You measure a cubic container and find it to be about 10 cm on each side. From what you conclude that it holds 1000 cc. However, your measurement is accurate only to within +- 0.1 cm. (So you can be sure that the side is between 9.9 and 10.1 cm.)
A) What are x, f(x), and a in this problem?
B) Use the tangent line approximation to estimate the error in your value of 1000 for the volume.
C) Give the percent error in your measurement for the length of the side.
D) Give the percent error in your estimate of the volume.
A) In this problem, x represents the side length of the cubic container (in cm), f(x) represents the volume of the container (in cc), and a represents the accuracy of the measurement (in cm).
B) To estimate the error in the value of 1000 cc for the volume, we can use the tangent line approximation. This involves finding the derivative of the volume function with respect to x and evaluating it at the measured value of x. The derivative of the volume function will give us the rate at which the volume changes with respect to a change in the side length.
Let's assume the volume function V(x) is given by V(x) = x^3, where x is the side length. Taking the derivative of V(x) with respect to x, we get:
V'(x) = 3x^2
Now, using the measured value of x (10 cm), we can evaluate the derivative at x = 10 to find the rate of change of volume:
V'(10) = 3(10)^2 = 300
From the tangent line approximation, the error in the value of 1000 cc can be estimated by calculating the difference between the actual volume and the approximation obtained from the tangent line. Let's denote the approximate volume as V_approx:
V_approx = V(10) + V'(10)(x - 10)
Since we are measuring the side length accurate to within +/- 0.1 cm, we can use this range to estimate the error. Thus, the error in the volume can be approximated as:
Error = |V(x) - V_approx|
C) To calculate the percent error in the measurement for the length of the side, we can use the known accuracy range (±0.1 cm). The percent error formula is given by:
Percent error = (actual value - measured value) / actual value * 100
In this case, the actual value is 10 cm (as given), and the measured value is accurate within ±0.1 cm.
D) To calculate the percent error in the estimate of the volume, we can use the formula:
Percent error = (actual value - estimated value) / actual value * 100
In this case, the actual value is 1000 cc (as given), and the estimated value is obtained from the tangent line approximation.