Get step by step solutions for your textbook problems from www math4u us 1 A drug response curve describes the level of medication in the bloodstream after a drug is administered A surge function S t At p e kt is often used to model the response curve reflecting an initial surge in the drug level and then a more gradual decline If for a particular drug A 0 01 p 4 k 0 07 and is measured in minutes estimate the times corresponding to the inflection points and explain their significance If you have a graphing device use it to graph the drug response curve 2 If the initial amount Ao of money is invested at an interest rate r compounded n times a years is nt r A Ao 1 n If we let n we refer to the continuous compounding of interest Use l Hopital s Rule to show rt that if interest is compounded continuously then the amount after t years A Ao e 3 A farmer wants to fence an area of 1 5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle How can he do this so as to minimize the cost of the fence 4 A box with a square base and open top must have a volume of 32 000 cm3 Find the dimensions of the box that minimize the amount of material used 5 If 1200 cm of material is available to make a box with a square base and an open top find the largest possible volume of the box 6 A rectangular storage container with an open top is to have a volume of 10 m The length of its base is twice the width Material for the base costs 10 per square meter Material for the sides costs 6 per square meter Find the cost of materials for the cheapest such container 7 Find correct to two decimal places the coordinates of the point on the curve y tan x that is closest to the point 1 1 8 Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r 9 Find the area of the largest rectangle that can be inscribed in the ellipse x2 a2 y2 b2 1 10 Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle 11 Find the dimensions of the rectangle of largest area that has its base on the x axis and its other two vertices above the x axis and lying on the parabola y 8 x2 12 Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r 13 Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs 14 A right circular cylinder is inscribed in a sphere of radius r Find the largest possible volume of such a cylinder Get step by step solutions for your textbook problems from www math4u us 15 A right circular cylinder is inscribed in a cone with height h and base radius r Find the largest possible volume of such a cylinder 16 A right circular cylinder is inscribed in a sphere of radius r Find the largest possible surface area of such a cylinder 17 A Norman window has the shape of a rectangle surmounted by a semicircle Thus the diameter of the semicircle is equal to the width of the rectangle See Exercise 56 on page 23 If the perimeter of the window is 30 ft find the dimensions of the window so that the greatest possible amount of light is admitted 18 The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm If the area of printed material on the poster is fixed at 384 cm2 find the dimensions of the poster with the smallest area 19 A poster is to have an area of 180 in2 with 1 inch margins at the bottom and sides and a 2 inch margin at the top What dimensions will give the largest printed area 20 A piece of wire 10 m long is cut into two pieces One piece is bent into a square and the other is bent into an equilateral triangle How should the wire be cut so that the total area enclosed is a a maximum b A minimum 21 A cylindrical can without a top is made to contain V cm3 of liquid Find the dimensions that will minimize the cost of the metal to make the can 22 A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building 23 A cone shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB Find the maximum capacity of such a cup 24 A cone shaped paper drinking cup is to be made to hold 27 cm3 of water Find the height and radius of the cup that will use the smallest amount of paper 25 A cone with height is inscribed in a larger cone with height H so that its vertex is at the center of the 1 base of the larger cone Show that the inner cone has maximum volume when h H 3 26 An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object If the rope makes an angle with a plane then the magnitude of the force is mW F msin q cos q where mis a constant called the coefficient of friction For what value of q is F smallest 27 If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms then the power in watts in the external resistor is Get step by step solutions for your textbook problems from www math4u us E2R R r 2 If E and r are fixed but R varies what is the maximum value of the power P 28 For a fish swimming at a speed relative to the water the energy expenditure per unit time is proportional to v3 It is believed that migrating fish try to minimize the total energy required to swim a fixed distance If the fish are swimming against a current u u v then the time required to swim a distance L is L v u and the total energy required to swim the distance is given by L E v av3 v u where a is the proportionality constant a Determine the value of …