Respuesta :
Answer:
43.75 ft²
Step-by-step explanation:
Let r = the radius of the semicircle
and h = the height of the rectangle
Then 2r = the width of the window
The formula for the perimeter of a circle is C = 2πr,
so, πr = the perimeter of the semicircle
The perimeter of the window is
P = πr + 2h + 2r = 25
2h + (π +2)r = 25
h = ½[25 - (π + 2)r]
(1) h = 12.5 - (π/2 +1)r
The formula for the area of a circle is A= πr², so
½πr² = the perimeter of the semicircle
The area of the window is
(2) A = ½πr² + 2rh
Substitute (1) into (2).
A = ½πr² + 2r[12.5 - (π/2 +1)r] = ½πr² + 25r - (π +2)r²
A = 25r - (π + 2 - π/2)r²
(3) A = -(π/2 + 2)r² + 25r
This is the equation for a downward opening parabola.
One way to find the vertex is to set the first derivative equal to zero.
dA/dr = -2(π/2 + 2)r + 25 = 0
-(π + 4)r + 25 = 0
-(π + 4)r = -25
r = 25/(π + 4)
(4) r ≈ 3.50 ft
The maximum area occurs when r = 3.50 ft.
Substitute (4) into (1).
h = 12.5 - (π/2 +1)(3.50) = 12.5 - (2.571× 3.50) = 12.5 - 9.00 = 3.50
(4) h = 3.50 ft
Substitute (4) into (2)
A = 1.571(3.50)² + 2×3.50×3.50 = 19.25 + 24.50
A = 43.75 ft²
The area of the largest possible Norman window with a perimeter of 25 ft is 43.75 ft².
The area of the largest possible Norman window with a perimeter of 25 ft is 43.75 ft².
What is the maximum area?
If r and h are the radius of the semicircle and height of the rectangle respectively, it means;
2r is the width of the window
Formula for the perimeter of a semi circle is; C = πr
Thus the perimeter of the window is expressed as;
P = πr + 2h + 2r = 25
making h the subject gives;
h = 12.5 - (¹/₂π +1)r ------(1)
Formula for the area of the semi-circle is; A = ½πr²
The area of the window is;
A = ½πr² + 2rh -------(2)
Putting the expression of h for h in eq 2 gives;
A = ½πr² + 2r[12.5 - (¹/₂π +1)r]
A = ½πr² + 25r - (π +2)r²
A = 25r - (π + 2 - π/2)r²
A = -(¹/₂π + 2)r² + 25r
Let us find the vertex by finding the first derivative and equating to zero.
dA/dr = -2(¹/₂π + 2)r + 25 = 0
dA/dr = -(π + 4)r + 25 = 0
dA/dr = -(π + 4)r
Since Perimeter = dA/dr, then;
-(π + 4)r = -25
solving for r gives;
r ≈ 3.50 ft
The maximum area will occur when r = 3.50 ft.
Putting 3.5 for r in eq 1 gives;
h = 12.5 - (¹/₂π + 1)(3.50) = 12.5 - (2.571 × 3.50)
h = 12.5 - 9.00
h = 3.50 ft
put r = 3.5 and h = 3.5 in eq 2 to get;
A = 1.571(3.50)² + 2×3.50×3.50 = 19.25 + 24.50
A = 43.75 ft²
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