An error in the measurement of the radius of a circle results in an error in the computation of its area. Find and interpret the sensitivity of area (a) of a circle to the measurement of radius, r, when the radius r=2 meters.
a = πr^2
da = 2πr dr
The error is ∆a which for small errors is approximately da.
So plug and chug.
Now, what does it all mean?
To find the sensitivity of the area of a circle to the measurement of its radius, we need to calculate the derivative of the area with respect to the radius. The sensitivity, also known as the partial derivative, will give us the rate of change of the area with respect to a small change in the radius.
The formula for the area of a circle is given by A = πr², where A is the area and r is the radius. To find the sensitivity of the area (A) to the radius (r), we differentiate the area formula with respect to the radius:
dA/dr = d(πr²)/dr
Using the power rule of differentiation, we can bring down the exponent of 2:
dA/dr = 2πr
Now, we can evaluate the sensitivity at the given radius r = 2 meters:
dA/dr = 2π(2)
= 4π
Therefore, the sensitivity of the area of a circle to the measurement of its radius when the radius is 2 meters is 4π.
Interpreting the sensitivity: A sensitivity of 4π means that for every unit increase in the radius of the circle, the area will increase by approximately 4π square units. In this case, since the radius is measured in meters, the sensitivity represents the increase in area in square meters.