + 18 = -9 + 17x
Combining like terms, we have:
-4x - 17x = -9 - 18
-21x = -27
Dividing both sides by -21, we get:
x = -27 / -21
Simplifying further, we find:
x = 9 / 7
Now that we have the x-coordinate, we can substitute it back into either of the original equations to find the corresponding y-coordinate. Let's use the first skater's equation y = -4x + 18:
y = -4(9/7) + 18
Simplifying, we get:
y = -36/7 + 18
To add the fractions, we need a common denominator:
y = -36/7 + 18 * 7/7
y = -36/7 + 126/7
y = (126 - 36) / 7
y = 90 / 7
Therefore, the possible point of collision for the two skaters is (9/7, 90/7).