Answer:
a) A = 64(2s - 1)
b) A = 1,856 cm²
Explanation:
Given the side of the frame, 8s, and that the border that frames the picture is 4 on either side = 8:
a) Find an expression for the area of the frame, in factored form.
A = area of the square frame
A₁ = Total area (including the picture inside the frame) = (8s)²
A₂ = Area of the picture inside the frame = 4 + 4 = (8s - 8)²
To find the area of the frame, subtract the area of the picture from the total area.
A = A₁ - A₂
A = (8s)² - (8s - 8)²
Perform the necessary exponential operations:
A = 64s²- (64s² - 64s - 64s + 64)
A = 64s²- (64s² - 128s + 64)
A = 64s²- (64s² - 128s + 64)
Distribute -1 into the parenthesis:
A = 64s²- 64s² + 128s - 64
Combine like terms:
A = 0 + 128s - 64
A = 128s - 64
Factor out 64:
A = 64(2s - 1)
The expression for the area of the frame in factored form is: A = 64(2s - 1).
b) Determine the area of the frame when s = 15cm
Using the same expression from part A:
A = 64(2s - 1)
Substitute s = 15 into the expression:
A = 64[2(15) - 1]
A = 64(30 - 1)
A = 64(29)
A = 1,856 cm²
Therefore, the area of the frame that has a side of 15 cm is: A = 1,856 cm²