3. You're so thrilled by your geometric and designing capabilities (see problems 1 and 2) that you decide to design a thin dinner plate that on your blueprint covers the region between the r-...
3. You're so thrilled by your geometric and designing capabilities (see problems 1 and 2) that you decide to design a thin dinner plate that on your blueprint covers the region between the r-axis and the curve To impress the friends, you decide to make two versions of the plate and exhibit them by holding them up on a single finger. In order to do this, you need to calculate the center of mass of each. (a) (5 points) One with uniform density. (b) (5 points) The other with density p(x)-H, (You will need to adjust the formulas on p. 563 to account for the non-uniform density; see the set-up on p. 562.) (e) (3 points) Sketch the blueprint of your plate in the ry-plane and label the centers of mass you calculated. By referring to the densities of each plate, explain to your friends why the relative position of one center of mass in comparison to the other makes physical sense. 4. Tired from the long week, with the amusement park and the wok and dinner plate designs, you retreat to be in peace and study parametric equations. A curious leaflike curve called the "folium of Descartes" catches your eye; it is defined by the parametric equations 3t2 3t (a) (4 points) Show that if the point (a, b) lies on the curve, then so does (b,a); that is, the curve is symmetric with respect to the line y r. (Hint: if ti is the time giving the point (a, b), then consider the time t =-.) (b) (3 points) Sketch the curve. (c) (5 points) Find all the points on the curve where the tangent lines are horizontal or vertical. (For the horizontal tangents, justify using calculus. For the vertical tangents, you can use the symmetry from part (a).)
3. You're so thrilled by your geometric and designing capabilities (see problems 1 and 2) that you decide to design a thin dinner plate that on your blueprint covers the region between the r-axis and the curve To impress the friends, you decide to make two versions of the plate and exhibit them by holding them up on a single finger. In order to do this, you need to calculate the center of mass of each. (a) (5 points) One with uniform density. (b) (5 points) The other with density p(x)-H, (You will need to adjust the formulas on p. 563 to account for the non-uniform density; see the set-up on p. 562.) (e) (3 points) Sketch the blueprint of your plate in the ry-plane and label the centers of mass you calculated. By referring to the densities of each plate, explain to your friends why the relative position of one center of mass in comparison to the other makes physical sense. 4. Tired from the long week, with the amusement park and the wok and dinner plate designs, you retreat to be in peace and study parametric equations. A curious leaflike curve called the "folium of Descartes" catches your eye; it is defined by the parametric equations 3t2 3t (a) (4 points) Show that if the point (a, b) lies on the curve, then so does (b,a); that is, the curve is symmetric with respect to the line y r. (Hint: if ti is the time giving the point (a, b), then consider the time t =-.) (b) (3 points) Sketch the curve. (c) (5 points) Find all the points on the curve where the tangent lines are horizontal or vertical. (For the horizontal tangents, justify using calculus. For the vertical tangents, you can use the symmetry from part (a).)