   Chapter 4.1, Problem 71E

Chapter
Section
Textbook Problem

Between 0°C and 30°C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula V = 999.87 − 0.06426 T + 0.0085043 T 2 − 0.0000679 T 3 Find the temperature at which water has its maximum density.

To determine

To find: The temperature at which water has maximum density.

Explanation

Given:

The volume of 1 kg of water at a temperature T between 0°C and 30°C is given by,

V=999.870.06426T+0.0085043T20.0000679T3

Calculation:

Consider the given function,

V(T)=999.870.06426T+0.0085043T20.0000679T3

Obtain the first derivative of V(T) .

V(T)=0.06426T+2×0.0085043T3×0.0000679T2=0.0002037T20.0170086T+0.06426T

Set V(t)=0 and obtain the critical number.

Use quadratic formula and solve for T .

T=b±b24ac2a=(0.0170086)±(0.0170086)24(0.0002037)(0.06426)2×(0.0002037)=(0.0170086)±0.0002369(0.0004074)T3.9665;79.5318

Therefore the critical numbers of the function are 3.665°C, 79

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