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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.48a
Chapter 3, Problem 3.9.48a

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?

Verified step by step guidance
1
First, recall the formula for the volume of a cylinder: \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder.
Since the problem involves the interior diameter, note that the diameter \( d = 2r \). Therefore, the radius \( r = \frac{d}{2} \).
To find the tolerance in the diameter, we need to consider the derivative of the volume with respect to the diameter. Start by expressing the volume in terms of the diameter: \( V = \pi \left(\frac{d}{2}\right)^2 h = \frac{\pi d^2 h}{4} \).
Differentiate the volume with respect to the diameter \( d \) to find \( \frac{dV}{dd} = \frac{\pi dh}{2} \). This represents the rate of change of volume with respect to changes in diameter.
Set up the equation for the tolerance: \( \frac{dV}{V} = \frac{1}{100} \). Substitute \( \frac{dV}{dd} \) and solve for the change in diameter \( dd \) that corresponds to a 1% change in volume. This will give you the required accuracy for measuring the diameter.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. Understanding this formula is crucial because the problem involves calculating the volume of a cylindrical tank. The accuracy of the diameter measurement directly affects the radius, which in turn impacts the volume calculation.
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Percentage Error

Percentage error is a measure of how inaccurate a measurement is, compared to the true value, expressed as a percentage. In this problem, the goal is to ensure the calculated volume is within 1% of the true volume. This concept helps in determining the acceptable range of error in the diameter measurement to achieve the desired accuracy in volume calculation.
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Differential Calculus

Differential calculus involves the study of how functions change when their inputs change. It is used here to determine how small changes in the diameter affect the volume of the cylinder. By using derivatives, one can estimate the change in volume with respect to a change in diameter, which is essential for calculating the required measurement accuracy.
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Related Practice
Textbook Question

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is


y = 37 sin[(2π/365)(x − 101)] + 25


and is graphed in the accompanying figure.


a. On what day is the temperature increasing the fastest?


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Textbook Question

Motion Along a Coordinate Line


Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.


a. Find the body’s displacement and average velocity for the given time interval.


s = (t⁴/4) − t³ + t², 0 ≤ t ≤ 3

Textbook Question

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.


a. f(x) = (1 − x)⁶

Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s


a. circumference?

Textbook Question

In Exercises 47 and 48, find an equation for


(a) the tangent line to the curve at P


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Textbook Question

Quadratic approximations


a. Let Q(x) = b₀ + b₁(x − a) + b₂(x − a)² be a quadratic approximation to f(x) at x = a with these properties:


i. Q(a) = f(a)

ii. Q′(a) = f′(a)

iii. Q″(a) = f″(a).


Determine the coefficients b₀, b₁, and b₂.