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Consider a closed rectangular box with a square base, as shown in the figure. Assume x is measured with an error of at most 0.8% and y is |dx[ measured with an error of at most 0.30%, so we have <0.008 and |dyl <0.003. y a. Use a differential to estimate the relative error in computing the box's volume V. V |ds| b. Use a differential to estimate the relative error in computing the box's surface area S. S

Consider a closed rectangular box with a square base, as shown in the figure. Assume x is measured with an error of at most 0.8% and y is |dx[ measured with an error of at most 0.30%, so we have <0.008 and |dyl <0.003. y a. Use a differential to estimate the relative error in computing the box's volume V. V |ds| b. Use a differential to estimate the relative error in computing the box's surface area S. S

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Explanation of the Exercise:** Consider a closed rectangular box with a square base as depicted in the figure. The dimensions of the box are such that: - \( x \) is measured with an error of at most 0.8%, hence \(\left| \frac{dx}{x} \right| \leq 0.008\). - \( y \) is measured with an error of at most 0.30%, hence \(\left| \frac{dy}{y} \right| \leq 0.003\). **Tasks:** a. Use differentials to estimate the relative error \(\left| \frac{dV}{V} \right|\) in computing the box's volume \( V \). b. Use differentials to estimate the relative error \(\left| \frac{dS}{S} \right|\) in computing the box's surface area \( S \). **Graphical Explanation:** The diagram shows a 3D box with a square base labeled with dimensions \( x \) for the base and \( y \) for the height. There are dashed lines indicating the edges of the box. **Questions for Completion:** a. \(\left| \frac{dV}{V} \right| \leq\) __________ % (Type an integer or a decimal. Round to two decimal places as needed.) b. \(\left| \frac{dS}{S} \right| \leq\) __________ % (Type an integer or a decimal. Round to two decimal places as needed.)
Transcribed Image Text:**Explanation of the Exercise:** Consider a closed rectangular box with a square base as depicted in the figure. The dimensions of the box are such that: - \( x \) is measured with an error of at most 0.8%, hence \(\left| \frac{dx}{x} \right| \leq 0.008\). - \( y \) is measured with an error of at most 0.30%, hence \(\left| \frac{dy}{y} \right| \leq 0.003\). **Tasks:** a. Use differentials to estimate the relative error \(\left| \frac{dV}{V} \right|\) in computing the box's volume \( V \). b. Use differentials to estimate the relative error \(\left| \frac{dS}{S} \right|\) in computing the box's surface area \( S \). **Graphical Explanation:** The diagram shows a 3D box with a square base labeled with dimensions \( x \) for the base and \( y \) for the height. There are dashed lines indicating the edges of the box. **Questions for Completion:** a. \(\left| \frac{dV}{V} \right| \leq\) __________ % (Type an integer or a decimal. Round to two decimal places as needed.) b. \(\left| \frac{dS}{S} \right| \leq\) __________ % (Type an integer or a decimal. Round to two decimal places as needed.)
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