Rose is making a postcard decorated with flecks of gold dust! The part of the postcard that Rose is decorating with flecks of gold dust this year is in a shape of isosceles trapezoid with top width equal to 4cm, the bottom width equal to 10 cm and both legs equal to 5 cm, shown as below. 5cm, 4cm 4cm 10cm 4cm 5 5cm Suppose that the density of gold dust on the postcard is given by p(y) mg/cm², where y represents Histance to the base of length 10 cm. a) Write a general Riemann sum that approximates the amount of gold dust on the postcard. Start by slicing into n slices; your final approximation should be in terms of n. Be sure to explain clearly how you are slicing and all the details of your work, including what your notation means. b) Write a definite integral giving the exact amount of gold dust in the postcard. C) This year, Rose would like to save money by using less gold dust. Which of the following density functions Rose should use? Briefly explain your reasoning. You do not need to evaluate any of the integrals. i. Rose should use p₁(y) = 4-y in order to save money. ii. Rose should use p2(y) = y in order to save money.
Rose is making a postcard decorated with flecks of gold dust! The part of the postcard that Rose is decorating with flecks of gold dust this year is in a shape of isosceles trapezoid with top width equal to 4cm, the bottom width equal to 10 cm and both legs equal to 5 cm, shown as below. 5cm, 4cm 4cm 10cm 4cm 5 5cm Suppose that the density of gold dust on the postcard is given by p(y) mg/cm², where y represents Histance to the base of length 10 cm. a) Write a general Riemann sum that approximates the amount of gold dust on the postcard. Start by slicing into n slices; your final approximation should be in terms of n. Be sure to explain clearly how you are slicing and all the details of your work, including what your notation means. b) Write a definite integral giving the exact amount of gold dust in the postcard. C) This year, Rose would like to save money by using less gold dust. Which of the following density functions Rose should use? Briefly explain your reasoning. You do not need to evaluate any of the integrals. i. Rose should use p₁(y) = 4-y in order to save money. ii. Rose should use p2(y) = y in order to save money.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Rose is making a postcard decorated with flecks of gold dust! The part of the postcard that Rose is decorating with flecks of gold dust this year is in the shape of an isosceles trapezoid with a top width equal to 4 cm, a bottom width equal to 10 cm, and both legs equal to 5 cm, as shown below.
[Diagram of an isosceles trapezoid]
- Top width: 4 cm
- Side length: 5 cm
- Bottom width: 10 cm
- Height sections: Two 4 cm sections between top and bottom edges are indicated with dotted lines, flanked by right angles.
Suppose that the density of gold dust on the postcard is given by \( \rho(y) \) mg/cm², where \( y \) represents distance to the base of length 10 cm.
**a)**
Write a general Riemann sum that approximates the amount of gold dust on the postcard. Start by slicing into \( n \) slices; your final approximation should be in terms of \( n \). Be sure to explain clearly how you are slicing and all the details of your work, including what your notation means.
**b)**
Write a definite integral giving the exact amount of gold dust in the postcard.
**c)**
This year, Rose would like to save money by using less gold dust. Which of the following density functions should Rose use? Briefly explain your reasoning. You do not need to evaluate any of the integrals.
i. Rose should use \( \rho_1(y) = 4 - y \) in order to save money.
ii. Rose should use \( \rho_2(y) = y \) in order to save money.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3b47e7c-f6ee-4dde-9ef1-0f8b8a2d2829%2Fa1084b1e-8f76-4415-967b-a1f03d43336f%2Fy345g7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Rose is making a postcard decorated with flecks of gold dust! The part of the postcard that Rose is decorating with flecks of gold dust this year is in the shape of an isosceles trapezoid with a top width equal to 4 cm, a bottom width equal to 10 cm, and both legs equal to 5 cm, as shown below.
[Diagram of an isosceles trapezoid]
- Top width: 4 cm
- Side length: 5 cm
- Bottom width: 10 cm
- Height sections: Two 4 cm sections between top and bottom edges are indicated with dotted lines, flanked by right angles.
Suppose that the density of gold dust on the postcard is given by \( \rho(y) \) mg/cm², where \( y \) represents distance to the base of length 10 cm.
**a)**
Write a general Riemann sum that approximates the amount of gold dust on the postcard. Start by slicing into \( n \) slices; your final approximation should be in terms of \( n \). Be sure to explain clearly how you are slicing and all the details of your work, including what your notation means.
**b)**
Write a definite integral giving the exact amount of gold dust in the postcard.
**c)**
This year, Rose would like to save money by using less gold dust. Which of the following density functions should Rose use? Briefly explain your reasoning. You do not need to evaluate any of the integrals.
i. Rose should use \( \rho_1(y) = 4 - y \) in order to save money.
ii. Rose should use \( \rho_2(y) = y \) in order to save money.
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