Question

Tolerance (a) About how accurately must the interior diameter of a 10 -m high cylindrical storage tank be measured to calculate the tank's volume to within 1\(\%\) of its true value? (b) About how accurately must the tank's exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5\(\%\) of the true amount?

Short answer

To calculate the tank's volume to within 1% of its true value, the interior diameter must be measured to an accuracy of about 0.50%. To calculate the amount of paint required to within 5% of the true amount, the exterior diameter needs to be measured to an accuracy of about 2.50%.

Step-by-step solution

Understand the Cylinder Parameters

In order to solve the problem, first understand what a cylinder's volume (\( V \)) and surface area (\( A \)) are. The volume of a cylinder is given by \( V = \pi r^{2}h \), where \( r \) is the radius and \( h \) is the height, while the surface area (which is the area that needs to be painted in this case) is determined by \( A = 2\pi rh \).

Calculate the percentage change for the volume

The relative change in volume with respect to radius dV/dr, if multiplied by r/V, results in the corresponding relative change in radius. Setting this equal to 0.01 (corresponding to 1\%), one can solve for dr/r, which gives the relative precision needed in measuring the radius. Since radius and diameter are directly proportional, this also gives the relative precision in the diameter measurement. Using the cylinder volume formula and computed relative changes, dr/r = -0.01. This will allow you to determine that the diameter must be measured to within about 0.50% accuracy.

Calculate the percentage change for the surface area (paint)

Similarly, the relative change in surface area with respect to radius dA/dr, if multiplied by r/A, gives the corresponding relative change in radius. Setting this equal to 0.05 (corresponding to 5\%) and solving for dr/r provides the relative precision required in measuring the radius. As before, since radius and diameter are proportional, this also gives the relative precision in the diameter measurement. Using the cylinder surface area formula and computed relative changes, dr/r = 0.05. This will allow you to determine that the diameter must be measured to within about 2.50% accuracy.