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A group of lifeguards want to construct a rectangular swimming area along a beach with a string of floats. Only three sides of the area will need floats, since the beach will form the fourth side If the lifeguards use 300 m of floats, what is the maximum possible area? What should the length and width of the swimming area be?

Sagot :

zoequiroz2010

Answer:

The dimensions of the area are 75 feet by 150 feet

The maximum area = 75 × 150 = 11250 sq ft

Step-by-step explanation:

I hope this helped !

sqdancefan

Answer:

  • maximum area: 11,250 m²
  • dimensions: 150 m parallel to the beach, 75 m out from the beach

Step-by-step explanation:

Let x represent the length of the swimming area along the beach. After x amount of floats are used in that direction, the remaining (300-x) of floats will be split in two to form the ends of the swimming area. The total area is the product of these dimensions:

  A = LW

  A = x(300 -x)/2

You will recognize this as a factored quadratic with zeros at x=0 and x=300. The vertex of a quadratic is found halfway between the zeros, at ...

  x = (0+300)/2 = 150

In this quadratic, A = -1/2x^2 +150x, the leading coefficient is negative, so the graph opens downward and the vertex is a maximum.

The maximum area is ...

  A = (150)(300 -150)/2 = 11,250 . . . square meters

The dimension along the beach is 150 m; the dimension out from the beach is (300 -150)/2 = 75 m.

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Additional comment

This is an example of a "maximize the area for a given cost" problem that involves a rectangular area. The generic solution to such a problem is that the sum of costs in one direction is equal to the sum of costs in the orthogonal direction.

Here, half the total cost (length of float) is parallel to the beach, and the other half is perpendicular to the beach. If the area had any partitions, their cost would be added to the total for the direction of their layout.

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