Mathematics

Question

The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, shown by the dashed line around the window. Each square in the window has an area of 100 i n 2 . What is the area of the shade? Round your answer to the nearest whole number.

2 Answer

  • Area of the window:
    A ( window ) = 16  · 100 + 1/2 · 20² π = 1600 + 200 π = 2,228
    A ( window + shade ) = 48  · 40 + 1/2 · 24² π = 2,824.32
    A ( shade ) = 2,824.32 - 2,228 = 596.32 ≈ 596 in²
    Answer:
    Area of the shade is 596 in².
  • Answer:

    The total area is 3016 square inches, approximately.

    Step-by-step explanation:

    The complete problem is attached.

    The area of the window is separated in two parts, the semicircle and the square.

    We know that each square has an area of 100 square inches, which means that the whole big square has an area of 1600 square inches, because there are 16 total squares.

    Also, each side of each small square is [tex](100)^{2}[/tex] long, which is 10 inches. This means the diameter of the semicircle is 40 inches, and its area is

    [tex]A_{semicircle} =\frac{\pi r^{2} }{2}=\frac{3.14(20)^{2}}{2} = 628 in^{2}[/tex]

    Therefore, the area of the whole window is

    [tex]A_{window}=1600+628 = 2228 in^{2}[/tex]

    Now, the shade is 4 inches beyond the perimeter, that means each side the new rectangle is

    [tex]b=40+4+4=48\\h=40+4=44[/tex]

    Similarly, the radius of the new semicircle is

    [tex]r=20+4=24[/tex]

    Then, we just have to find the new combined area.

    [tex]A_{rectangle}=48(44)=2,112 in^{2} \\A_{new \ semicircle}=\frac{3.14(24)^{2} }{2}=904.32[/tex]

    So, the total area of the shade is

    [tex]A_{shade}=2112+904.32 = 3016 in^{2}[/tex]

    Therefore, the total area is 3016 square inches, approximately.

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