   Chapter 2.4, Problem 12E

Chapter
Section
Textbook Problem

A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given byT(w) = 0.lw2 + 2.155w + 20where T is the temperature in degrees Celsius and w is the power input in watts.(a) How much power is needed to maintain the temperature al 200°C?(b) If the temperature is allowed to vary from 200°C by up to ± 1°C, what range of wan age is allowed for the input power?(c) In terms of thee, ε, δ definition of lim x → a f ( x ) = L , what is x? What is f(x)? What is a? What is L.? What value of ε. is given? What is the corresponding va1ue of δ?

(a)

To determine

To find: The Power need to maintain the temperature at 200°C.

Explanation

Given:

Suppose the relationship is T(ω)=0.1ω2+2.155ω+20 and the temperature is

T=200°C. Since, 200=0.1ω2+2.155ω+20 (1)

Result used:

The solution for quadratic form of the equation is ax2+bx+c=0 as follows:

x=b±b24ac2a

Calculation:

Multiply both sides to (1) with 1000.

1000(0.1ω2+2.155ω+20)=1000×200100ω2+2155ω+20000=200000100ω2+2155ω+20000200000=0100ω2+2155ω+180000=0

By Quadratic equation formula,

ω=2155±215524100(180000

(b)

To determine

To find: The range of wattage is allowed for the input power.

(c)

To determine

To explain: The terms in ε,δ definition of limxaf(x)=L.

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