Final answer:
To find the slope dy/dx at x=5 for the given parametric equations, we see that when x=5, t=1. We then use the chain rule to differentiate y with respect to x, finding that the slope of the curve at x=5 is 30.
Step-by-step explanation:
The student has been given the parametric equations x=5t and y=3x²+1, with the rate of change of x with respect to t given as dx/dt=2. To find dy/dx at x=5, you would first use the given parametric equation to relate the value of t to x, then differentiate y with respect to x using the chain rule.
Step 1: Since x=5t, to find the value of t when x=5, we set 5t=5. Solving for t, we get t=1.
Step 2: Next, we need to differentiate y with respect to x. The derivative of y with respect to t (dy/dt) can be found by differentiating 3x²+1 regarding t. However, since we need dy/dx, we will use the chain rule. The chain rule in this context is dy/dx = (dy/dt) × (dt/dx). To find dt/dx, which is the reciprocal of dx/dt, we take the reciprocal of 2, giving 1/2.
First, find dy/dt:
Given y = 3x²+1, differentiate with respect to t to get dy/dt = 6x(dx/dt). Setting x=5 into this equation, dy/dt = 6×5×2 = 60.
Finally, apply the chain rule to get dy/dx:
dy/dx = dy/dt × dt/dx = 60 × (1/2) = 30.
Therefore, the slope of the curve dy/dx at x=5 is 30.