The measurement of the radius of a circle is 16 inches, with a possible error of .25 inches.
Use differentials to approximate the possible propagated error in computing the area of the circle and find the percent error.
I've gotten to 2Pi*16*.25 but don't know what to do past this point.
dr = .25 - the error in the radius
a = pi r^2
da = 2pi r dr
so, as you found, the variation in area is 8pi.
the relative error is thus
da/a = 8pi/(256pi) = 1/32 = 3.125%
To find the propagated error in computing the area of the circle, we can use differentials.
The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
In this case, the radius is given as 16 inches with a possible error of 0.25 inches. Let's represent the error as dr.
1. Write down the formula for the area of the circle in terms of the radius:
A = πr^2
2. Differentiate both sides of the equation with respect to r:
dA = (d/dx)(πr^2)
3. Apply the power rule of differentiation to the equation:
dA = 2πr(dr)
Now we need to substitute the values into the equation.
r = 16 inches (given)
dr = 0.25 inches (given)
4. Plug in the values into the equation:
dA = 2π(16)(0.25)
dA = 8π
So, the possible propagated error in computing the area of the circle is 8π square inches.
To find the percent error, we divide the propagated error by the actual area and multiply by 100:
5. Calculate the actual area of the circle using the given radius:
A = π(16)^2
A = 256π
6. Calculate the percent error:
Percent error = (dA / A) * 100
Percent error = (8π / 256π) * 100
Percent error = 3.125%
Therefore, the percent error in computing the area of the circle is approximately 3.125%.