Question

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7 A cylinder of height h metres is inscribed in a sphere of constant radius R metres.
a If the cylinder has radius r metres, show that r? = R? - ]h?.
b Show that the volume of the cylinder is given by V = Ih (4R? - h).
c Show that the volume of the cylinder is maximised when h = §v3R.
d Hence show that the ratio of the volume of the sphere to the maximum volume of the cylinder is V3: 1.

12 A canned fruit producer wishes to minimise the area of sheet metal used in manufacturing cylindrical cans of a given volume. Find the ratio of radius to height for the desired can.
10 A rectangle is inscribed in a quadrant of a circle of radius r so that two of its sides are along the bounding radii of the quadrant.
a If the rectangle has length x and width y, show that its area is given by A = yVr2 - y
b Show that the maximum possible area of the rectangle is †r?

Answer 1

7a) To show that r^2 = R^2 - h^2, one can use the Pythagorean theorem and the fact that the distance from the center of the sphere to the surface of the cylinder is h/2.

7b) To find the volume of the cylinder, one can use the formula for the volume of a cylinder, V = πr^2h, where r is the radius and h is the height.

7c) To find the maximum volume of the cylinder, one can differentiate the expression for the volume with respect to h, set the derivative equal to zero, and solve for h.

7d) The ratio of the volume of the sphere to the maximum volume of the cylinder can be found by dividing the volume of the sphere by the maximum volume of the cylinder.

To find the ratio of radius to height for the desired can, one can set up an expression for the surface area of the can and differentiate it with respect to the radius and height. Setting the partial derivatives equal to zero, one can find the critical points, and evaluate which one is the minimum.

10a) To find the area of the rectangle, one can use the formula for the area of a rectangle, A = xy, where x is the length and y is the width.

10b) To find the maximum area of the rectangle, one can differentiate the expression for the area with respect to x and y, set the partial derivatives equal to zero, and solve for x and y.

Explanation: