(e) A cylinder is placed within a cone as shown, so that the top of the cone and cylinder share the same plane and centre. The cone is of height 18cm and radius 3cm. What height must the cylinder have if its volume is to be maximum?
You must use calculus and show any derivatives that you need to find when solving this problem. You do not have to show that the height you find will give a cylinder of maximum volume.
4 (d) Find the value of k such that the tangent toy = —x when x = 3 and y = x2 when x = k are perpendicular. Remember that two straight lines are perpendicular if the product of their gradients is -1.
You must use calculus and show any derivatives that you need to find when solving this problem
ex (d) For what value(s) of x is the tangent to the graph of the function f(x) = parallel x +
(e) A curve is defined by the function f(x) =xe(x k)2. Find in terms of k, the x-coordinate(s) for which f ‘(x) = e(x k)2 You must use calculus and show any derivatives that you need to find when solving this problem.
(e) Sand forms a cone as it falls from a chute. It is increasing the volume of the cone at a rate of 10m1s’. In terms of its height h and base radius r, the volume of a cone is 1—nr2/.
The cone has a slant height making an angle of 30° with the horizontal. Find the rate at which the slant height is increasing when the base radius of the cone is 20cm.
height
4 Base Radiu?
Show any derivatives that you need to find when solving this problem.
(d) Find the equation of the normal to • = x3ln x at the point when x = e.
You must use calculus and show any derivatives that you need to find when solving this problem.
the x-axis. u must use calculus and show any derivatives that you need to find when solving this proble