Final answer:
To find the point where the tangent line has a slope of 1/2, you need to solve a quadratic equation derived from the derivative of the given curve. The point occurs when t = (1 + sqrt(5)) / 2.
Step-by-step explanation:
The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. To find the point where the tangent line has a slope of 1/2, we need to find the value of t that satisfies the equation.
First, differentiate the equations to find the derivative of x and y:
dx/dt = 6t
dy/dt = 3t^2 - 3
Next, substitute the values of dx/dt and dy/dt into the slope formula:
slope = (dy/dt) / (dx/dt) = (3t^2 - 3) / (6t) = 1/2
Simplify the equation:
3t^2 - 3 = (1/2) * 6t
6t^2 - 6 = 6t
6t^2 - 6t - 6 = 0
Finally, solve the quadratic equation to find the value of t:
t^2 - t - 1 = 0
Using the quadratic formula:
t = (1 + sqrt(5)) / 2
Therefore, the point on the curve where the tangent line has a slope of 1/2 is when t = (1 + sqrt(5)) / 2.