Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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how do I approach part c)
The problem involves determining how to cut a rectangular beam from a cylindrical log with a diameter of \( d = 20 \) cm. The challenge is to express and graph the cross-sectional area as a function of the angle \( \theta \). ### (a) Express the Cross-Sectional Area The problem requires expressing the cross-sectional area \( A(\theta) \) of the beam as a function of the angle \( \theta \): \[ A(\theta) = 400 \sin(\theta) \cos(\theta) \] ### (b) Graph the Function Four graphs are presented, showing \( A(\theta) \) as a function of \( \theta \): 1. **Top Left Graph:** A plot of \( A(\theta) \) versus \( \theta \) showing a peak at approximately 1.0 radians, with values on the vertical axis ranging from 0 to 300. 2. **Top Right Graph:** A decreasing curve starting at around 300 and approaching zero as \( \theta \) increases. 3. **Bottom Left Graph:** An increasing graph starting from zero and rising steeply. 4. **Bottom Right Graph:** A plot similar in shape to the top left graph, but with smaller scale on the vertical axis, ranging from 0 to 20. ### (c) Dimensions of the Beam with Largest Cross-Sectional Area Calculate the dimensions of the beam (width and depth) that result in the largest cross-sectional area. Answers should be rounded to two decimal places. #### Note - Users are encouraged to use a graphing calculator to better visualize and verify the functions. - Two interactive options at the bottom, "Read It" and "Watch It", are likely resources for additional help.
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Transcribed Image Text:The problem involves determining how to cut a rectangular beam from a cylindrical log with a diameter of \( d = 20 \) cm. The challenge is to express and graph the cross-sectional area as a function of the angle \( \theta \). ### (a) Express the Cross-Sectional Area The problem requires expressing the cross-sectional area \( A(\theta) \) of the beam as a function of the angle \( \theta \): \[ A(\theta) = 400 \sin(\theta) \cos(\theta) \] ### (b) Graph the Function Four graphs are presented, showing \( A(\theta) \) as a function of \( \theta \): 1. **Top Left Graph:** A plot of \( A(\theta) \) versus \( \theta \) showing a peak at approximately 1.0 radians, with values on the vertical axis ranging from 0 to 300. 2. **Top Right Graph:** A decreasing curve starting at around 300 and approaching zero as \( \theta \) increases. 3. **Bottom Left Graph:** An increasing graph starting from zero and rising steeply. 4. **Bottom Right Graph:** A plot similar in shape to the top left graph, but with smaller scale on the vertical axis, ranging from 0 to 20. ### (c) Dimensions of the Beam with Largest Cross-Sectional Area Calculate the dimensions of the beam (width and depth) that result in the largest cross-sectional area. Answers should be rounded to two decimal places. #### Note - Users are encouraged to use a graphing calculator to better visualize and verify the functions. - Two interactive options at the bottom, "Read It" and "Watch It", are likely resources for additional help.
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