At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
A triangle whose sides are all equal is called an equilateral triangle and its area is given by A=[tex]\frac{\sqrt{3} }{4}[/tex] [tex]x^{2}[/tex], where x is the side of the triangle.
The area of a rectangle is found by using the formula
A=lw, where l is the length and w is the width.
The given figure can be split into two shapes: a rectangle and an equilateral triangle as shown in the diagram. Let l be the length of the rectangle.
Add all of the sides of the window and equate their result to 16. Then solve for l.
x + x + l + x + l = 16
3x + 2l = 16
2l = 16 - 3x
l = [tex]\frac{16-3x}{2}[/tex]
The total area of the window will be the sum of the area of the equilateral triangle ABE and the area of rectangle BCDE.
A = [tex]\frac{\sqrt{3} }{4}[/tex][tex]x^{2}[/tex] + lx
Subsitute [tex]\frac{16-3x}{2}[/tex] for l into the obtained equation and simplify.
A =[tex]\frac{\sqrt{3} }{4}[/tex][tex]x^{2}[/tex] + ( [tex]\frac{16-3x}{2}[/tex])x
A = [tex]\frac{\sqrt{3} }{4} x^{2}[/tex] + 8x - [tex]\frac{3}{2} x^{2}[/tex]
Differentiate the obtained equation with respect to x and equate the first derivative to 0 to calculate the critical point x .
[tex]\frac{dA}{dx}[/tex] = [tex]\frac{\sqrt{3} }{4}[/tex] (2x) + 8 - [tex]\frac{3}{2}[/tex] (2x)
0 = [tex]\frac{\sqrt{3} }{2}[/tex] x - 3x + 8
x (3 - [tex]\frac{\sqrt{3} }{2}[/tex]) = 8
x = [tex]\frac{16}{6-\sqrt{3} }[/tex]≈3.74887
Again, differentiate the equation
[tex]\frac{dA}{dx}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex]x - 3x + 8 with respect to x.
[tex]\frac{d^{2}A }{dx^{2} }[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] - 3
The value of the second derivative is [tex]\frac{\sqrt{3} }{2}[/tex]−3, which is negative. This implies that the area will be maximum at the critical point.
Subsitute x = 3.74887 into the equation
l = [tex]\frac{16-3x}{2}[/tex] and simplify
l = [tex]\frac{16-(3)(.74887)}{2}[/tex] ≈ 2.3767
The maximum light will enter through the window when the triangular portion will have the side length of 3.74887 feet and the dimensions of the rectangular portion will be 3.74887-ft-by-2.3767-ft.
Learn more about surmounted related word problems by:
https://brainly.com/question/27547781
#SPJ4
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
