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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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HOW DO YOU SEE IT? The normal monthly high temperatures for Erie, Pennsylvania, are approximated by

H ( t ) =   56.72   +   23.62  sin ( 0.50 t   2.08 ) and the normal monthly low temperatures for Erie, Pennsylvania, are approximated by

L ( t ) =   41.89   +   21.52  sin ( 0.52 t     2.27 ) where t is the time (in months), with t = 1 corresponding to January. (Source: National Climatic Data Center)

Chapter 8.3, Problem 84E, HOW DO YOU SEE IT? The normal monthly high temperatures for Erie, Pennsylvania, are approximated by

(a) During what part of the year is the difference between the normal high and low temperatures greatest? When is it least?

(b) The sun is the farthest north in the sky around June 21, but the graph shows the highest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

(a)

To determine

The part of the year in which the difference between the normal high and low temperatures is greatest also to determine when is it least when the normal monthly high temperatures for Erie, Pennsylvania, are approximated by H(t)=56.72+23.62sin(0.50t2.08) and the normal monthly low temperatures for Erie, Pennsylvania, are approximated by L(t)=41.89+21.52sin(0.52t2.27); where t is the time (in months), with t=1 corresponding to January.

Calculus: An Applied Approach (MindTap Course List), Chapter 8.3, Problem 84E , additional homework tip  1

Explanation

Given Information:

The normal monthly high temperatures for Erie, Pennsylvania, are approximated by H(t)=56.72+23.62sin(0.50t2.08) and the normal monthly low temperatures for Erie, Pennsylvania, are approximated by L(t)=41.89+21.52sin(0.52t2.27); where t is the time (in months), with t=1 corresponding to January.

Consider the functions:

H(t)=56.72+23.62sin(0.50t2.08) and L(t)=41.89+21.52sin(0.52t2.27).

From the graph it can be observed that the difference of H(t) and T(t), that is H(t)T(t). The maximum value of occurs at the interval 6<t<7.

Substitute the value of t=7 in the function H(t)=56.72+23.62sin(0.50t2.08),

That is,

H(7)=56.72+23.62sin(0.50(7)2.08)=56.72+23.62sin(1.42)80

Substitute the value of t=7 in the function L(t)=41.89+21.52sin(0.52t2.27),

That is,

L(7)=41

(b)

To determine

The lag time of the temperatures relative to the position of the sun when the normal monthly high temperatures for Erie, Pennsylvania, are approximated by H(t)=56.72+23.62sin(0.50t2.08) and the normal monthly low temperatures for Erie, Pennsylvania, are approximated by L(t)=41.89+21.52sin(0.52t2.27); where t is the time (in months), with t=1 corresponding to January.

Calculus: An Applied Approach (MindTap Course List), Chapter 8.3, Problem 84E , additional homework tip  2

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