A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
What I have so far:
C = 1/2*pi*d, where d=x, or the width of the rectangle
P(rectangle) = 2y + x
So, the total perimeter:
P = 2y + x + 1/2*pi*x = 30ft
I'm not sure what to do next. How do I make this expression in terms of x?
Thank you
To solve the problem, you need to express the perimeter equation in terms of a single variable, such as x. Here's how you can proceed:
1. Start with the perimeter equation:
P = 2y + x + (1/2)πx = 30ft
2. Since the diameter of the semicircle is equal to the width of the rectangle (d = x), the radius of the semicircle would be (1/2)x.
3. Substitute the value of the radius in terms of x:
P = 2y + x + (1/2)πx = 30ft
P = 2y + x + (1/2)π(2y) = 30ft (substituting (1/2)x for y)
4. Simplify the expression by distributing π:
P = 2y + x + πy = 30ft
5. Combine like terms:
P = (2 + π)y + x = 30ft
Now, you have the perimeter equation expressed in terms of x and y. From here, you can proceed to solve for the dimensions of the window that admit the greatest possible amount of light.
To maximize the light admitted, you would want to maximize the area of the window. The area of the window can be expressed as the sum of the area of the rectangle and the semicircle:
Area = xy + (1/2)π(x/2)^2
To maximize the area, you can take the derivative of the area equation with respect to x, set it equal to zero to find critical points, and determine the dimensions that correspond to the maximum area.
Then, use those dimensions to calculate the actual values of x and y using the perimeter equation P = 30ft.
Alternatively, you can also use optimization techniques like the Method of Lagrange Multipliers to find the dimensions that maximize the area while satisfying the constraint of the given perimeter.
To make the expression in terms of x, we need to express the length y in terms of x using the given information about the shape of the window.
From the given information, we know that the diameter of the semicircle (which is equal to the width of the rectangle) is x. Therefore, the radius of the semicircle is half of the diameter, which is x/2.
The diagonal of the rectangle (which also corresponds to the height of the rectangle) is equal to the diameter of the semicircle, which is x. So, using the Pythagorean theorem, we can find the length y in terms of x:
y^2 + (x/2)^2 = x^2
Simplifying the equation:
y^2 + (x^2/4) = x^2
Multiplying through by 4 to eliminate the fraction:
4y^2 + x^2 = 4x^2
Rearranging the equation to express y^2 in terms of x:
4y^2 = 3x^2
Dividing through by 4:
y^2 = (3/4)x^2
Taking the square root of both sides to solve for y:
y = √(3/4)x
Now we have expressions for the lengths of the sides of the window, with the perimeter equation in terms of x:
P = 2y + x + (π/2)x = 30 ft
Substituting the expression for y:
P = 2(√(3/4)x) + x + (π/2)x = 30 ft
area = A = pi x^2/8 + x y
so
y = [A - pi x^2/8]/x
so
30 = 2 [A - pi x^2/8]/x + x + .5 pi x
30 x = 2 A - pi x^2/4 + x^2 + .5 pi x^2
(1+pi/4)x^2 - 30 x = -2 A
parabola, find vertex for max
1.785 x^2 - 30 x = -2 A
x^2 - 42 x = -1.12 A
complete square
x^2 -16.8 x + 70.6 = -1.12 A + 70.6
(x-8.4)^2= -1.12 A + 70.6
= -1.12 (A -63.3)
so x = 8.4 at vertex
and A = 63.3
so find y from
y = [A - pi x^2/8]/x
= [ 63.3 - pi 70.6/8] /8.4
=4.23
Check my arithmetic !
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