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Cloud [144]2 years ago

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If you are asking if your work is good,thn yes,you are right

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A Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the outer perimeter of such a window must

Feliz [49]

The base length that will maximize the area for such a window is 168.03 cm. The exact largest value of x when this occurs is 233.39 cm

Suppose we make an assumption that:

(x) should be the width of the rectangle base;
(h) should be the height of the rectangle

Also, provided that the diameter of the semi-circle appears to be the base of the rectangle, then;

the radius $\mathbf{r = \dfrac{x}{2}}$

and, the perimeter of the window can now be expressed as:

$\mathbf{x + 2h + \pi r = x + 2h + \dfrac{\pi x }{2}}$

$\mathbf{= \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h}$

Given that the perimeter = 600 cm

∴

$\mathbf{ \Big ( 1 + \dfrac{\pi}{2}\Big) x + 2h= 600}$

$\mathbf{ h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x}$

Since h > 0, then:

$\mathbf{ h = 300 - \Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) x>0}$

By rearrangement and using the inverse rule:

$\mathbf{ x< \dfrac{ 300}{\Big( \dfrac{1}{2} + \dfrac{\pi}{4}\Big) } }$

$\mathbf{ x= \dfrac{ 1200}{\Big( 2 +\pi \Big) } }$

$\mathbf{ x= 233.39 \ cm }$

Thus, the largest length x = 233.39 cm

However, the area of the window is given as:

$\mathbf{A(x) = xh + \dfrac{1}{2} \pi r^2}$

$\mathbf{A = x \Big [ 300 - \Big ( \dfrac{1}{2}+\dfrac{1}{4} \Big) x \Big ] +\dfrac{1}{2}\pi \Big(\dfrac{x}{2} \Big )^2}$

$\mathbf{A (x) = 300x - \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ cm^2}$

Now, at maximum, when the area A = 0. Taking the differentiation, we have:

$\mathbf{\dfrac{d}{dx} 300x - \dfrac{d}{dx} \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) x^2 \ =0}$

$\mathbf{ 300 - 2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \ =0}$

Making x the subject of the formula, we have:

$\mathbf{x = \dfrac{1200}{4 +\pi}}$

x = 168.03 cm

Taking the second derivative:

$\mathbf{\dfrac{d}{dx} \Big [300 -2x \Big( \dfrac{1}{2} + \dfrac{\pi}{8}\Big) \Big]}$

$\mathbf{= -2 \Big( \dfrac{1}{2}+\dfrac{\pi}{8}\Big )$

Therefore, we can conclude that the maximum area that exists for such a window is 168.03 cm

Learn more about derivative here:

brainly.com/question/9964510?referrer=searchResults

6 0

2 years ago

Gelneren [198K]

Step-by-step explanation:

8.9x-5 and -6.8x+8

8.9-6.8 + -5+8

2.1x+3

4 0

2 years ago

Two similar triangles have have areas of 75 m2 and 12 m2. find the ratio of the perimeters

soldi70 [24.7K]

we know that having similar triangles so the ratio of sides,area and volume are same.

$s/s=s^2/s^2$

we know that $s^2$ will be the area of triangle.

$s^2/s^2=75/12$

so $s/s=\sqrt{75/12}$

so the ratio of the perimeter of two triangles will be

$(s+s+s)/(s+s+s)=(\sqrt{75} +\sqrt{75} +\sqrt{75} )/(\sqrt{12}+\sqrt{12} +\sqrt{12} )$

ratio of the perimeter= $\sqrt{225} /\sqrt{36}$

8 0

2 years ago

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If m = 3 and b = 6, what is the correct equation for Line of Best Fit?

Alchen [17]

Answer:

B

Step-by-step explanation:

The equation is Y = mx + b so just plug in the m and the b

6 0

3 years ago

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If x=100 what is x squared divided by 2

posledela

100 divided by 2 equals to 50

7 0

3 years ago

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