The measurement of the radius of a circle is 16 inches, with a possible error of .25 inches.
Use differentials to approximate the possible propagate error in computing the area of the circle.
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%
a = pi r^2
da = 2pi r dr
now just plug in your numbers.
Well, let's clown around with some math, shall we?
The formula for the area of a circle is A = πr^2, where r is the radius. Now, we want to find the possible propagated error in computing the area when the radius has a possible error of 0.25 inches.
To do this, we need to use differentials. Let's say the radius is r, and the error in the radius is Δr. So, our new radius is r + Δr.
Now, let's differentiate the formula for the area with respect to r:
dA/dr = 2πr
This gives us the rate of change of the area with respect to the radius. Now, we can use differentials to approximate the propagated error in the area, ΔA:
ΔA ≈ (dA/dr) * Δr
Plugging in the values, we have:
ΔA ≈ (2πr) * (Δr)
Since r = 16 inches and Δr = 0.25 inches, we can calculate:
ΔA ≈ (2π * 16) * (0.25)
ΔA ≈ 25π * 0.25
ΔA ≈ 6.25π
So, the approximate propagated error in computing the area of the circle with a radius of 16 inches (and a possible error of 0.25 inches) is approximately 6.25π square inches.
But hey, remember that this is just an approximation, so don't take it too "circley" serious!
To approximate the possible propagated error in computing the area of a circle with a given radius and possible error, we can use differentials.
The formula for calculating the area of a circle is A = πr^2, where A is the area and r is the radius.
Let's denote the radius as r = 16 inches and the possible error as Δr = 0.25 inches.
To find the propagated error in the area, we need to find the differential of A with respect to r.
dA = (∂A/∂r) · dr
Taking the derivative of the area formula with respect to r:
∂A/∂r = 2πr
Substituting the given radius:
∂A/∂r = 2π(16)
Simplifying:
∂A/∂r = 32π
Now we can calculate the propagated error in the area by multiplying the derivative by the possible error:
ΔA = (∂A/∂r) · Δr
Substituting the values:
ΔA = 32π · 0.25
Approximating the value of π to 3.14:
ΔA ≈ 32(3.14)(0.25)
Simplifying:
ΔA ≈ 25.12
Therefore, the approximate possible propagated error in computing the area of the circle is approximately 25.12 square inches.
To approximate the propagated error in computing the area of a circle, we can use differentials.
The formula for the area of a circle is A = πr², where A is the area and r is the radius. To find the propagated error, we need to find the differential of A with respect to r.
First, we can find the differential of A using the power rule of differentiation. Since A = πr², we differentiate both sides of the equation with respect to r:
dA/dr = d(πr²)/dr
Using the power rule, we bring down the exponent and multiply it by the coefficient:
dA/dr = 2πr
Next, we can find the differential of r. Since the radius has a possible error of 0.25 inches, we can express this as dr = 0.25 inches.
Finally, we can use these differentials to approximate the propagated error in computing the area of the circle. Using the differential equation, we substitute the values of dA/dr and dr:
ΔA = dA/dr * Δr
where ΔA is the propagated error in the area and Δr is the possible error in the radius. Substituting the values we found earlier:
ΔA = (2πr) * (0.25)
In this case, the radius is given as 16 inches, so substituting r = 16:
ΔA = (2π * 16) * 0.25
Calculating this expression will give you the approximate propagated error in computing the area of the circle.