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3 years ago

Janna drank 0.7 liter of water before soccer pratice and 0.70 liter of water after practice compare using 2decimal < ,=.>

Mathematics

1 answer:

Ahat [919]3 years ago

8 0

=======its equal cause the zeros only count when it before a number for example 0.305 or 0.052==============

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There are 5 types of candy sold in a store, at least 6 candies of each type. in how many ways can one buy n candies so that no t

Brrunno [24]

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3 years ago

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

Mai is visiting Paris to see the Eiffel Tower. She is 80 feet away when she spots it. To see the top, she has to look up at an a

cluponka [151]

The picture shows how this would look as a triangle.
In this case, we would use tan. x is the opposite side, and 80 is adjacent.
tan(85.7) = x/80
13.299.. = x/80
x = 1,064 feet