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3 years ago

Enter the missing numbers in the box to complete the table of equivalent ratios of length to widths. PLEASE HELP

Mathematics

1 answer:

Goryan [66]3 years ago

8 0

All you do is multiply the left column to get the right column
or
to divide the right column to get the left column.

So you want to go from left to right in the first row. 6*4 = 24
Second Row is complete.
Third row is Right to left so you would divide by 4 which will give 12
Fourth row multiply by 4 ; 4 * 15 = 60

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Find the extreme values of f subject to both constraints f(x,y) = 2x^2+3y^2-4x-5, x^2 + y^2 <=16

Softa [21]

$f(x,y)=2x^2+3y^2-4x-5$
$f_x=4x-4=0\implies x=1$
$f_y=6y=0\implies y=0$

$f(x,y)$ has only one critical point at $(x,y)=(1,0)$ . The function has Hessian

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}4&0\\0&6\end{bmatrix}$

which is positive definite for all $(x,y)$ , which means $f(x,y)$ attains a minimum at the critical point with a value of $f(1,0)=-7$ .

To find the extrema (if any) along the boundary, parameterize it by $x=4\cos t$ and $y=4\sin t$ , with $0\le t$ . On the boundary, we have

$f(x(t),y(t))=F(t)=2(4\cos t)^2+3(4\sin t)^2-4(4\cos t)-5=32\cos^2t+48\sin^2t-16\cos t-5$
$F(t)=35-16\cos t-8\cos2t$

Find the critical points along the boundary:

$F'(t)=16\sin t+16\sin2t=16\sin t+32\sin t\cos t=16\sin t(1+2\cos t)=0$
$\implies t=0,\dfrac{2\pi}3,\pi,\dfrac{4\pi}3$

Respectively, plugging these values into $F(t)$ gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when $F(t)=47$ .

Now, solve for $x,y$ for both cases:

$t=\dfrac{2\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=2\sqrt3\end{cases}$

$t=\dfrac{4\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=-2\sqrt3\end{cases}$

so $f(x,y)$ has two absolute maxima at $(x,y)=(-2,\pm2\sqrt3)$ with the same value of 47.

5 0

3 years ago

The product of two integers is 84. If one integer is (-12), find the other.

Papessa [141]

Answer:

-7

Step-by-step explanation:

-12x = 84

/-12 /-12

x = -7

8 0

3 years ago

Read 2 more answers

Help me please i need answer

marishachu [46]

Answer: 57.75 m

Explanation: First do 5 times 0.25. (The 0.25 is the same as 1/4.) Now take that answer and multiply it by 11. Your equation should look like this: 11*(5 1/4).

* = times

I hope this helps! :)

5 0

3 years ago

Read 2 more answers

Describe all equation of motion

wel

Answer:

First equation

from definition of acceleration:

$a = \frac{change \: in \: velocity}{time} \\ \\ a = \frac{v - u}{t} \\ \\ at = v - u\\ { \boxed{v = u + at}}$

Second equation

from displacement:

$s = (average \: velocity) \times time \\ s = ( \frac{v + u}{2} ) \times t \\ \\ 2s = (v + u)t$

but v is u + at :

$2s = \{(u + at) + u \} \times t \\ \\ 2s = (2u + at)t \\ 2s = 2ut + {at}^{2} \\ s = \frac{2ut}{2} + \frac{a {t}^{2} }{2} \\ \\ { \boxed{s = ut + \frac{1}{2} {at}^{2} }}$