Give the answer to the following calculation with the correct number of significant digits. Report your answer in scientific notation.
(6.17 x 10-1 + 4.9 x 10-2) x (3.95 x 10-2 + 6.454 x 10-3)
i know how to solve for the answer, i just don't know the correct number of sig figs! please help!!!!
4.9*10^(-2) is known to two significant figures, so the answer should be given in to significant figures.
Note that this is only a rule of thumb. In general you need to work as follows. You have some function like:
f(x1,x2,x3,x4)
E.g., in this case
f(x1,x2,x3,x4) = (x1 + x2)x(x3 + x4)
and x1, x2, x3 and x4 are known to finite precision. If x1 = 6.17 x 10(-1) then that essentially means that x1 could be anything in the interval
6.165x 10(-1) to 6.175x 10(-1)
You can thus say that:
x1 = 6.17 x 10(-1) ± 0.005 x 10^(-1)
0.005 x 10^(-1) is thus the error in x1, let's call this sigma1. Similarly you can find sigma2 ...sigma4, the errors in the other variables.
To compute the error in f, you proceed as follows. You just change the value for each variable, one by one, by its error and look at by how much the value of the function changes:
sigmaf1 = f(x1+sigma1,x2,x3,x4) -
f(x1,x2,x3,x4)
sigmaf2 = f(x1,x2+sigma2,x3,x4) -
f(x1,x2,x3,x4)
etc.
The error in f is then given by:
sigmaf = sqrt[sigmaf1^2 + sigmaf2^2 + sigmaf3^2 + sigmaf4^2]
From the value of sigmaf you can then see how many significant digits of f can be given.
To determine the correct number of significant digits, you need to consider the rules:
1. Non-zero digits are always significant.
2. Any zeros between significant digits are also significant.
3. Leading zeros (zeros before the first non-zero digit) are not significant.
4. Trailing zeros (zeros after the last non-zero digit) are significant only if there is a decimal point present.
Now let's break down the calculation step-by-step:
(6.17 x 10^-1 + 4.9 x 10^-2) x (3.95 x 10^-2 + 6.454 x 10^-3)
First, let's add the numbers inside the parentheses:
(6.17 x 10^-1 + 4.9 x 10^-2) = 6.17 x 10^-1 + 0.49 x 10^-1 = 6.66 x 10^-1
(3.95 x 10^-2 + 6.454 x 10^-3) = 3.95 x 10^-2 + 0.006454 x 10^-2 = 3.956454 x 10^-2
Now let's multiply the two results:
(6.66 x 10^-1) x (3.956454 x 10^-2) = 26.36128 x 10^-3
To express the answer in scientific notation with the correct number of significant digits, we need to round it:
The first non-zero digit after the decimal point is 2. Therefore, we round the number to have two significant digits.
26.36128 x 10^-3 = 26 x 10^-3 = 2.6 x 10^-2
So the final answer, expressed in scientific notation with the correct number of significant digits, is 2.6 x 10^-2.