( ) Applications of Derivatives Assignment Part 1

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Applications of Derivatives Assignment Part 1
Max-Min Problems
1. Find the maximum area that can be enclosed with 3600 m of fencing used to create a
rectangular region subdivided into 4 equal sub regions as shown in the diagram.
2. “t” hours after a drug is administered to a patient, the concentration k (t ) of the drug
in the body is given by:
2t
t +4
How long after the drug is administered is the concentration at a maximum?
k (t ) =
2
3. A box with a square base and no top is to have a volume of 4 m3. Find the
dimensions of the most economical box.
4. Find the largest rectangle that can be inscribed in an ellipse with equation
x2 y 2
+
=1
9
4
5. Design a 1 litre oil can shaped like a right cylinder, which uses the least amount of
material.
6. A drilling rig 12 km offshore is to be connected to a refinery onshore,20 km down
the coast from the rig. If the underwater pipe costs are $40,000 per km and land
based pipe costs $30,000 per km, find where best to lay the pipe to minimize the cost.
Velocity and Acceleration Problems
7. A particle along a straight line path is moving according to the equation. (s is in
metres, t is in seconds)
s = t 3 − 9t 2 + 15t
(a) Give the equations for velocity and acceleration
(b) Determine when the particle is at rest.
(c) Determine the distance traveled in the first 10 s. Use a diagram to illustrate
the trip.
8. A dynamite blast propels a coyote straight up with an initial velocity of 160 m/s.
(a) How high will the coyote go?
(b) Find its velocity halfway up.
(c) Find its acceleration.
(d) Find its velocity when it hits the ground.
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