The measurement of the radius of a circle is found to be 14 inches, with a possible error of 1/4 inch. Use differentials to approximate the possible error and the percentage error in computing the area of a circle.
a = pi r^2
da = 2pi r dr
= 2pi * 14 * 1/4
= 7pi
%err = 7pi/196pi = 1/28
A box is measured to have a length of 15 inches, a width of 8 inches and a depth of 4 inches. From these measurements, we would calculate the volume to be 480 cubic inches. If there is a possible error of up to 0.12 inch in each of the measurements use differentials to estimate the maximum possible error in the calculated volume. (Use V=lbh where V=volume, 1= length(1), 2= width (b), 3= depth (h) )
To solve this problem, we can use differentials to approximate the possible error and the percentage error in computing the area of a circle.
First, let's identify the given information:
- The measurement of the radius of the circle is 14 inches.
- The possible error in the measurement is 1/4 inch.
Now, let's calculate the possible error in the radius:
Since the possible error in the measurement of the radius is 1/4 inch, we can say that the radius can vary between 14 - 1/4 = 13.75 inches and 14 + 1/4 = 14.25 inches.
Next, let's find the differential of the area of the circle:
The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius. Taking the differential of this formula, we get dA = 2πr dr.
Now, we can substitute the values we have:
- Radius, r = 14 inches
- Differential of the radius, dr = (14.25 - 13.75) = 0.5 inches (since the radius can vary within this range)
Plugging these values into the differential formula, we get:
dA = 2π(14) (0.5)
= 14π(0.5)
= 7π
So, the differential of the area of the circle, dA, is 7π square inches.
To find the possible error in the area, we take the absolute value:
Possible Error = |dA| = |7π|
Now, let's calculate the percentage error in computing the area of the circle:
Percentage Error = (Possible Error / Actual Area) × 100%
We know that Actual Area = πr^2, where r = 14 inches.
Plugging in the values, we get:
Percentage Error = (|7π| / (π(14)^2) ) × 100%
= (|7π| / (196π) ) × 100%
= (7 / 196) × 100%
= (7/2)%
Therefore, the possible error in computing the area of a circle is |7π| square inches, and the percentage error is (7/2)%.