Assume that x = 5.2 ± 0.1 and y = 3.8 ± 0.2. Calculate the following expressions, including the errors. Keep only one significant digit in the error.
A) p = xy
B) u = x/y2
C) v = √(x3y5)
5.3 * 4.0 = 21.2 biggest
5.2 * 3.8 = 19.76 middle
difference = 1.44 which is 1 to one sig figure
19.8 +/- 1
etc
B) u = x/y2
assume you mean u = x / y^2
for this to get the biggest use big x and small y
5.3 / (3.8 -.2)^2 = 5.3 / 3.6^2
If you want to be really fussy you can use average of biggest and smallest +/- half of difference
To calculate the expressions p, u, and v, we will use the rules for error propagation. Error propagation is a method for estimating the error in a calculation based on the errors in the input values.
Let's calculate each expression step by step:
A) p = xy
To calculate p, we simply multiply the values of x and y:
p = 5.2 * 3.8 = 19.76
To calculate the error in p, we use the formula for relative error:
Relative Error in p = (Relative Error in x + Relative Error in y) * p
The Relative Error in x is given by the ratio of the error in x to the value of x:
Relative Error in x = (0.1 / 5.2) = 0.0192
Similarly, the Relative Error in y is given by the ratio of the error in y to the value of y:
Relative Error in y = (0.2 / 3.8) = 0.0526
Now, we can calculate the Relative Error in p:
Relative Error in p = (0.0192 + 0.0526) * 19.76 = 1.001
Since we want to keep only one significant digit in the error, the error in p is approximately 1.
Therefore, p = 19.76 ± 1.
B) u = x/y^2
To calculate u, we divide the value of x by the square of y:
u = 5.2 / (3.8)^2 = 5.2 / 14.44 = 0.3596
To calculate the error in u, we use the formula for relative error:
Relative Error in u = (Relative Error in x + 2 * Relative Error in y) * u
The Relative Error in x is the same as before: 0.0192
The Relative Error in y is now multiplied by 2 to account for the squaring:
Relative Error in y = 2 * (0.2 / 3.8) = 0.1053
Now, we can calculate the Relative Error in u:
Relative Error in u = (0.0192 + 2 * 0.1053) * 0.3596 = 0.125
Keeping only one significant digit in the error, the error in u is approximately 0.1.
Therefore, u = 0.3596 ± 0.1.
C) v = √(x^3 * y^5)
To calculate v, we take the square root of the product of x^3 and y^5:
v = √(5.2^3 * 3.8^5) = √(140.608 * 1759.8728) = √247499.8977 = 497.5
To calculate the error in v, we use the formula for relative error:
Relative Error in v = (3 * Relative Error in x + 5 * Relative Error in y) * v
The Relative Error in x is the same as before: 0.0192
The Relative Error in y is also the same as before: 0.0526
Now, we can calculate the Relative Error in v:
Relative Error in v = (3 * 0.0192 + 5 * 0.0526) * 497.5 = 1.302
Keeping only one significant digit in the error, the error in v is approximately 1.
Therefore, v = 497.5 ± 1.
To summarize:
A) p = 19.76 ± 1
B) u = 0.3596 ± 0.1
C) v = 497.5 ± 1