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1. (a) A butterfly lands on a flower, 2 cm above the top of a chamber that has nectar. The chamber has a vertical cross section as shown in the diagram at right, and the horizontal K - - - - - H 2 cm cross sections are ellipse. Let ((h) be the length, in cm, of the narrowest part (the "minor axis") of the ellipse at a height of h cm e (h) 3 cm above the bottom of the nectar chamber. Further, the widest part of each ellipse h cm (coming out of the plane in the picture shown), the "major axis", is double the length of the minor axis. For each of the following parts, you should give an expression involving one or more integrals to represent the requested quantity. You do not need to compute a value. Be sure to include units. i. Nectar fills the entire chamber. Give an expression for the total volume of nectar. Note that the area of an ellipse with semi-minor axis length a and semi-major axis length b is Tab. (The semi-major axis is half of the major axis.) ii. The density of nectar is given by o(h) mg/cm , where h is height from the bottom of the chamber in centimeters. The chamber of the flower is completely filled with nectar. Give an expression for the total mass of nectar in this chamber of the flower. iii. Give an expression for the total amount of work the butterfly must do in order to drink all the nectar from this part of the flower. (b) Nearby, 20 mg of gnats and debris have gotten stuck together in a spider web. The spider, 5 cm above the clump of gnats, pulls the clump up on a strand of its web. The density of the strand of the web is 0.05 mg/cm, and as the spider pulls up the clump, some gnats escape (along with some debris falling off) at a constant rate of 1 mg per centimeter it is pulled up. How much work does the spider have to do to pull the clump of gnats to itself