Problem Breakdown:

Part (a):

Given: $x(t) = \sqrt{t} \ln(t) - 4, \quad y(t) = 5\sqrt{t}, \quad t > 1$ This describes a curve in the $xy$-plane.

1. i) Tangent Slope $k$ at the Point $(2e^{-4}, 5e)$:

   • First, find the derivatives $x'(t)$ and $y'(t)$ to determine the slope of the tangent line at the point.
   • The slope $k$ is given by: $k = \frac{dy}{dx} = \frac{y'(t)}{x'(t)}$ evaluated at the $t$ value corresponding to $x = 2e^{-4}$ and $y = 5e$.

2. ii) Determine the $t$ value for which the tangent slope is $53$:

   • Solve the equation $k = 53$ for $t$ using the expression found for $k$.

Part (b):

The equation $4(xy^2 + y^2)^{1/4} + xy - 20 = 0$ implicitly defines $y$ as a function of $x$ near the point $(0, 5)$.

To find $y'(0)$ using implicit differentiation:

1. Differentiate the equation with respect to $x$.
2. Substitute $x = 0$ and $y = 5$ into the differentiated equation to solve for $y'(0)$.

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I'll proceed with these calculations now.### Part (a) Results

1. Tangent Slope $k$ at the Point $(2e^{-4}, 5e)$:

   • The parameter $t$ corresponding to this point is approximately: $t = e^{2 \cdot W(e^{-4} + 2)}$ where $W$ is the Lambert W function.
   • The slope $k$ of the tangent line at this $t$ value is: $k = \frac{5 e^{-W(e^{-4} + 2)}}{2 \left( e^{-W(e^{-4} + 2)} W(e^{-4} + 2) + e^{-W(e^{-4} + 2)} \right)}$

2. Parameter $t$ for which the Tangent Slope is $53$:

   • The parameter $t$ that gives a tangent slope of $53$ is: $t = e^{-\frac{101}{53}}$

Part (b) Solution Outline:

I will now proceed to Part (b) and determine $y'(0)$ using implicit differentiation.It appears there was an issue when attempting to solve for $y'(0)$ in part (b). I’ll revisit the implicit differentiation setup and check for any constraints that might affect the solution, ensuring it’s properly calculated.It seems that the implicit differentiation approach with the current setup isn’t yielding a solution for $y'(0)$. This could indicate either that there is no solution in the defined region or a more complex behavior in the function around this point.

If you have further constraints or additional details, please let me know. Alternatively, I can explore alternative methods or verify any assumptions to proceed.