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Setler [38]
3 years ago
13

Use Lagrange multipliers to find the maximum and minimum values of

Mathematics
1 answer:
marusya05 [52]3 years ago
5 0

The Lagrangian is

L(x,y,\lambda)=x+8y+\lambda(x^2+y^2-4)

It has critical points where the first order derivatives vanish:

L_x=1+2\lambda x=0\implies\lambda=-\dfrac1{2x}

L_y=8+2\lambda y=0\implies\lambda=-\dfrac4y

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

From the first two equations we get

-\dfrac1{2x}=-\dfrac4y\implies y=8x

Then

x^2+y^2=65x^2=4\implies x=\pm\dfrac2{\sqrt{65}}\implies y=\pm\dfrac{16}{\sqrt{65}}

At these critical points, we have

f\left(\dfrac2{\sqrt{65}},\dfrac{16}{\sqrt{65}}\right)=2\sqrt{65}\approx16.125 (maximum)

f\left(-\dfrac2{\sqrt{65}},-\dfrac{16}{\sqrt{65}}\right)=-2\sqrt{65}\approx-16.125 (minimum)

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He counted 206 wheels and 170 pedals. How many bicycles and tricycles does he have.
noname [10]
Bicycle has 2 wheels and 2 pedals
 tricycle has 3 wheels and 2 pedals

EQ 1 : 2b + 2t = 170 pedals

 EQ 2: 2b +3t = 206 wheels

subtract  EQ 1 from EQ 2

2b +3t = 206 - 2b + 2t = 170 = t=36

 there were 36 tricycles

36 x 2 = 72 pedals

170 pedals - 72 = 98 pedals left
98/2 = 49 bicycles

36 Tricycles and 49 bicycles




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lozanna [386]
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irga5000 [103]

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Step-by-step explanation:

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Sin 70°cos 10°-cos 70°sin 10=√3/2
hoa [83]
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matrenka [14]

Answer:

The length of the ramp must be at least 14.71 feet

Step-by-step explanation:

Given

BC = 2.3ft

\theta = 9^\circ

<em>See attachment for complete figure</em>

Required

Length of the ramp (z)

To do this, we consider sin of angle A (\theta).

So, we have:

sin\theta = \frac{Opposite}{Hypotenuse}

This gives:

sin(9)= \frac{2.3}{z}

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