5. A rectangular field is to be bounded by a fence on three sides and by a straight stream on the fourth side. Find the dimensions of the field with the maximum area that can be enclosed by 1000 feet of fencing. y RARRRStream = NRRY tep 1: What is the quantity we are trying to maximize?
5. A rectangular field is to be bounded by a fence on three sides and by a straight stream on the fourth side. Find the dimensions of the field with the maximum area that can be enclosed by 1000 feet of fencing. y RARRRStream = NRRY tep 1: What is the quantity we are trying to maximize?
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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I need Help answering both question 5 and 6 . I don’t know how to do them.
Transcribed Image Text:Sure! Here is the transcription of the document along with an explanation of the diagrams:
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**5.** A rectangular field is to be bounded by a fence on three sides and by a straight stream on the fourth side. Find the dimensions of the field with the maximum area that can be enclosed by 1000 feet of fencing.
*(Diagram: A rectangle with dimensions labeled as `y` for the length, and `x` for the width on each side adjacent to the stream. The stream is depicted as a wavy line on the side opposite `y`.)*
**Step 1:** What is the quantity we are trying to maximize?
___________________________
**Step 2:** What is the formula that relates the variables and the quantity we are trying to maximize?
___________________________
**Step 3:** The amount of fencing is the constraint. Write an equation for this constraint. Remember that no fencing is needed for the side with the stream.
___________________________
**Step 4:** Solve for one of the variables in the constraint equation.
___________________________
**Step 5:** Substitute into the function we are trying to maximize, and simplify.
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**6.** An open box is to be made from a sheet which is 30 inches by 30 inches by cutting out squares of equal size from the four corners and bending up the edges. Let `h` be the side of the square. Find the value of `h` that will produce the large volume.
*Write the equation of the function to be maximized. Remember that the volume of a rectangular solid is \( V = lwh \) in terms of `h`.*
\( V(h) = \) ______________
*(Diagram: A square sheet labeled as 30 inches by 30 inches, with small squares labeled `h` at each corner to be cut out. The remaining part of the sheet is folded to form an open box with dimensions \(30-2h\) by \(30-2h\) by `h`.)*
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**Explanation of Diagrams:**
- The first diagram illustrates a rectangular field with the longer dimension `y` parallel to a stream and the shorter dimension `x` perpendicular, requiring fencing only on three sides.
- The second diagram illustrates the process of creating an open box by cutting squares of size `h` from each corner of a square sheet and folding it upwards. The box dimensions are then
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