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konstantin123 [22]
3 years ago
9

12. Find the radius of a circle with a circumference of 457 centimeters.

Mathematics
1 answer:
irga5000 [103]3 years ago
6 0

Step-by-step explanation:

Circumference=d\pi\\Circumference=2r\pi

Since we need to find the radius, then we will rearrange the second formula.

Radius=\frac{C}{2\pi }

Radius=\frac{457}{2\pi }

Radius=\frac{457}{6.28}

Radius= 72.8 (rounded)

So, the radius of a circle that has a circumference of 457 is about 72.8 cm

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The equations that must be solved for maximum or minimum values of a differentiable function w=​f(x,y,z) subject to two constrai
sammy [17]

The Lagrangian is

L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(4x^2+4y^2-z^2)+\mu(2x+4z-2)

with partial derivatives (set equal to 0)

L_x=2x+8\lambda x+2\mu=0\implies x(1+4\lambda)+\mu=0

L_y=2y+8\lambda y=0\implies y(1+4\lambda)=0

L_z=2z-2\lambda z+4\mu=0\implies z(1-\lambda)+2\mu=0

L_\lambda=4x^2+4y^2-z^2=0

L_\mu=2x+4z-2=0\implies x+2z=1

Case 1: If y=0, then

4x^2-z^2=0\implies4x^2=z^2\implies2|x|=|z|

Then

x+2z=1\implies x=1-2z\implies2|1-2z|=|z|\implies z=\dfrac25\text{ or }z=\dfrac23

\implies x=\dfrac15\text{ or }x=-\dfrac13

So we have two critical points, \left(\dfrac15,0,\dfrac25\right) and \left(-\dfrac13,0,\dfrac23\right)

Case 2: If \lambda=-\dfrac14, then in the first equation we get

x(1+4\lambda)+\mu=\mu=0

and from the third equation,

z(1-\lambda)+2\mu=\dfrac54z=0\implies z=0

Then

x+2z=1\implies x=1

4x^2+4y^2-z^2=0\implies1+y^2=0

but there are no real solutions for y, so this case yields no additional critical points.

So at the two critical points we've found, we get extreme values of

f\left(\dfrac15,0,\dfrac25\right)=\dfrac15 (min)

and

f\left(-\dfrac13,0,\dfrac23\right)=\dfrac59 (max)

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180(n-2)=s

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Divide each side by 180

180n/180 = s/180 + 360/180

n = s/180 +2

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