Filling a reservoir A reservoir with a capacity of 2500 m 3 is filled with a single inflow pipe. The reservoir is empty when the inflow pipe is opened at t = 0. Letting Q( t ) be the amount of water in the reservoir at time t , the flow rate of water into the reservoir (in m 3 /hr) oscillates on a 24-hr cycle (see figure) and is given by Q ′ ( t ) = 20 ( 1 + cos π t 12 ) . a. How much water flows into the reservoir in the first 2 hr? b. Find the function that gives the amount of water in the reservoir over the interval [0. t], where t ≥ 0. c. When is the reservoir full?
Filling a reservoir A reservoir with a capacity of 2500 m 3 is filled with a single inflow pipe. The reservoir is empty when the inflow pipe is opened at t = 0. Letting Q( t ) be the amount of water in the reservoir at time t , the flow rate of water into the reservoir (in m 3 /hr) oscillates on a 24-hr cycle (see figure) and is given by Q ′ ( t ) = 20 ( 1 + cos π t 12 ) . a. How much water flows into the reservoir in the first 2 hr? b. Find the function that gives the amount of water in the reservoir over the interval [0. t], where t ≥ 0. c. When is the reservoir full?
Filling a reservoir A reservoir with a capacity of 2500 m3 is filled with a single inflow pipe. The reservoir is empty when the inflow pipe is opened at t = 0. Letting Q(t) be the amount of water in the reservoir at time t, the flow rate of water into the reservoir (in m3/hr) oscillates on a 24-hr cycle (see figure) and is given by
Q
′
(
t
)
=
20
(
1
+
cos
π
t
12
)
.
a. How much water flows into the reservoir in the first 2 hr?
b. Find the function that gives the amount of water in the reservoir over the interval [0. t], where t ≥ 0.
An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides
to do it in three sections. The first section runs from point P to point Q, and costs $48 per mile to lay, the
second section runs from point Q to point R and costs $54 per mile, the third runs from point R to point S
and costs $44 per mile. Looking at the diagram below, you see that if you know the lengths marked x and y,
then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline.
Please show your answers to 4 decimal places.
2 Miles
x =
1 Mile
R
10 miles
miles
y =
miles
An open-top rectangular box is being constructed to hold a volume of 150 in³. The base of the box is made
from a material costing 7 cents/in². The front of the box must be decorated, and will cost 11 cents/in².
The remainder of the sides will cost 3 cents/in².
Find the dimensions that will minimize the cost of constructing this box. Please show your answers to at
least 4 decimal places.
Front width:
Depth:
in.
in.
Height:
in.
Find and classify the critical points of z = (x² – 8x) (y² – 6y).
Local maximums:
Local minimums:
Saddle points:
-
For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. Enter DNE if
there are no points for a classification.
University Calculus: Early Transcendentals (4th Edition)
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