Select all of the appropriate descriptions for the object below.СAT

Consider the system of differential equations dxdt=−4ydydt=−4x. Convert this system to a second order differential equation in y

koban [17]

$\dfrac{\mathrm dy}{\mathrm dt}=-4x\implies x=-\dfrac14\dfrac{\mathrm dy}{\mathrm dt}\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}$

Substituting this into the other ODE gives

$-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}=-4y\implies y''-16y=0$

Since $x(t)=-\dfrac14y'(t)$ , it follows that $x(0)=-\dfrac14y'(0)=4\implies y'(0)=-16$ . The ODE in $y$ has characteristic equation

$r^2-16=0$

with roots $r=\pm4$ , admitting the characteristic solution

$y_c=C_1e^{4t}+C_2e^{-4t}$

From the initial conditions we get

$y(0)=5\implies 5=C_1+C_2$

$y'(0)=16\implies-16=4C_1-4C_2$

$\implies C_1=\dfrac12,C_2=\dfrac92$

So we have

$\boxed{y(t)=\dfrac12e^{4t}+\dfrac92e^{-4t}}$

Take the derivative and multiply it by -1/4 to get the solution for $x(t)$ :

$-\dfrac14y'(t)=\boxed{x(t)=-\dfrac12e^{4t}+\dfrac92e^{-4t}}$

7 0

4 years ago

Mr castle shares out 64 marbles between emily and asif in the ratio 7:1 how many marbles does each child get

Novay_Z [31]

Answer:

<h2>Emily: 72 marbles</h2><h2>Asif: 8 marbles</h2>

<em>Step-by-step explanation:</em>

first lets find in how many shares is the total amount of marbles divided into:

7+1=8

now, lets find how many marbles are in each share:

64/8=8

now that we know that each share of marbles contains 8 marbles, we can find how many each child got:

Emily got 7 shares: 8*7= 72 marbles

Asif got 1 share: 8*1= 8 marbles

Hence,

<h3><em>Emily got 72 marbles, while Asif got 8 marbles</em></h3>

7 0

3 years ago

Clearing fractions<br><br> SOLVE

lana66690 [7]

You can multiply by -30 to clear fractions.
.. 47 ≤ 10x +60

Now, subtract 60 and divide by 10
.. -13 ≤ 10x
.. -1.3 ≤ x

4 0

4 years ago

HELP!!!!!!!!! I DONT UNDERSTAND THIS

ValentinkaMS [17]

Answer:

$x=86^{\circ}$

$y=94^{\circ}$

$z=54^{\circ}$

Step-by-step explanation:

First, let's look at the largest triangle (two smaller triangles are combined) to solve for $z$ . Since the sum of the angles in a triangle adds up to $180^{\circ}$ , we can write the equation:

$20+74+32+z=180$

$z=180-126$

$z=54^{\circ}$

Looking at the smaller triangle on the left, $y$ (exterior angle) is the sum of the two opposite interior angles of the triangle on the left:

$y=74+20$

$=94^{\circ}$

Since $x$ is the exterior angle of the triangle on the right, it is equivalent to the sum of the opposite interior angles of that triangle:

$x=32+z$

$=32+54$

$=86^{\circ}$

Hope this helps :)

6 0

3 years ago

A rectangle has a a perimeter of 72 ft. The length and width are scaled by a factor 3.5.

jarptica [38.1K]

Formula for perimeter of rectangle:
P=2(L+w)
P=2L+2w
Here Perimeter, p= 72 ft
And scale factor of length and width=3.5 which means rectangle is enlarged.
So perimeter of rectangle= 3.5 x 72
P=252ft

Answer: 252 ft

8 0

3 years ago

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Select all of the appropriate descriptions for the object below.СAT​

Select all of the appropriate descriptions for the object below.СAT