The value of x that would make the area of the Norman window at least 1920 square inches is approximately 48.6.
The area of the Norman window is given by the sum of the area of the rectangle and the semicircle.
Let the length of the rectangle = x.
The area of the rectangle = x(10 - x)
The area of the semicircle = (1/2)π(x/2)^2.
Therefore, the area of the Norman window is A = x(10 - x) + (1/2)π(x/2)^2
Simplifying this expression:
A = -x^2 + 10x + (1/8)πx^2
To find the values of x that would make the area of the Norman window at least 1920 square inches, we can set up the following inequality:
-x^2 + 10x + (1/8)πx^2 ≥ 1920
Multiplying both sides by -8/π:
(8/π)x^2 - (80/π)x - 15360/π ≥ 0
Using the quadratic formula to solve for x:
x = [80/π ± sqrt((80/π)^2 + 4 * (8/π) * (15360/π))] / (2 * 8/π)
Simplifying this expression:
x = [40/π ± sqrt(1600/π^2 + 61440/π^2)] / (1/π)
x = [40/π ± sqrt(63040/π^2)] / (1/π)
x = [40/π ± 8√(98)/π] / (1/π)
x = 40 ± 8√(98)
Therefore, the values of x that would make the area of the Norman window at least 1920 square inches approximate to:
x ≈ -3.6 or 48.6
Since x represents the length of the rectangle, its value must be positive.
Therefore, the value of x that would make the area of the Norman window at least 1920 square inches is approximately:
x ≈ 48.6
Thus, we can conclude that the value of x is approximately 48.6.