Cable Configuration and Distance Optimization

The video tutorial explains how to minimize the cable length needed to connect Centerville to Springfield and Shelbyville. It uses a Y-shaped configuration and applies mathematical concepts like the Pythagorean theorem and derivatives to find the optimal point for cable splitting. The tutorial also verifies the solution using the second derivative test, concluding with the minimum cable length required.

What is the main objective of the cable configuration problem?

• To connect Centerville to Springfield and Shelbyville with the least cable.

• To determine the population of Centerville.

• To find the shortest path between Springfield and Shelbyville.

• To calculate the cost of cable installation.

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CCSS.5.G.A.1

Where is Centerville located on the coordinate plane?

• (0, 5)

• (9, 0)

• (0, -5)

• (0, 0)

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CCSS.HSG.MG.A.1

What shape is formed by the cable configuration?

• A straight line

• A circle

• A triangle

• A Y shape

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CCSS.HSG.GPE.B.7

Which theorem is used to derive the distance formula for the cable?

• Pythagorean theorem

• Fundamental theorem of calculus

• Remainder theorem

• Binomial theorem

What is the expression for the distance from the split point to Springfield?

• sqrt(x^2 + 25)

• x^2 + 25

• x + 5

• 9 - x

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CCSS.HSF.LE.B.5

What is the function D(x) used for in this problem?

• To calculate the cost of cable

• To represent the total cable length

• To find the shortest path between towns

• To determine the population of towns

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CCSS.8.F.A.3

What is the critical number found for minimizing D(x)?

• x = 25/3

• x = 5/sqrt(3)

• x = 9

• x = 0

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CCSS.8.F.A.3

Which test is used to verify the minimum cable length?

• First derivative test

• Second derivative test

• Integral test

• Limit test

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CCSS.3.MD.D.8

What is the approximate total cable length required?

• 25 units

• 15.5 units

• 17.6603 units

• 20 units

At which point on the x-axis does the cable split to minimize length?

• (5/sqrt(3), 0)

• (0, 5)

• (0, 0)

• (9, 0)