Math 1210-001 Lab 7
Spring 2016; due March 24 at the beginning of lab
Name _____________________________________________UID______________________________
This lab contains section 3.4 and 3.5 material.
1) (12 points) (see WebWork #2) A rectangle is inscribed with its base on the xKaxis and its upper
corners on the parabola y = 12 K x2 . What are the dimensions of such a rectangle having the greatest
possible area?
2) (12 points) A rectangular poster is to contain 200 square inches of print. The margins at the top and
bottom of the poster are to be 2 inches, and they are to be 1 inch on each side. What size paper should be
used to minimize the total area of the paper?
3) (12 points) (see WebWork #4) A rancher wants to fence in a rectangle area of 6$104 square meters,
and then divide it in half with a fence down the middle parallel to one side. What is the shortest total length
of fence the rancher can use?
4) (12 points) A huge conical tank is to be made from a circular piece of sheet metal of radius 10 meters,
by cutting out a sector with vertex angle q and then welding together the straight edges of the remaing
piece (as in the figure below). Find q so that the resulting cone has the largest possible volume
5) (12 points) (See WebWork #10) It takes a certain power P to keep a plane moving along at speed v.
The power needs to overcome air drag, which increases as the speed increases, and it needs to keep the
plane in the air which gets harder as the speed decreases. So assume the power required is given by
d
P = c v2 C 2
v
where c, d are positive constants. (They depend on your plan, your altitude, and the weather, among other
things.)
a) Find the value of v that minimizes the power P required to keep the plane flying. (Your solution
depends on c, d.)
b) Suppose that you have a certain amount of fuel and that the fuel flow required to deliver a certain
power is proportional to that power. What is the speed v that will maximize your range (i.e. the distance
you can travel at that speed before your fuel runs out)?
c) What is the ratio of the speed that maximizes the distance vs the speed that minimizes the required
power?
6) (10 points) Sketch the graph of f x =
4
using Calculus: Find all asymptotes, all critical points,
x2 K 4
the intervals on which f is increasing/decreasing, concave up/concave down, local extrema, and inflection
points.
2
3
7) (10 points) Sketch the graph of f x = x K 2
using Calculus: Find all asymptotes, all critical
points, the intervals on which f is increasing/decreasing, concave up/concave down, local extrema, and
inflection points.
8) (10 points) Sketch the graph of f x =
x
2
using Calculus: Find all asymptotes, all critical
1Cx
points, the intervals on which f is increasing/decreasing, concave up/concave down, local extrema, and
inflection points.
K2 x2 C 2 x C 8
9) (10 points) Sketch the graph of f x =
using Calculus: Find all asymptotes, all
xK1
critical points, the intervals on which f is increasing/decreasing, concave up/concave down, local extrema,
and inflection points.