K asked in Science & MathematicsMathematics · 1 month ago

Please help with MATH HOMEWORK ?

i’ve tried on my own but i don’t understand this at all!! please help me!! thank you in advance to anyone who helps <3

1. Develop an equation for the area of the neighbor’s corral, in terms of x only, A(x)

2. Once you have the area function, A(x), use algebra to determine the dimensions for x that will maximize the area of the neighbor’s corral, and determine the actual maximum area. 

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1 Answer

  • 1 month ago
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    PART 1:

    The corral is a rectangle, so the area is length times width.

    Let's assume the width is the side marked 'x' in the diagram.

    width = x

    The length is not marked, however, you know the total fencing is 400 meters and you have two sections of 'x' meters each for the side fences. What's an expression for the long fence along the length?

    length = 400 - 2x

    So what's an expression for the area in terms of x?

    A(x) = x(400 - 2x)

    Or if you like, you can expand it and put the exponents in descending order:

    A(x) = 400x - 2x²

    A(x) = -2x² + 400x

    PART 2:

    From algebra you should recognize this function is a quadratic. Visually, if you graph it, it is a downward facing parabola. The vertex will be where the function has the maximum area.

    So how do you find the vertex of a parabola?

    One quick way is to use the formula for the line of symmetry. When you have a quadratic of the form y = ax² + bx + c, the line of symmetry is:

    x = -b/(2a)

    If you need help remembering that, it's essentially the quadratic formula, but without the ±√ part.

    Plug in your values:

    a = -2

    b = 400

    x = -400/(2(-2))

    x = -400/-4

    x = 100

    Looking at your diagram, you can see that means:

    width(x) = 100 meters

    length(400-2x) = 200 meters

    The area is the product:

    Maximum area = 100 * 200 = 20,000 m² (or 2 hectares).

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