Final answer:
The equation of the tangent line is: y - 380.25 = 39(x - 19.5).
Step-by-step explanation:
To find an equation of the tangent line to the curve y = x² that is parallel to the line y = 39x, we first need to identify the slope of the given line. The slope of the line y = 39x is 39.
Since parallel lines have the same slope, the tangent line to the curve will also have a slope of 39.
Next, we need to find the point on the curve y = x² where the slope of the tangent line is 39.
To do this, we find the derivative of y = x², which gives us 2x.
Setting the derivative equal to 39, we get 2x = 39, solving which yields x = 19.5.
Now, we calculate the y-value on the curve at x = 19.5, which equals (19.5)² = 380.25.
Thus, the point of tangency is (19.5, 380.25). Using the point-slope form of a line, the equation of the tangent line is y - 380.25 = 39(x - 19.5).