Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 34
Exercise 3.6
1. Find two positive numbers whose product is 100 and whose sum is minimum.
2. Find two numbers whose product is −16 and the sum of whose squares is minimum.
3. Forwhatnumber doestheprincipal fourthrootexceed twicethenumberbythelargest
amount.
4. Find the dimensions of a rectangle with area 1000 m 2 whose perimeter is as small as
possible.
5. A box with a square base and open top must have the volume of 32,000 cm 3 . Find
the dimensions of the box that minimize the amount of material used.
6. If 1200 cm 2 of material is available to make a box with a square base and an open
top, find the largest possible volume of the box.
7. A farmer wishes to fence off two identical adjoining rectangular pens, each with 900
square feet of area, as shown in the following Figure.
y
x
What are x and y so that the least amount of fence is required
8. Find the point on the line y = 4x+7 that is closest to the origin.
9. Find the point on the line 6x+y = 9 that is closest to the point (−3,1).
10. Find the point on the parabola y = x 2 that is closest to the point (0,5).
11. Find the point on the parabola x+y 2 = 0 that is closest to the point (0,−3).
12. Find the dimensions of the rectangle of largest area that can be inscribed in a circle
of radius r.
13. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a
circle of radius r.
14. 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.
15. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible
surface area of such a cylinder.
16. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible
volume of such a cylinder.
17. Show that the rectangle with maximum perimeter that can be inscribed in a circle is
a square.