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Please EXPLAIN the steps on how to solve these problems. Please provide detailed steps.

A rectangle picture is 12 by 16 inches. If a frame of uniform width contains an area of 165 square inches, what is the width of the frame?

The length of a rectangular garden is 6 feet more than its width. A walkway is 3 feet wide surrounds the outside of the garen. The total area of the walkway is 288 square feet. Find the dimensions of the garden.

  • Trig -

    The first problem:
    Sketch this problem to make it easier to visualize. There is a rectangular frame with the inside dimensions of 12 * 16 inches. The frame has a width of W. You need to divide the frame into pieces and then write an equation for those pieces to equal 165 in^2. One approach would be to extend the lines of the picture to form 4 rectangles of the frame along the edges and 4 squares at the corners.

    W
    -------------------------
    | | | | W
    -------------------------
    | | | |
    | | 12 | |
    | | | |
    | | 16 | |
    -------------------------
    | | | | W
    -------------------------
    W

    There are 2 rectangles that are W * 16.
    There are 2 rectangles that are W * 12.
    There are 4 corner squares that are W * W.
    Add these 8 items and set the total to 165. Then solve for W. You will need to use the quadratic formula.

  • Trig -

    Sorry, the diagram didn't survive the font change.

  • Trig -

    For the rectangle problem, let w be the frame width. 165 in^2 is considered to be the frame area, not including the picture hole in the middle.
    165 + (12*16) = (12+2w)(16+2w)
    = 192 + 56w + 4w^2
    4w^2 + 56w -165 = 0
    (2w -5)(2w+33) = 0
    w = 5/2 inch (ignore the negative root)

    Use a similar approach for the second problem. Someone will be glad to critique your work.

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