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Answer

To determine the coefficients P1 , P2 , P3 of the parabola given by

f (x) = P1 x2 + P2 x + P3

that passes through the specified density values, we can set up a system of equations
based on the given density values at temperatures.

Given Conditions:

1. f (300) = 0.616

2. f (400) = 0.525

3. f (500) = 0.457
These conditions can be expressed as:

f (300) : P1 (3002 ) + P2 (300) + P3 = 0.616

f (400) : P1 (4002 ) + P2 (400) + P3 = 0.525
f (500) : P1 (5002 ) + P2 (500) + P3 = 0.457

Step 1: Setting up the Equations

Let's calculate 3002 , 4002 , and 5002 :

3002 = 90000
4002 = 160000
5002 = 250000

Now we rewrite our system of equations:

1. 90000P1 + 300P2 + P3 = 0.616 --- (1)

2. 160000P1 + 400P2 + P3 = 0.525 --- (2)

3. 250000P1 + 500P2 + P3 = 0.457 --- (3)

Step 2: Forming the Augmented Matrix

We can represent the above equations in an augmented matrix form:

90000 300 1 ∣ 0.616

160000 400 1 ∣ 0.525
250000 500 1 ∣ 0.457

Step 3: Row Reduction

Use Gaussian elimination to solve the system. The goal is to transform the augmented
matrix into row echelon form.

1. Subtract appropriate multiples of the first row from the second and third rows to

eliminate P1 .
Step 4: Solving for Coefficients

After row reduction, we will end up with a simpler system which we can solve for

P1 , P2 , and P3 .
Step 5: Plugging into the Equation

Once we find the values of P1 , P2 , and P3 , we can compute f (200) :

Finding Density at T = 200°C

Finally, substitute x = 200 into the equation:

f (200) = P1 (2002 ) + P2 (200) + P3

Conclusion

By following these steps and performing the necessary calculations, you will determine

the coefficients of the parabola and find the density at the temperature of 200°C .

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