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 196 in². round your answers to the nearest whole number.
part 1 out of 2
what is the area of the window? use 3.14 for π. round your answer to the nearest whole number.
the area of the window is square in².

Explanation:

Step1: Find side length of square

Each square has area \( 196 \, \text{in}^2 \). Let side length be \( s \). Then \( s^2 = 196 \), so \( s = \sqrt{196} = 14 \, \text{in} \).

Step2: Analyze window shape

The window has a rectangular part (with 5 columns and 4 rows of squares) and a semicircular part (top).

• Rectangular area: Number of squares in rectangle is \( 5 \times 4 = 20 \). Area of rectangle: \( 20 \times 196 = 3920 \, \text{in}^2 \).
• Semicircular part: Diameter of semicircle is \( 5 \times 14 = 70 \, \text{in} \), so radius \( r = \frac{70}{2} = 35 \, \text{in} \). Area of semicircle: \( \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times 35^2 \).
• Calculate \( 35^2 = 1225 \), then \( \frac{1}{2} \times 3.14 \times 1225 = 1.57 \times 1225 = 1923.25 \, \text{in}^2 \).

Step3: Total window area

Add rectangular and semicircular areas: \( 3920 + 1923.25 = 5843.25 \), rounded to nearest whole number is \( 5843 \). Wait, wait—wait, maybe I misread the square count. Wait, the diagram: let's recheck. Wait, the bottom is formed by squares, and the top is a semicircle. Wait, maybe the number of squares in the rectangle: looking at the diagram, the rectangular part has 5 columns (width) and 4 rows (height) of squares? Wait, no—wait, each square is 14x14. Wait, maybe the rectangular part is 5 columns (each 14 in) and 4 rows (each 14 in), so length \( 5 \times 14 = 70 \), height \( 4 \times 14 = 56 \). Then rectangular area is \( 70 \times 56 = 3920 \) (same as before, since 5x4 squares: 514=70, 414=56, 7056=3920). Then semicircle: diameter is 70 (same as length of rectangle), so radius 35. Semicircle area: 0.53.1435²=0.53.141225=1923.25. Total area: 3920 + 1923.25 = 5843.25 ≈ 5843. Wait, but maybe the square count is different? Wait, the diagram: the shaded part (window) has a rectangle with 5 columns (x-axis) and 4 rows (y-axis) of squares, and a semicircle on top (which is like a mirror of the semicircular indent? Wait, no—the problem says "a semicircular shape at the top. The bottom is formed by squares of equal size". Wait, maybe I got the semicircle wrong. Wait, maybe the semicircle is on the top, and the bottom rectangular part: let's count the squares. The diagram shows a grid with 5 columns (vertical) and 4 rows (horizontal) for the rectangle, and a semicircle on the right? Wait, no, the diagram: the left part is a rectangle with 5 columns (each square) and 4 rows, and the right part (top) is a semicircle? Wait, no, the dashed line: maybe the window has a rectangular part (5 columns, 4 rows) and a semicircular part (which is a semicircle with diameter equal to the width of the rectangle). Wait, maybe the number of squares in the rectangle is 5 (width) 4 (height) = 20 squares, each 196 in², so 20196=3920. Then the semicircle: diameter is 514=70, radius 35, area 0.53.1435²=1923.25. Total 3920+1923.25=5843.25, round to 5843. But wait, maybe I messed up the square count. Wait, maybe the rectangular part is 5 columns and 4 rows, but let's check again. Alternatively, maybe the height of the rectangle is 4 squares (each 14 in), so height 56 in, width 514=70 in. Then rectangular area 7056=3920. Semicircle area: 0.5π35²=1923.25. Total 3920+1923.25=5843.25≈5843.

Answer:

\( 5843 \)