Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 35
18. (a) Show that of all the rectangles with a given area, the one with smallest perimeter
is a square.
(b) Show that of all the rectangles with a given perimeter, the one with largest area
is a square.
19. At 7 : 00am. one ship was 60 miles due east from a second ship. If the first ship
sailed west at 20 miles per hour and the second ship sailed southeast at 30 miles per
hour, when were they closest together
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
Answer to Exercise 3.6
1. 10,10 2. −4,4 3.
1
16
4. 10 √ 10×10 √ 10 5. 40×40×20 6. 4,000 cm 3
7. 15 √ 3×20 √ 3 8. ( − 28,
)
7
17 17
9. ( − 45,
)
63
37 37
10.
( ) (
−√ 3
2
, 9 2
,
3√
2
, 9 2
11. (−1,−1) 12. √ 2r × √ 2r 13. Base √ 3r, height 3r/2 14.
4
27 πr2 h
15. πr 2( 1+ √ 5 ) 16. 4πr 3 /3 √ 3 19. 8 : 09am.
20. (a) Use all of the wire for the square (b) 40 √ 3/ ( 9+4 √ 3 ) m for the square
Exercise 3.7
1. Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.
(a) f(x) = x 2 −4x+1, [0,4] (b) f(x) = x 3 −3x 2 +2x+5, [0,2]
(c) f(x) = sin2πx, [−1,1]
(d) f(x) = x √ x+6, [−6,0]
2. Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean
Value Theorem for the interval [0,8].
y
)
y = f(x)
1
1
x