c
Math 151 WIR, Fall 2013, Benjamin
Aurispa
Math 151 Week in Review 12
Sections 5.5, 5.7, 6.1
1. A farmer wants to fence in a rectangular area along a river and divide it into 3 equal-sized pens with
fencing perpendicular to the water. He is not going to build a fence along the river. If he has 120 ft
of fencing available to use, find the maximum area of the field.
2. Find the closest point on the line y = −3x + 4 to the point (1, 5).
3. A box with an open top is to be constucted from 30 m2 of cardboard. The length of the box is to be
twice its width. Find the dimensions of the box that will maximize the volume.
4. A company is designing a cylindrical can which will have a volume of 2000 cm3 . Find the dimensions
of the can that would minimize the surface area.
5. Find the dimensions of the largest rectangle that can be inscribed in the area bounded by the top half
of the ellipse 4x2 + y 2 = 64 and the x-axis. What is the maximum area?
6. A 20-m wire is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent
into a square. How should the wire be cut (if at all) so that the total area enclosed is a maximum?
minimum?
7. Find f (x) for the following.
(a) f ′ (x) = x2 x2 −
3
x6
+
1
x
1
+ cos x + ex , f (0) = 7
(b) f ′ (x) = 5 + 2
x +1
√
x + x5/4
′′
(c) f (x) =
, f ′ (1) = 5, f (0) = 4
x
(d) f ′′ (x) = x − 4 − ex , f (0) = 3, f (1) = 2
8. An object is thrown down from a 500 ft ledge with a speed of 15 ft/s.
(a) Find the distance of the object above the ground at time t.
(b) When and with what velocity does the object hit the ground?
9. A car traveling at a speed of 30 mph (44 ft/s) applies a constant acceleration of 8 ft/s2 . How far will
the car travel before reaching a speed of 60 mph (88 ft/s)?
10. An object in motion has acceleration given by a(t) =< sin t + t3 + 1, et + 4 >. Find the position vector
function given that v(0) =< 0, 5 > and r(0) =< 3, 0 >.
11. Calculate the following sums.
(a)
4
X
i=1
(2i2 − 3i)
(b)
35
X
i=5
(3i − 3i+1 )
(c)
n→∞
n
X
2 2i 3
i=1
n
(i3 + 4)
i=3
12. Represent the following sum using sigma notation.
13. Calculate lim
20
X
n
1
4 9 16 25
400
+ +
+
+ ··· +
7 8
9
10
25
(d)
n
X
i=1
(3i + 2)2