Question

A crystal growth furnace is used in research to determine how best to manufacture crystals used in electric 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 by the following equation, where T is the temperature in degrees Celsius and w is the power input in watts. T(w) = 0.1w2 + 2.157w + 20

(a) How much power is needed to maintain the temperature at 197°C? (Give your answer correct to 2 decimal places.) X watts
(b) If the temperature is allowed to vary from 197°C by up to ±1°C, what range of wattage is allowed for the input power? (Give your answer correct to 2 decimal places.) X watts (smaller value) X watts (larger value)

Answer 1

Answer:

a) We need 32.65 W of power to maintain the temperature at 197°C.

b) The range of wattage allowed is (32.53W, 32.76W)

Explanation:

a) In order to find the needed wattage to maintain a temperature of 197°C for the crystals, we need to start by substituting that value into the provided function:

$T(x)=0.1w^{2}+2.157w+20$

$197=0.1w^{2}+2.157w+20$

and set the equation equal to zero by subtracting 197 from both sides, so we get:

$0.1w^{2}+2.157w-177=0$

To accurately solve this equation we can make use of the quadratic formula, which in this case will be:

$w=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$

in this case:

a=0.1

b=2.157

c=-177

(which come from the original equation according to their position in the equation with the form $ax^{2}+bx+c=0$

so we can substitute them into the formula like this:

$w=\frac{-2.157\pm\sqrt{(2.157)^2-4(0.1)(-177)} }{2(0.1)}$

We can solve this formula by directly plugging it into the calculator. That way you will get an exact answer. If you have troubles inputing this into the calculator, you can follow order of operations to solve it (parenthesis, exponentials, multiplication and division, addition and subtraction) but use as many decimal numbers as you can in the middle operations so you get a precise answer.

When inputing it into the calculator we get two answer:

w= -54.22W and w=32.65W

we use the positive answer, since that means that the power is being inputed into the system, a negative answer would mean you are retrieving power from the system which would cool the system down.

So we need 32.65W of power to maintain the temperature at 197°C

b) We need to follow the same procedure to find the answer for b. The only change is that the temperature may vary from  196°C to 198°C  (197±1)°C

so the equations to solve change to:

$0.1w^{2}+2.157w-176=0$  for 196°C

(in this case the "c" on the quadratic formula changes to -176)

when solving this equation we get an answer of:

w=32.53W

and

$0.1w^{2}+2.157w-178=0$ for 198°C

(in this case the "c" in the quadratic formula changes to -178)

when solving this equation we get an answer of:

w=32.76W

so the powers must be between 32.53W and 32.76W for the temperature to stay in an acceptable range.