Question

(b) if the original square had a side length of 20 inches, find its area and the area of the new rectangle. how many square inches larger is the squares area?
width of rectangle: 18 in
length of rectangle: 25 in
area of square: 400 in
area of rectangle: 375 in
the squares area is 25 square inches larger than the area of the rectangle.
(c) if the square had a side length of x inches, show that its area will always be four square inches more than the area of the new rectangle
width of rectangle: in
length of rectangle: $x-2$ in

Explanation:

Part (b)

Step1: Calculate square area

Area = side length squared.
$A_{square} = 20^2 = 400$ square inches

Step2: Calculate rectangle area

Area = width × length.
$A_{rectangle} = 15 \times 25 = 375$ square inches

Step3: Find area difference

Subtract rectangle area from square area.
$400 - 375 = 25$ square inches

Step1: Define square area

Area = side length squared.
$A_{square} = x^2$

Step2: Define rectangle area

Area = width × length.
$A_{rectangle} = (x-2)(x+2)$

Step3: Expand rectangle area

Use difference of squares: $(a-b)(a+b)=a^2-b^2$.
$(x-2)(x+2) = x^2 - 4$

Step4: Find area difference

Subtract rectangle area from square area.
$x^2 - (x^2 - 4) = 4$

Answer:

Area of the square: 400 square inches
Area of the rectangle: 375 square inches
The square's area is 25 square inches larger than the rectangle's area.

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Part (c)