Duncan's person is making him a tent. She plans to make it a rectangular prism in shape, with...

Question:

Duncan's person is making him a tent. She plans to make it a rectangular prism in shape, with plastic tubing making a frame on the four vertical edges and the four edges on the top. At least one of the sides (not the top) of the tent will be a square, let s be the length (in inches) of each side of that square.

  • a. She already has the fabric she needs, but she still needs to purchase the tubing, which costs {eq}$1.49 {/eq} per foot. Duncan will be most comfortable in a tent with a volume of {eq}720 {/eq} cubic inches.
    • i. Find a formula for {eq}C(s) {/eq}, the cost (in dollars) of the tubing needed to build a tent when an edge of the square side of the tent is {eq}s {/eq} inches.
    • ii. In the context of this problem, what is the domain of {eq}C(s) {/eq}?
    • iii. Find the dimensions that will give this volume while minimizing the cost (if such dimensions exist), and the cost associated to those dimensions.
    • iv. Find the dimensions that will give this volume while maximizing the cost (if such dimensions exist), and the cost associated to those dimensions.
  • b. Later she discovers that she does have {eq}8 {/eq} feet of tubing, and decides that instead of making a tent with a particular volume, she will make one using all of this tubing.
    • i. Find a formula for {eq}V(s) {/eq}, the volume (in cubic inches) of the tent when an edge of the square side is {eq}s {/eq} inches.
    • ii. In the context of this problem, what is the domain of {eq}V(s) {/eq}?
    • iii. Find the dimensions that will use the tubing while minimizing the volume (if such dimensions exist), and the volume associated to those dimensions.
    • iv. Find the dimensions that will use the tubing while maximize the volume (if such dimensions exist), and the volume associated to those dimensions.

Optimization- Geometrical Applications:

The volume of the rectangular prism depends on the measure of the edges on the sides and on the top of the prism. For a fixed volume {eq}V_0 {/eq} we get a relation between the lengths of the vertical and top edges, so we can get the cost depending on just one of the variables so we can minimize the cost using derivatives, if that maximum exists.

Similarly, for a fixed amount of tubing while allowing the volume to change, we get a relation between the lengths of the edges. We can get the volume depending on just one of the edge's length so we can maximize the volume using derivatives, if that maximum exists.

Answer and Explanation: 1

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For Part (a): Given that the vertical edges measure {eq}s {/eq} inches each, let say that two of the top edges measure {eq}l {/eq} inches. The...

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Optimization and Differentiation

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Chapter 10 / Lesson 5
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Optimization is the process of applying mathematical principles to real-world problems to identify an ideal, or optimal, outcome. Learn to apply the five steps in optimization: visualizing, definition, writing equations, finding minimum/maximums, and concluding an answer.


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