Suppose a tank in Oklahoma City leaks toxic chemicals. The area affected is a circle centered on the tank. When the spill is first noticed, the circle has a radius of 50 feet. A response team arrives 80 minutes later, by which time the affected area has grown to a circle with a radius of 70 feet.

Required:
How rapidly was the radius of the spill changing in ft/min between the time the spill was noticed and the arrival of the response team?

Answer:

Hence, the [tex]y(t)=e^{-2 t}\left(\frac{75}{74} \cos 6 t+\frac{75}{148} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t))\end{array}[/tex] and approximately value of [tex]y(t)[/tex] is [tex]-0.844[/tex].

Given :

 [tex]my''+cy'+ky=F(t), y(0)=0, y'(0)=0,[/tex]

Where  [tex]m=2[/tex] kilograms

           [tex]c=8[/tex] kilograms per second

           [tex]k=80[/tex] Newtons per meter

          [tex]F(t)=20\sin (6t)[/tex] Newtons

Explanation :

(1)

Solve the initial value problem. [tex]y(t)[/tex]

   [tex]my''+cy'+ky=F(t), y(0)=0, y'(0)=0,[/tex]

[tex]\Rightarrow 2y''+8y'+80y=20\sin (6t)[/tex]

[tex]\Rightarrow y''+4y'+40y=10\sin (6t)[/tex]

Auxilary equations :[tex]F(t)=0[/tex]

[tex]\Rightarrow r^2+4r+40=0[/tex]

[tex]\Rightarrow r=\frac{-4\pm\sqrt{4^2-4\times 1\times 40}}{2\times 1}[/tex]

[tex]\Rightarrow r=\frac{-4\pm\sqrt{16-160}}{2}[/tex]

[tex]\Rightarrow r=\frac{-4\pm\sqrt{-144}}{2}[/tex]

[tex]\Rightarrow r=\frac{-4\pm12i}{2}[/tex]

[tex]\Rightarrow r=-2\pm6i[/tex]

The complementary solution is [tex]y_c=e^{-2t}\left(c_1\cos 6t+c_2\sin 6t\right)[/tex]

The particular Integral, [tex]y_p=\frac{1}{f(D)}F(t)[/tex]

[tex]y_{y} &=\frac{1}{D^{2}+4 D+40} 25 \sin (6 t) \\\\ y_{y} &=\frac{25}{-6^{2}+4 D+40} \sin (6 t) \quad\left(D^{2} \text { is replaced with }-6^{2}=-36\right) \\\\y_{y} &=\frac{25}{4 D+4} \sin (6 t) \\\\y_{y} &=\frac{25}{4(D+1)} \sin (6 t) \\\\y_{y} &=\frac{25(D-1)}{4(D+1)(D-1)} \sin (6 t) \\\\y_{y} &=\frac{25(D-1)}{4\left(D^{2}-1\right)} \sin (6 t) \\\\y_{y} &=\frac{25(D-1)}{4(-36-1)} \sin (6 t) \\\\y_{y} &=-\frac{25}{148}(D-1) \sin (6 t) \\y_{y} &=-\frac{25}{148}\left(\frac{d}{d t} \sin (6 t)-\sin (6[/tex]

Hence the general solution is :[tex]y=y_c+y_p=e^{-2t}(c_1\cos 6t+c_2\sin 6t)-\frac{25}{148}(6\cos 6t-\sin 6t)[/tex]

Now we use given initial condition.

[tex]y(t) &=e^{-2 t}\left(c_{1} \cos 6 t+c_{2} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t)) \\\\\y(0) &=e^{-\alpha 0)}\left(c_{1} \cos (0)+c_{2} \sin (0)\right)-\frac{25}{148}(6 \cos (0)-\sin (0)) \\\\0 &=\left(c_{1}\right)-\frac{25}{148}(6) \\\\c_{1} &=\frac{75}{74} \\\\y(t) &=e^{-2 t}\left(c_{1} \cos 6 t+c_{2} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t)) \\\\[/tex]

[tex]y^{\prime}(t)=-2 e^{-2 t}\left(c_{1} \cos 6 t+c_{2} \sin 6 t\right)+e^{-2 t}\left(-6 c_{1} \sin 6 t+6 c_{2} \cos 6 t\right)-\frac{25}{148}(-36 \sin (6 t)-6 \cos (6 t)) \\\\y^{\prime}(0)=-2 e^{0}\left(c_{1} \cos 0+c_{2} \sin 0\right)+e^{0}\left(-6 c_{1} \sin 0+6 c_{2} \cos 0\right)-\frac{25}{148}(-36 \sin 0-6 \cos 0) \\\\0=-2\left(c_{1}\right)+\left(6 c_{2}\right)-\frac{25}{148}(-6) \\\\0=-2 c_{1}+6 c_{2}+\frac{75}{74} \\\\0=-2\left(\frac{75}{74}\right)+6 c_{2}+\frac{75}{74} \\\\[/tex][tex]\begin{array}{l}0=-\frac{150}{74}+6 c_{2}+\frac{75}{74} \\\\\frac{150}{74}-\frac{75}{74}=6 c_{2}\end{array}[/tex]

[tex]\begin{array}{l}c_{2}=\frac{25}{148}\\\\\text { Substitute } c_{1} \text { and } c_{2} \text { in } y(t) \text { . Then }\\\\y(t)=e^{-2 t}\left(\frac{75}{74} \cos 6 t+\frac{75}{148} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t))\end{array}[/tex]

(2)

[tex]y(t)=e^{-2 t}\left(\frac{75}{74} \cos 6 t+\frac{75}{148} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t)) \\\\y(t)=\left(\frac{75}{74} e^{-2 t} \cos 6 t+\frac{75}{148} e^{-2 t} \sin 6 t\right)-\frac{25}{148}(6 \cos (6 t)-\sin (6 t)) \\\\y(t)=\frac{75}{74}\left(e^{-2 t}-1\right) \cos 6 t+\frac{25}{148}\left(3 e^{-2 t}+1\right) \sin 6 t \\\\|y(t)| \leq \frac{75}{74} e^{-2 t}-1|\cos 6 t|+\frac{25}{148}\left|3 e^{-2 t}+1\right||\sin 6 t| \\\\[/tex]

[tex]|y(t)| \leq \frac{75}{74}\left|e^{-2 t}-1\right|+\frac{25}{148}\left|3 e^{-2 t}+1\right| \\\\\lim _{t \rightarrow \infty} y(t) \leq \lim _{t \rightarrow \infty}\left\{\frac{75}{74}\left|e^{-2 t}-1\right|+\frac{25}{148}\left|3 e^{-2 t}+1\right|\right\} \\\\\lim _{t \rightarrow \infty} y(t)=\left\{\frac{75}{74}\left|\lim _{t \rightarrow \infty}\left(e^{-2 t}-1\right)\right|+\frac{25}{148}\left|\lim _{t \rightarrow \infty}\left(3 e^{-2 t}+1\right)\right|\right\} \\[/tex]

[tex]\lim _{t \rightarrow \infty} y(t)=\left\{\frac{75}{74}(-1)+\frac{25}{148}(1)\right\}=-\frac{75}{74}+\frac{25}{148}=-\frac{-150+25}{148}=-\frac{125}{148} \approx-0.844[/tex]

Answer: 7500

Step-by-step explanation:

multiply 150 by 50

Answer:

Angles in a triangle sum to 180.

So

x= 180 - 75 - 43

x=62°

Answer:

x=62°

Step-by-step explanation:

x= 180 - 75 - 43

Answer:

Part 1:

y= 23

Part 2:

x=20

Step-by-step explanation:

Part 1:

cos(y) = 48/52

y=cos^-1(48/52)

y=22.619864948~23

Part 2:

48^2+x^2=52^2

x^2=2704-2304

x=sqrt(400)

x=20 or -20.

A negative length makes no sense, so x=20

Answer:

y = 23º

x = 20 cm

Step-by-step explanation:

Cos∅ = adj/hyp

cosy = 48/52

∅ = arccos48/52

use calculator

∅ = 22.619864948040426

Rounded

∅ = 23º

----------------------------------

a² + b² = c²

x² + 48² = 52²

x² = 52² - 48²

x² = 400

x = 20 cm

Answer:

if you just wanted the equations fixed: y= -3x + 8 & y= 1/3x - 11/3

but if you wanted x then x = 7/2

Step-by-step explanation:

3x + y = 8 *subtract 3x from both sides*

y = -3x + 8

x - 3y = 11 *add 3y to both sides*

x = 3y + 11 subtract 11 from both sides*

x - 11 = 3y * divide both sides by 3*

1/3x - 11/3 = y --> y = 1/3x - 11/3

and to solve for x i just used ma thway but if you did want to solve it you'd need to put the equations equal to each other --> -3x + 8 = 1/3x - 11/3

Answer:

x=52   y=116

Step-by-step explanation:

because they give you the angle 116.

those two are actually equal.

y=116

since that is true, you can do 180-116=64.

now, you subtract 64-12

which is 52.

Answer:

V = 310.5 cm³

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Geometry

Volume of a Prism Formula: V = lwh

Step-by-step explanation:

Step 1: Define

Identify variables

l = 13.8 cm

w = 4.5 cm

h = 5 cm

Step 2: Solve for V

Answer:

b

Step-by-step explanation:

3/3 = 1

Answer:

C

Step-by-step explanation:

Answer:

< 1 = 92º

< 2 = 24º

< 3 = 35º

< 4 = 35º

< 5 = 81º

Step-by-step explanation:

< 1 = 180 - 88 = 92

< 3 = 64 corresponding angle due to parallel lines

< 2 = 180 - (92 + 64) = 24

< 4 = 180 - (64 - 81) = 35

< 5 = 180 - (64 + 35) = 81

Answer:

Error term

Step-by-step explanation:

Given :

In a regression analysis, the actual values of Y may be found above or below the regression line. These deviations are called the _____. slope error term variance standard error of the mean standard deviation

To find :

Fill in the blanks.

Now, In a regression analysis, the actual values of Y may be found above or below the regression line. These deviations are called the Error term.

Because we use the error term in the regression analysis difference between the actual value of [tex]Y[/tex] and the approximate value of [tex]Y[/tex] which is above of actual value or below of actual value.

 As [tex]Y[/tex]is an actual value and [tex]Y_0[/tex] is an approximate value then,

[tex]|Y-Y_0|[/tex] is an error.

It is a vertical distance between the actual value and the approximate value  of [tex]Y[/tex].

So, the error term is always positive.

Answer:

d. 63%

Step-by-step explanation:

percent, will be white?

A41% b50% c59% d63%

The probability of white = P (w) = 41/65= 0.63

The  probability of yellow = P (y)= 24/65= 0.369=0.37

The probability of choosing white is 0.63 . When rounded to  nearest percent gives

0.63*100/100

=0.63*100 percent

= 63 percent

= 63%

the probability of getting   the next marble white is the same as the probability of getting a white.

Answer:

$3.74

Step-by-step explanation:

we already know the cost of the banana's and it gives us the total price including apples. So $5.24 - $1.50 = $3.74, and the 3.74 is the cost of apples

Answer:

About 292.72

Step-by-step explanation:

We can imagine a circle circumscribed around the triangle, then solve it as we normally do using the apothem. By dividing 360 by the number of sides (3) we know that the central angle constructed by the apothem and radiuses would be 120. Divided by 2 (because we are using a right triangle to solve it) is 60. Now with this you can either use trig ratios (sin, cos, or tan) or use special right triangles (30-60-90 or 45-45-90). I used special rights. With this I know my apothem is [tex]\frac{13\sqrt{3} }{3}[/tex]. Perimeter is 26 times 3 which is 78. The formula for area of regular polygons is 1/2(perimeter times apothem). When we plug our values in the answer is about 292.72. You are lucky that I just had my test on this, I hope it helps.

Here is my work since this is very visual for me:

Answer:

[tex] \displaystyle 5\sqrt{2}[/tex]

Step-by-step explanation:

we have a right angle isosceles triangle

in order to figure out the length of each leg we can consider Pythagoras theorem given by

[tex] \displaystyle {a}^{2} + {b}^{2} = {c}^{2} [/tex]

remember that,

isosceles triangle has two equal legs so a=b and given that the the hypotenuse is 10

substitute:

[tex] \displaystyle {a}^{2} + {a}^{2} = {10}^{2} [/tex]

simplify addition:

[tex] \displaystyle {2a}^{2}= 100[/tex]

simplify square:

divide both sides by 2:

[tex] \displaystyle {a}^{2}= 50[/tex]

square root both sides:

[tex] \displaystyle {a}^{}= \sqrt{50}[/tex]

[tex] \displaystyle {a}^{}= 5\sqrt{2}[/tex]

hence,

the length of each leg is 5√2

Answer: Hello the scatter plot related to your question is missing attached below is the scatter plot

answer :  The model fails to capture the relationship between the variables accurately, and there may exist nonlinear relationship ( C )

Step-by-step explanation:

The conclusion that can be drawn based upon the scatter chart is that The model fails to capture the relationship between the variables accurately, and there may exist nonlinear relationship

A scatter plot helps in observing the relationship within different numeric variables but the scatter plot attached fails in the showing the actual relationship

Answer:

1/20

Step-by-step explanation:

According to G0ogle 5 percent equals 1/20 in lowest terms.

Answer:

5% equals 5/100 which is 1/20 in lowest terms.

Step-by-step explanation:

5% is basically equivalent to 5/100. 5/100 in lowest terms is 1/20 since you divide the numerator and denominator by 5.

I hope this helps, have a nice day.

Answer:

look at the picture i have sent

Answer:

The cities are 35 cm apart in map.

The scale on a map is 5 cm : 8 km.

Step-by-step explanation:

mrk me brainliest please

Answer:

[tex](6\sqrt{3},\,\frac{5\pi}{6})[/tex]

Step-by-step explanation:

The radius  r  can be found from the relationship

 [tex]r^2=x^2+y^2\\r^2=(-9)^2+(3\sqrt{3})^2\\r^2=81+27=108\\r=\sqrt{108}\\r=6\sqrt{3}[/tex]

The point is in Quadrant II (-, +), so use the inverse cosine function to find the angle.

[tex]\cos{\theta}=\frac{x}{r}=\frac{-9}{6\sqrt{3}}\\\cos{\theta}=-\frac{9}{6\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}\\\cos{\theta}=-\frac{9\sqrt{3}}{6\cdot3}\\\cos{\theta}=-\frac{\sqrt{3}}{2}\\\\\cos^{-1}\frac{-\sqrt{3}}{2}}=\frac{5\pi}{6}[/tex]

See the attached image.

Answer:

10x

20

Step-by-step explanation:

expression

10x

for x = 2

10 (2)

20

Answer:

the answer is 6.

Step-by-step explanation:

Answer:

6.

Step-by-step explanation:

3+3=6 = 2×3=6

you can do draw 3 circles 2 times and add it all together.

Answer:

u=−x2−x+1

Step-by-step explanation:

Let's solve for u.

2x−u−1x^2−3x+3=2

Step 1: Add x^2 to both sides.

−x2−u−x+3+x2=2+x2

−u−x+3=x2+2

Step 2: Add x to both sides.

−u−x+3+x=x2+2+x

−u+3=x2+x+2

Step 3: Add -3 to both sides.

−u+3+−3=x2+x+2+−3

−u=x2+x−1

Step 4: Divide both sides by -1.

−u/−1=x2+x−1

−1/u=−x2−x+1

Answer:

u=−x2−x+1

Answer:

-7

Step-by-step explanation:

The full coordinate is (-7,3) the dot is on the line between -6 and -8

Answer:

-7

Step-by-step explanation:

x = 3

y = -7

the answer is just -7

Answer:

AC² = 10.8²+14.4² = 324

AC = √324= 18

area = AC.DC÷2

area=18×24÷2

area=216

Answer:

216 centimeters squared

Step-by-step explanation:

working is above

Answer:

C) 96i – 280j

Step-by-step explanation:

Multiplying vector by constant:

When a vector is multiplied by a constant, each component of the vector is multiplied by this constant.

In this question:

u = -12i + 35j

-8u = -8(-12i + 35j) = (8*12i - 8*35j) = 96i - 280j.

The answer is C) 96i – 280j

Answer:

The first one is supplementary

The second one is complementary

The third one is vertical

Step-by-step explanation:

Answer:

First answer is supplementary.

second answer is complementary.

Third answer is Vertical.

Answer:

C. 18

Step-by-step explanation:

4-6 paris = about 23

10-12 pairs = about 5

28 - 5 = 18

Answer:

The answer is less than that is the answer

Answer:

discount=MP-SP

=Rs420-Rs405

=Rs15

Step-by-step explanation: