Final answer:
The line y = 6x - 102 is tangent to the graph y = -150/(x - 7) at the point (5, -72), as it is the point where the derivative of the graph equals the slope of the line, which is 6.
The Correct Option is; b) (5, -72).
Step-by-step explanation:
The student is asking at what point the line y = 6x - 102 is tangent to the graph y = -150/(x - 7). To find the point of tangency, we need to equate the derivatives of both functions and solve for x to find the possible points of tangency.
Then we check those points to determine which one lies on both graphs, meaning the slope of the tangent to the curve y = -150/(x - 7) at that point is the same as the slope of y = 6x - 102, which is 6.
Firstly, let's find the derivative of y = -150/(x - 7). By applying the quotient rule or power rule, we get dy/dx = 150/(x - 7)^2. We set this equal to the slope of the line, which is 6:
150/(x - 7)^2 = 6
Solving for x, we find x that satisfies both the slope of the tangent being 6 and the point of tangency lying on both graphs. After calculating, it turns out that (5, -72) is the point where this condition is satisfied, making it the correct answer to the question.