Calc problem?

A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing $22 per foot along three sides and fencing costing $14 per foot along the fourth side.

Find the minimum total cost.

Round your answer to the nearest integer.

Comments

  • Let x be the width of the garden in feet (on the fenced side).

    Let 3000/x be the length of the garden in feet.

    The cost of surrounding the lot will be:

    $14 for one width

    $22 for two lengths and one width

    A function for the cost for a given width is:

    f(x) = 14x + 22(x + 2 * 3000/x)

    f(x) = 14x + 22x + 84000/x

    f(x) = 36x + 84000/x

    Take the derivative:

    f'(x) = 36 - 84000/x²

    Set that to zero:

    36 - 84000/x² = 0

    Multiply both sides by x²:

    36x² = 84000

    Divide both sides by 36:

    x² = 84000/36

    x² = 7000/3

    x = ±√(7000/3)

    We can ignore the negative value.

    x ≈ 60.553

    Plug that back in to f(x) and you'll have minimum cost.

    Answer:

    About $4,360

Sign In or Register to comment.