Tolerance (a) About how accurately must the interior...

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?

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?

Key Concepts

Measurement Tolerance

Measurement tolerance refers to the acceptable limits within which a physical measurement can deviate from a true or specified value without significantly impacting the desired outcome. In these scenarios, tolerances are set for the volume calculation (1%) and paint estimation (5%), and the challenge is to determine the precision required in measuring the tank's diameters.

Differential Calculus

Differential calculus is used to approximate the change in a function based on a small change in its input values through derivatives. In these problems, the derivative of the volume or surface area with respect to the diameter is used to relate a small variation in the diameter measurement to the corresponding error in the computed value.

Cylindrical Geometry

Cylindrical geometry pertains to the study of the properties and measures associated with cylinders, such as volume and lateral surface area. Understanding the formulas for these properties, like the volume formula V = (?d²/4)h and the lateral surface area formula A = ?dh, is essential to identify how errors in dimensions affect these calculated values.

Error Propagation

Error propagation involves determining how uncertainty in measured quantities affects the uncertainty in a derived quantity. In these problems, small changes in the diameter measurement lead to changes in the computed volume or surface area, and the relationship between the measurement error and the resulting error must be quantified.

Relative Error

Relative error is the ratio of the absolute error to the true value, often expressed as a percentage. It provides a measure of the error's size in relation to the magnitude of the quantity being measured, which is crucial when establishing how accurately a measurement like a diameter needs to be taken for a specific tolerance requirement.

Transcript

0:00 Two parts.

00:01 Part a says about how accurately must the interior diameter of a 10m high cylindrical storage tank be measured to calculate the tank's volume to within 1 % of its true value.

00:10 So we know that the equation we ultimately have to set up is that the error that we have has to be less than or equal to 1 % or 0 .1 times the true volume of the tank.

00:22 So let's find dv.

00:25 We know the volume equals pi r squared h but because we're operating talking about the diameter which we're going to call d d equals 2 r so we're going to replace r squared with d over 2 squared so v equals pi d over 2 squared times h and h is going to be 10 so now what we're going to do is we're going to do is we're going to to clean this up, we're going to get 10 pi d squared over 4 equals 5 pi d squared over 2.

01:13 And that is our volume.

01:17 I'm going to circle in blue so we know where it is.

01:21 So now to find dv, all we're going to do is first differentiate with respect to d.

01:27 And then, of course, move to the other side.

01:28 So we get 5 pi d d and then this is going to be less than or equal to 0 .01 times our volume, which we have right here, 5 pi d squared over 2.

01:52 Now what we can do is divide both sides by 5 pi d.

01:59 We just get dd on this side and we get rid of this and just have a single d left.

02:05 So we have dd is less than 0 .01 over 2d.

02:16 And this is going to be the percentage of within we need to measure, but we have to multiply it by 100 to get the percent.

02:26 So 0 .01, 0 .01 divided by 2 is 0 .005 times 100 is 0 .5.

02:34 So within 0 .5%.

02:40 That's part a.

02:42 So part b asks 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...

Please give Ace some feedback