Letter to a Friend 3. (a) Compute the arclength of the curve with para- materization (x, y) = (a cos³ 0, a sin³ 0), 0≤ 0 < 2π, where a > 0 is a constant with units of feet. Explain your reasoning. (b) Let P be a point in the first quadrant on the above curve that is s feet from the point A(a,0) as measured along the curve. Express the distance from P to the x-axis as measured along the tangent line to the curve at P in terms of s. Explain your reasoning. Letter to a Friend 4. (a) Interpret the meaning of the integral ["h(w)k' (w) dw as it relates to the parametrically defined curve (x, y) = (k(t), h(t)), a ≤ t ≤ b, where x, y, and t are measured in feet. Supplement your explanation with a graph. Include interpretations of each of the following in your explanation: The differential dw. . The product k' (w)dw. The product h(w) k'(w)dw. The role of the integral sign. • The integral fh(w)k' (w) dw. Include units in each part of your explanation. Assume that the function k is continuously differentiable and increasing over the interval [a, b] and that h is continuous. (b) The function s = g(t), 2 ≤t≤ 5, expresses the reading on your tripmeter (in miles) in terms of the number of hours past noon during a Sunday drive. The function m=h(t), 2 ≤ t ≤ 5, gives your gas mileage (in miles/gallon) at time t during the same trip. Your car has 8 gallons of gas at 4:00pm and your tripmeter reads 60 miles at 3:00pm. (i) Without using the function g¹, find a two-term expression for the function G = w(t) that gives the number of gallons of gas in your car at time t. Explain your reasoning. Include an interpretation of each of the bullet points in part (a) as they apply in this particular situation. (ii) Choose specific functions for g and h, with g a cubic polynomial and h a trigonometric function. Explain why your functions are reasonable. (iii) Use desmos to graph both the function G = w(t) and the parametri- cally defined curve with the closest relation to w. Explain how the two graphs are related.