Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Answer both questions for the values of w and h

**Optimization Problem: Maximizing Area** Farmer John has 1200 feet of fencing and wishes to use it to fence a rectangular plot divided into two subplots as in the figure below. Suppose we want to determine the dimensions \( w \) and \( h \) of the plot with maximum area. ![Rectangular plot divided into two subplots](image link here, if any) If Farmer John uses all the fencing, then \( 3h + 2w = 1200 \) and so the area of the region is \[ A(w) = w \left( \frac{1200 - 2w}{3} \right) \] Determine the values of \( w \) and \( h \) that give the maximum area. --- **The exact value of \( w \) is:** [Input box for \( w \) value] --- **The exact value of \( h \) is:** [Input box for \( h \) value] --- ### Diagram Explanation: - The diagram shows a large rectangle divided into two equal smaller rectangular sections by a vertical line. - The total width of the large rectangle is denoted by \( w \). - The height of the large rectangle is denoted by \( h \). - The perimeter constraint is given by the equation \( 3h + 2w = 1200 \). ### Steps to Solve the Problem: 1. **Setting up the Perimeter Equation:** Since the total fencing used around the border and the dividing line is 1200 feet, we have the equation \( 3h + 2w = 1200 \). 2. **Express \( h \) in terms of \( w \):** \[ h = \frac{1200 - 2w}{3} \] 3. **Formulate the Area Function:** Substituting \( h \) in terms of \( w \) into the area formula \( A = w \cdot h \): \[ A(w) = w \left(\frac{1200 - 2w}{3} \right) = \frac{1200w - 2w^2}{3} \] 4. **Maximizing the Area:** Find the value of \( w \) that maximizes \( A(w) \). This involves differentiating \( A(w) \) with respect to \( w \
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Transcribed Image Text:**Optimization Problem: Maximizing Area** Farmer John has 1200 feet of fencing and wishes to use it to fence a rectangular plot divided into two subplots as in the figure below. Suppose we want to determine the dimensions \( w \) and \( h \) of the plot with maximum area. ![Rectangular plot divided into two subplots](image link here, if any) If Farmer John uses all the fencing, then \( 3h + 2w = 1200 \) and so the area of the region is \[ A(w) = w \left( \frac{1200 - 2w}{3} \right) \] Determine the values of \( w \) and \( h \) that give the maximum area. --- **The exact value of \( w \) is:** [Input box for \( w \) value] --- **The exact value of \( h \) is:** [Input box for \( h \) value] --- ### Diagram Explanation: - The diagram shows a large rectangle divided into two equal smaller rectangular sections by a vertical line. - The total width of the large rectangle is denoted by \( w \). - The height of the large rectangle is denoted by \( h \). - The perimeter constraint is given by the equation \( 3h + 2w = 1200 \). ### Steps to Solve the Problem: 1. **Setting up the Perimeter Equation:** Since the total fencing used around the border and the dividing line is 1200 feet, we have the equation \( 3h + 2w = 1200 \). 2. **Express \( h \) in terms of \( w \):** \[ h = \frac{1200 - 2w}{3} \] 3. **Formulate the Area Function:** Substituting \( h \) in terms of \( w \) into the area formula \( A = w \cdot h \): \[ A(w) = w \left(\frac{1200 - 2w}{3} \right) = \frac{1200w - 2w^2}{3} \] 4. **Maximizing the Area:** Find the value of \( w \) that maximizes \( A(w) \). This involves differentiating \( A(w) \) with respect to \( w \
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