Which set of parametric equations corresponds to this rectangular equation? y=2x-(x^2 / 10)
A. x=4t, y=4t-(t^2 /10)
B. x=2t, y=8t-(4t^2/10)
C. x=2t, y=4t-(4t^2/10)
D. x=.5t, y=2t-(t^2/10)
E. x=t, y=t-(t^2/10)
Looks like C to me
To determine which set of parametric equations corresponds to the given rectangular equation, we need to express x and y in terms of a common parameter, typically denoted as t.
In the given rectangular equation, y = 2x - (x^2 / 10), we can rearrange it to eliminate the fraction:
10y = 20x - x^2
Rearranging further, we have:
x^2 - 20x + 10y = 0
Now, let's compare the given options with this equation to find the matching parametric equations.
A. x = 4t, y = 4t - (t^2 / 10)
To check if this matches the original equation, let's substitute these values into the rearranged equation:
(4t)^2 - 20(4t) + 10(4t - (t^2 / 10))
16t^2 - 80t + 40t - (t^2)
16t^2 - 40t = 0
The equation does not match, so option A is not correct.
B. x = 2t, y = 8t - (4t^2 / 10)
Substituting these values into the rearranged equation:
(2t)^2 - 20(2t) + 10(8t - (4t^2 / 10))
4t^2 - 40t + 80t - 4t^2
40t = 0
The equation simplifies to 40t = 0, which is not equivalent to the original equation. Therefore, option B is incorrect.
C. x = 2t, y = 4t - (4t^2 / 10)
Substituting these values into the rearranged equation:
(2t)^2 - 20(2t) + 10(4t - (4t^2 / 10))
4t^2 - 40t + 40t - 4t^2 = 0
The equation becomes 0 = 0, which is true for all values of t. Therefore, option C satisfies the original equation.
D. x = 0.5t, y = 2t - (t^2 / 10)
Substituting these values into the rearranged equation:
(0.5t)^2 - 20(0.5t) + 10(2t - (t^2 / 10))
0.25t^2 - 10t + 20t - t^2
-0.75t^2 + 10t = 0
The equation does not match, so option D is incorrect.
E. x = t, y = t - (t^2 / 10)
Substituting these values into the rearranged equation:
(t)^2 - 20(t) + 10(t - (t^2 / 10))
t^2 - 20t + 10t - t^2
-10t = 0
The equation simplifies to -10t = 0, which is not equivalent to the original equation. Therefore, option E is incorrect.
Based on the analysis, the correct set of parametric equations that corresponds to the given rectangular equation is:
C. x = 2t, y = 4t - (4t^2 / 10)