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3 years ago

3y-6 divided by 9 = 4-2 divided by -3

Mathematics

2 answers:

antoniya [11.8K]3 years ago

4 0

I believe the answer is 16/9

Sergeeva-Olga [200]3 years ago

4 0

Y=4/3 hope this helps

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A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 56 ft

harkovskaia [24]

Answer:

Dimensions of the window in order to allow maximum light is $x=\dfrac{56}{4+\pi}$ and $y=\dfrac{56}{4+\pi}$

Step-by-step explanation:

Consider following Norman window, assuming ABCD as rectangle and arc AD as semicircle with center at E and radius r. (Refer attachment)

Given that perimeter of window is 56 ft. Therefore perimeter of window is given as,

$Perimerter = AB + BC + CD + arc\:AD$

Calculate arc AD as follows,

Let, x denote radius of semi circle. That is, r=x

Since AD is the diameter of semi circle

So AD = 2 r = 2 x.

Now perimeter of semicircle is equal to circumference of semicircle, so calculate circumference of semicircle as follows

Circumference of circle is $C=2\pi r$ . So half of it will be

$C=\dfrac{2\pi r}{2}$

$C=\dfrac{2\pi x}{2}$

$C=\pi x$

So, $arc\:AD = \pi x$

Calculate AB , BC and AD as follows,

Consider rectangle ABCD,

Since, AD is the diameter of semi circle which is also one of the side of the rectangle.

So AD = 2 r = 2 x.

Since AD is parallel to BC. Therefore, AD=BC=2x. Also length of rectangle be y

$\therefore BC=2x,AB=CD=y$

Substituting the value,

Perimeter = AB + BC + CD + arc AD

$56=y+2x+y+\pi x$

$56=2y+2x+\pi x$

To calculate value of y,subtracting 2x and $\pi x$ on both sides,

$56-2x-\pi x=2y$

Dividing by 2,

$28-x-\dfrac{\pi x}{2}=y$

Now calculate area of window.

Area = Area of rectangle + Area of semicircle

From diagram,

$Area = width\times length + \dfrac{1}{2}\times\pi r^{2}$

$Area = 2x\times y + \dfrac{1}{2}\times\pi x^{2}$

$Area = 2xy + \dfrac{\pi x^{2}}{2}$

Substituting value of y in above equation,

$Area = 2x\left (28-x-\dfrac{\pi x}{2} \right ) + \dfrac{\pi x^{2}}{2}$

Simplifying,

$Area = 56x-2x^{2}-\pi x^{2}+ \dfrac{\pi x^{2}}{2}$

$Area = 56x-2x^{2}- \dfrac{\pi x^{2}}{2}$

In order to find the maximum of area function, differentiate the equation with respect to x and find the critical points.

Applying difference rule of derivative,

$\dfrac{dA}{dx}=\dfrac{d}{dx}\left(56x\right)-\dfrac{d}{dx}\left ( 2x^{2} \right )-\dfrac{d}{dx}\left ( \dfrac{\pi x^{2}}{2} \right )$

Applying constant multiple rule of derivative,

$\dfrac{dA}{dx}=56\dfrac{d}{dx}\left(x\right)-2\dfrac{d}{dx}\left ( x^{2} \right )-\dfrac{\pi}{2}\dfrac{d}{dx}\left (x^{2}\right )$

Applying power rule of derivative,

$\dfrac{dA}{dx}=56\left(1x^{1-1}\right)-2\left(2x^{2-1}\right)- \dfrac{\pi}{2}\dfrac{d}{dx}\left (2x^{2-1}\right )$

$\dfrac{dA}{dx}=56\left(1\right)-2\left(2x\right)- \dfrac{\pi}{2}\dfrac{d}{dx}\left (2x\right )$

$\dfrac{dA}{dx}=56-4x-\pi x$

Now find the critical number by solving as follows,

$\dfrac{dA}{dx}=0$

$56-\left(4+\pi\right)x =0$

$56=\left(4+\pi \right)x$

$\dfrac{56}{4+\pi} =x$

Since there is only one critical point, directly substitute the value of x into equation of A. If value of A is greater than 0, then the area is maximum at critical point.

$Area = 56\left (\dfrac{56}{4+\pi} \right )-2\left (\dfrac{56}{4+\pi} \right )^{2}- \dfrac{\pi \left (\dfrac{56}{4+\pi} \right )^{2}}{2}$

Calculating the above expression,

$Area = \dfrac{1568}{4+\pi }$

So area is greater than 0.

Now calculate value of y,

$28-\dfrac{56}{4+\pi}-\dfrac{\pi}{2}\left ( \dfrac{56}{4+\pi} \right )=y$

$\dfrac{56}{4+\pi}=y$

Hence dimensions are $x=\dfrac{56}{4+\pi}$ and $y=\dfrac{56}{4+\pi}$

3 0

4 years ago

The winner of a raffle will receive 3/4 of the $530.40 raised from raffle ticket sales.How much money will the winner get?

Setler79 [48]

$money\ for\ winner\ =\frac{3}{4}*raffle\ ticket\ sales\\\\
money\ for\ winner=\frac{3}{4}*530.40\\\\
money\ for\ winner=\frac{3}{4}*530.40=\frac{1591.2}{4}=397,8$\\\\
Winner\ will\ get\ 397.8\$.

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