solve question 4 only..with complete details

Statement Swimsuit development took a massive leap forward in 2009 with the introduction of 'Speedo LZR' suits with polyurethane suits such as the 'Arena X-Glide'. It was hypothesized that the suits reduced the effective cross- sectional body area of the swimmer and the drag coefficient when moving through water, which would lead to a reduction in the drag force. The increase in performance may have been reduced by the fatiguing effect of the relatively stiff swimsuits, which would impact more in the longer events with a large number of turns. To test this hypothesis, suppose you suggest the mathematical model for the velocity if of the swimmer propelling in the f-direction through water as 1(P, C)II = (2)". where P is the metabolic power exerted by the swimmer to propel her body, C is the drag coefficient, and k and n are empirical constants such that k, n 2 0. The reasoning for this model is that the swimmer can increase her swimming speed by increasing her exerted metabolic power or reducing the drag coefficient wearing the X-Glide suit, or by a combination of both. You want to test how effective is each of these. Tasks 1. Find an expression for a bivariate function f(x, y) = = in terms of P and C, where x and y are the fractional change in speed resulting from change in metabolic power and the drag coefficient respectively. What are the mathematical and physical limitations on the rate of expenditure of metabolic energy with time? 2. Calculate the gradient of f (x, y) in terms of P and C. 3. Find and plot the equation of the tangent plane to the surface z = f(x, y) and the parametric equations of the normal line at the point ro = i + / + k for a suitable value of parameter t. 4. Suppose that the power exerted by the swimmer is directly proportional to her body mass index (B), which in turn depends on the swimmer's height (h) and mass (m) as B(m, h) = . The mass and height would change with the swimmer's age (A). If 1 is the constant of proportionality relating power to the body mass index, draw a tree diagram and find an expression for the rate of change in speed with age in terms of x, m, h, A and 1