The measurement of the side of a square is found to be 12 cm with a possible error of 0.05 cm. Use differentials to estimate the propagated error ΔA in computing the area of the square. Please do not include the ± symbol in your answer. Round your final answer to one decimal place.
A = x^2
A + dA = (x+dx)^2 = x^2 + 2 x dx + dx^2
if dx is small
A + dA = x^2 + 2 x dx
so
dA = 2 x dx
To estimate the propagated error in computing the area of the square, we can use differentials.
Let's start by finding the formula for the area of a square. The formula for the area of a square is given by A = side^2.
Now, let's differentiate both sides of the equation implicitly with respect to the side length (s).
dA = 2s * ds
Where dA represents the change in the area (propagated error), s is the side length, and ds represents the change in the side length (possible error).
We are given that the side of the square is 12 cm with a possible error of 0.05 cm. So s = 12 cm and ds = 0.05 cm.
Substituting these values into the equation for the propagated error:
dA = 2 * 12 cm * 0.05 cm
= 1.2 cm^2
Therefore, the propagated error in computing the area of the square is approximately 1.2 cm^2.
To estimate the propagated error ΔA in computing the area of the square, we can use differentials.
The area of a square is given by A = s^2, where s is the side length.
In this case, the side length of the square is measured to be 12 cm with a possible error of 0.05 cm.
Let's call the side length s, and the error in the side length ds (which is given as 0.05 cm).
So, we have s = 12 cm and ds = 0.05 cm.
To find the propagated error ΔA in computing the area, we need to consider how a small change in s affects the area A.
Differentiating both sides of the equation A = s^2 with respect to s, we get:
dA = 2s * ds
Substituting the values of s = 12 cm and ds = 0.05 cm, we can find the propagated error ΔA as follows:
ΔA = 2 * 12 cm * 0.05 cm
= 1.2 cm^2.
Rounding the final answer to one decimal place, the propagated error ΔA in computing the area of the square is 1.2 cm^2.