Related Rates (Section4.1)A spherical balloon, measured in two different unit systems (inches and centimeters),is inflated so that its radiusgrows over time. LetV=43r3be the volume of the balloon.If the radius ris changing at a known rate drdt, use related rates to find dVdt.Suppose you want to determine dVdtat the exact moment r=4. Evaluate the expression carefully,considering any potential unit conversions or overlooked details.Reflect on how you would communicate the results of your calculations to Alexandra, ensuring sheunderstands the practical relevance of these rates for her research. Linear Approximations (Section4.2)Alexandra also wants a quick method to estimate changes in a function without doing a full recalculation. Sheprovides you with the functionf(x)=x+83Use a linear approximation (differentials)to estimate f(x) near x=0. Specifically, find the linearizationL(x)atx=0.Suppose x changes from 0to0.5. Use L(x)to approximate the new function value and compare itto theexact value off(0.5).Share how a quick approximation like this could be beneficial in a real-world research setting, such asadjusting parameters without recalculating everything from scratch.