The side of a square is measured to be 16 ft with a possible error of ±0.1 ft. Use differentials to estimate the error in the calculated area. Include units in your answer.
A = x^2
dA = 2 x dx
x = 16
dx = .1
so dA = 2 * 16 * .1
Well, isn't that a square-ly question! Let's get calculating.
The formula for the area of a square is A = s^2, where "s" represents the length of a side. Given that the side is measured as 16 ft with a possible error of ±0.1 ft, we can express the possible side lengths as 16 + 0.1 and 16 - 0.1.
Now, let's differentiate the formula with respect to "s" to find the differential expression for the area: dA = 2s ds.
Substituting the original side length value of 16, we get: dA = 2(16) ds = 32 ds.
The error, or differential, in the calculated area will then be: ΔA = 32 ds.
To find the value of "ds", we can take half of the possible error in the side length, which is 0.1/2 ft = 0.05 ft.
Plugging this value into the equation for the error in the area, we get: ΔA = 32(0.05) = 1.6 ft^2.
So, the estimated error in the calculated area is 1.6 square feet. That's a little wiggle room for our square!
To estimate the error in the calculated area, we can use differentials.
First, let's find the exact area of the square. The formula for the area of a square is A = s^2, where s is the side length. Given that the side length is measured to be 16 ft, the exact area is A = 16^2 = 256 sq ft.
Next, let's find the differential of the area with respect to the side length. The differential of the area, dA, is given by dA = 2s ds, where ds is the change in the side length.
Since the side length has a possible error of ±0.1 ft, we can assume that ds = ±0.1 ft.
Substituting the values into the differential equation, we have dA = 2(16)(±0.1) = ±3.2 sq ft.
Therefore, the estimated error in the calculated area is ±3.2 sq ft.
To estimate the error in the calculated area of the square, we can use differentials.
The formula for the area of a square is A = s^2, where A represents the area and s represents the side length.
Given that the side of the square is measured to be 16 ft with a possible error of ±0.1 ft, we can express the side length as s = 16 ft ± 0.1 ft.
Using differentials, we can find the differential of the area, which represents the change in the area for a small change in the side length, denoted as dA .
To find dA, we take the differential of the area formula:
dA = (d/ds)(s^2) * ds
Differentiating the area formula with respect to s, we get:
dA = 2s * ds
Substituting in the given value of s = 16 ft and ds = ±0.1 ft, we can calculate the error in the area:
dA = 2(16 ft)(±0.1 ft)
Simplifying:
dA = ±3.2 ft^2
Therefore, the estimated error in the calculated area is ±3.2 square feet.