This is the first question of a 4 part problem. I just need help with how to start it.
above is the 1st question to below problem:
In some experiments, tathered data suggest that volume of gas and temperature are related quantities. In one particular experiment, at a temp. of -30 degrees C, the volume of a gas was measured to be 380 cm3, while at a temp of 30 degrees C this same gas had a volume of 470 cm3. Assumer that the relationship between volume and temp. is linear. (question for this is Give a linear equation that expresses volume,V, in terms of Temp, T.)
We were told to do this without a calculator and w/a calculator.
Please help! I just don't know how to do the first question. If i get this one maybe I can figure out the others.
SOLUTION This solution is FREE courtesy of BrainMass!
We have to fit these values into a straight line of the form
y = mx + c
where m and c are constants to be determined.
(We will take y = V and x = T )
(x1,y1)=(T1,V1) =(-30,380) and
(x2,y2)= (T2,V2) = (30,470)
We can do this by least squares fitting. To fit a given set of data in to a straight line of the form y = mx + c, the normal equations for the constants m and c are, (we will determine m and c from the following equations)
Sum(y) = n*c + m * Sum(x) ......(1)
and Sum(xy) = c * Sum(x) + m * Sum(x^2) ......(2)
where n is the number of data points (x1,y1), (x2,y2) etc.
here n = 2
Sum(y) is the sum of all the y values, here Sum(y) = 850
Sum(x) = (-30)+ (30) = 0
Sum(xy) = -30*380 + 30*470 = 2700
Sum(x^2) = (-30)^2 + (30)^2 = 1800
Using these equations in (1) and (2), we get two equations
(1) gives, 850 = 2 c
(2) gives 2700 = 1800 * m
from this, c = 425 and m = 1.5
Therefore the required linear equation is y = 1.5 * x + 425
Or, V = 1.5 * T + 425
You can verify this equation by putting the values of T in this and obtain the corresponding V value. Remember that this procedure is valid only if the relation between variables is linear.© BrainMass Inc. brainmass.com December 24, 2021, 4:48 pm ad1c9bdddf>