The side of a square is measured to be 10 ft with a possible error of ±0.1 ft. Use differentials to estimate the error in the calculated area. Include units in your answer.
To estimate the error in the calculated area of the square, we can use differentials. Let's start by finding the formula for the area of a square.
The area of a square is given by the formula A = s^2, where s is the length of its side.
Given that the side of the square is measured to be 10 ft with a possible error of ±0.1 ft, we can express the side length as s = 10 ft.
Now, we can take the differential of the area formula to find how much the area changes when the side length changes by a small amount. The differential of A = s^2 is dA = 2s ds.
Using the given side length of s = 10 ft, we have s = 10 ft and ds = ±0.1 ft.
Now we can substitute the values into the differential equation to calculate the differential of the area (dA).
When the side length is increasing by 0.1 ft, we have:
dA = 2s ds
dA = 2(10 ft)(0.1 ft)
dA = 2 ft^2
When the side length is decreasing by 0.1 ft, we have:
dA = 2s ds
dA = 2(10 ft)(-0.1 ft)
dA = -2 ft^2
Therefore, the estimated error in the calculated area of the square is ±2 ft^2.