Solve the following equation by factoring.
y²-8y=0

The solution to the initial value problem is determined as y = e³ˣ.

The auxiliary equation is determined as follows;

y'' - 7y' + 12y = 0

m² - 7m + 12 = 0

m² - 3m - 4m + 12 = 0

m(m - 3) - 4(m - 3) = 0

m = 3 or 4

[tex]y = c_1e^{3x} + c_2e^{4x}\\\\y(0) = 1\\\\1 = c_1e^{3\times 0} + c_2e^{4\times 0}\\\\1 = c_1 + c_2 \ --(i)\\\\y' = 3c_1e^{3x} + 4c_2e^{4x}\\\\y'(0) = 3\\\\3 = 3c_1e^{3\times 0} + 4c_2e^{4\times 0}\\\\3 = 3c_1 + 4c_2 \ \ --(ii)[/tex]

solve (i) and (ii)

3 = 3c₁ + 4(1 - c₁)

3 = 3c₁ + 4 - 4c₁

3 = -c₁ + 4

c₁ = 1

c₂ = 0

[tex]y = e^{3x}[/tex]

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Answer:

a = -3

b = 9

c = -3

Step-by-step explanation:

[tex]a. (2^{5} )(2)^{-8} =2^{a} \\2^{(5-8)} =2^{a} \\2^{-3}=2^{a}[/tex]

[tex]a=-3[/tex]

[tex]b.\frac{4^{7} }{4^{-2} } =4^{b} \\4^{(7-(-2))} } =4^{b} \\4^{(7+2)} =4^{b}[/tex]

[tex]4^{9} =4^{b}[/tex]

[tex]b=9[/tex]

[tex]c. \frac{7^{9} }{7^{c} } =7^{12}[/tex]

[tex]7^{9-c} =7^{12}[/tex]

[tex]9-c=12[/tex]

[tex]c=9-12[/tex]

[tex]c=-3[/tex]

Hope this helps

Answer:

Yes, x = 0 is a solution to the given equation

Step-by-step explanation:

Hi Student!

The goal of this question is to determine x = 0 is a solution of the expression that was provided.  The first step that we must take is input 0 into all of the x's that we have in the expression.  Then we just simplify both sides and determine if the end expression is true and if it is then x = 0 is a solution.

Plug in the values

Simplify both sides

Looking at the final expression, we can see that 1 is indeed equal to 1 and since the expression is true, we can say that x = 0 is a solution of the expression that was provided in the problem statement.

Answer:

A,D,F

Step-by-step explanation: Because it not a fraction or decimal in it its an whole number

The equation of the line that is perpendicular to the line and passes through the point (15,-5) is y = -x + 10

The complete question is

What is the y-intercept of the equation of the line that is perpendicular to the line y = x + 10 and passes through the point (15, –5)?

A perpendicular means a line which intersects the other line at 90 degree.

Line given y = x+10

slope = 1

Slope of the line perpendicular to the above line = -1

and it passes through the point  (15, –5)

The equation will become

y = mx +c

-5 = -1 * 15 + c

15-5 = c

c = 10

Therefore the equation is y = -x +10

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Answer: d

Step-by-step explanation:

The 108 decibel sound is 6.31 times more intense than a 100 decibel sound option (B) is correct.

It is another way to represent the power of numbers, and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.

[tex]\rm a^b = c\\log_ac =b[/tex]

We have:

The loudness L of sound in decibels is given by

[tex]\rm L=10log(\dfrac{I}{I_0})[/tex]

Here I is the intensity of the sound.

[tex]\rm I_0=10^{-12} \ W/m^2[/tex]

Loudness for 108 decibel sound:

[tex]\rm 108=10log(\dfrac{I}{10^-^1^2})[/tex]

I = 0.06309

Similarly, for:

100 decibel sound:

i = 0.01

I/i = 0.06309/0.01 = 6.31

I = 6.31i

Thus, the 108 decibel sound is 6.31 times more intense than a 100 decibel sound option (B) is correct.

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The height is represented by h = -16t² + 4t + 20 and There is only one solution, because the solutions indicate time and time cannot be negative.

We are given the vertical model of the diver's jump as;

h = -16t² + vt + s

where;

h is height

t is time

v is velocity

s is distance

A) We are given;

Distance; s = 20 ft

velocity; v = 4 ft/s

Thus;

h = -16t² + 4t + 20

B) The solution of the model is gotten from;

-16t² + 4t + 20 = 0

Solving this quadratic equation gives;

t = 1.25 seconds or -1 second

C) B. There is only one solution, because the solutions indicate time and time cannot be negative.

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Answer:

x ≤ 0

Step-by-step explanation:

The number line's direction is left, any number that is left on a number line behind zero has a less value. Greater than or equal!

Answer:

5 hours 3 minutes

Step-by-step explanation:

Add all of the times together, but remember that minutes only goes up to 60 minutes. So when you add all the minutes and you get 63, subtract 60 and add an hour, and you are left with 3 minutes.

The graph that represents function h is the graph in; Option C

The parent function is; f(x) = tan (x)

The transformed function is; h(x) = -tan(¹/₂x)

Now, from general tangent function we know that; y = tan(Bx)

Thus, period = π/(¹/₂)

From the given transformation that produced the h(x) function and when we look at the given graphs, the one that represents the function h is Option C.

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Step-by-step explanation:

normally the perimeter of a rectangle is

2×length + 2×width

in this case here one of the sides in that calculation has to be replaced by the outer circumference of the semicircle.

but which one ? you did not say, whether the semicircle is attached to a length or to a width of the rectangle.

the circumference of a circle is

2×pi×r

r being the radius (= half of the diameter).

so, the outer circumference of a semicircle is

2×pi×r/2 = pi×r

since we don't know the side of the rectangle, I am giving you both calculations here, and you have to pick the right one that fits your original problem definition.

if it is attached to a length (25 ft), then the radius of the semicircle is

25/2 = 12.5 ft

and the whole perimeter is

length + semicircle + 2×width =

= 25 + pi×12.5 + 2×29 =

= 25 + 39.26990817... + 58 = 122.2699082... ft

if it is attached to a width (29 ft), then the radius of the semicircle is

29/2 = 14.5 ft

and the whole perimeter is

2×length + semicircle + width =

= 2×25 + pi×14.5 + 29 =

= 50 + 45.55309348... + 29 = 124.5530935... ft

Answer:

Depending on the unit your answer needs to be in, it would be 88 cubic feet or 3.25926 cubic yards

If your question asks you to round to the nearest hundredth it would be 3.26 cubic yards.

Step-by-step explanation:

To find the volume of soil:

Volume= Length x Width x Depth/Height

Volume= 44ft x 2ft x 1ft

Volume= 88 cubic feet or 3.25926  (there are 27 cubic feet in a cubic yard, 88/27=3.25926)

hope this helps! :)

Answer:

A

Step-by-step explanation:

the domain is the definition of the valid x (input) values.

the range is the definition of the valid y (result) values.

as the function is defined by listing the specific points, the function only consists of individual points and not of connecting lines.

therefore the domain and range cannot be continuous intervals (otherwise the function would suddenly also include points not in the original list).

so, B and D are wrong.

and A is right, because the domain contains the x values, and the range the y values.

Answer:

$473

Step-by-step explanation:

We divide 5676 by 12 and we get 473

$473 they lose each month (?)

Step-by-step explanation:

calculator

the exact value of sin A  to the nearest ten- thousandth is 0. 6625

a² + b² = c²

The opposite side is unknown, so use the pythagorean theorem to find it

c = hypotenuse = 4

a= opposite site = ?

b= adjacent side = 3

Substitute into the formula

4² = a² + 3²

16 = a² + 9

a² = 16 -9 = 7

Find the square root

a =√7 = 2. 65

To find Sin A, use

Sin A = opposite side ÷ hypotenuse

Sin A = 2. 65 ÷ 4 = 0. 6625

Thus,  the value of sin A is 0. 6625

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Answer:

[tex] { - 7y}^{2} + 20y + 32[/tex]

Step-by-step explanation:

1.) -7y times y = -7y(squared)

2.) -7y times -4 = 28y

3.) -8 times y = -8y

4.) -8 times -4 = 32

5.) Combine like terms

[tex] { - 7y}^{2} + 28y - 8y + 32 = { - 7y}^{2} + 20y + 32[/tex]

Answer:

  about 101°

Step-by-step explanation:

The bearing back to the parking lot is the reverse of the bearing from the parking lot to the lake. That bearing can be found two ways. One of them uses the bearing and distance of each leg of the hike. The other uses what Zach remembers about the map.

__

The location of the lake from the parking lot can be found by adding the vectors to the campsite from the lot and to the lake from the campsite. That sum will be ...

  12.4∠315° +7.0∠180° ≈ 8.94∠281.4°

A suitable calculator can be used to find this sum, as shown in the second attachment. (Angles are displayed by the calculator in the range (-180°, 180°].)

__

The direction of the parking lot from the lake will be the opposite direction. It can be found by subtracting 180° from the angle:

  bearing to parking lot = 281.4° -180° = 101.4°

Michael and Zack should follow a compass direction of about 101°.

__

Northwest is a compass bearing of 315°. Points south of that bearing will have a smaller angle. Zach remembers the direction of the lake to be about 30° less than that angle, so at 315° -30° = 285°.

The bearing back to the parking lot is the opposite direction, so ...

  285° -180° = 105°

Based on Zack's memory, they should follow a compass direction of about 105°.

They will come within about 560 meters of the parking lot using this heading.

_____

Additional comments

When doing these vector calculations by hand, a couple of different methods can be used. One is to convert the vectors to component form, add the components, then convert back to polar form. The other is to make use of the Law of Cosines and the Law of Sines to solve the relevant triangles.

Here, the lake-lot distance is the third side of a triangle with two sides and the included angle known. The Law of Cosines applied to this situation is ...

  c² = a² +b² -2ab·cos(C) . . . . . generic Law of Cosines equation

  c² = 12.4² +7.0² -2(12.4)(7.0)cos(45°) ≈ 80.0063

  c ≈ √80.0063 ≈ 8.9446

Using the Law of Sines, we find the obtuse angle of the triangle to be ...

  sin(L)/12.4 = sin(C)/8.9446 . . . . . L and C are vertex angles in the diagram

  L = 180° -arcsin(12.4/8.9446×sin(45°)) ≈ 101.4°

__

If translation to rectangular coordinates is used, this can be done using the usual translation ...

  (x, y) = (r·cos(θ), r·sin(θ)) . . . . rectangular coordinates for r∠θ

The value of θ used can be the bearing angle (measured clockwise from north), or it can be the corresponding Cartesian plane angle (measured counterclockwise from +x). As long as the angles are consistently measured, the math works either way.

__

We have assumed that "northwest" is a bearing of 315°. If the compass rose is divided into 32 segments, the name "northwest" will be given to any bearing within 5° of this value. This makes our detailed vector calculations be somewhat approximate, and means that Zach's memory of the required bearing may be sufficiently accurate after all. Or not.

The possible base and height of a second triangle are 12 and 7, respectively

An inverse variation is represented as:

Base * Height = k

Where k is the constant of variation.

In this case, k is the area

So, we have:

Base * Height  = Area

For the first triangle, we have:

14 * 6 = Area

Evaluate

Area = 84

Substitute Area = 84 in Base * Height  = Area

Base * Height  = 84

Express 84 as the product of two numbers

Base * Height  = 12 * 7

By comparison;

Base =12 and Height = 7

Hence, the possible base and height of a second triangle are 12 and 7, respectively

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Answer:

A

Explanation:

Formula for the volume of a cone :Volume (cone) = 1/3πr²h

=========================================================

Given :

⇒ r = 4 units

⇒ h = 21 units

=========================================================

Solving :

⇒ Volume (cone) = 1/3 × π x 4² × 21

⇒ Volume (cone) = 16 × 7 × π

⇒ Volume (cone) = 112π units²

Some possible ethical issues to this type of testing that they are subjected to unpleasant and hazardous procedures, as well as stress and injury.

Any well-defined method that yields an observable outcome that cannot be precisely predicted in advance is referred to as a experiment. To avoid any ambiguity or surprise, a experiment must be properly defined.

A) Because human experimentation is immoral, animal research has been employed as a research approach.

However, using animals is unethical because they are subjected to unpleasant and hazardous procedures, as well as stress and injury. The use of non-animal methods to substitute animal methods as much as possible will provide both ethical and technical benefits. Researchers should make the use of non-animal methods for a top priority experiments.

B) Even though the purpose of an experiment is to ensure success in rabbits, this is incorrect in people because action is what will operate properly.

The absence of thyroid-stimulating hormone in rabbit serum is a mistaken notion. As a result, it is preferable to test the drug further, and it was a success. As a result, there will be no dysfunctionality for humans.

Thus, some possible ethical issues to this type of testing that they are subjected to unpleasant and hazardous procedures, as well as stress and injury.

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Answer:

81

Step-by-step explanation:

Hello!

We can plug in 6 for x in x² + 8x - 3

When x = 6, the expression evaluates to 81.

Answer:

see the attachment photo!

There are two negative roots to the provided polynomial function

f(x) = x⁷ – 2x⁴ + 7x² + 2x – 2

Polynomial is the combination of variables and constants systematically with "n" number of power in ascending or descending order.

[tex]\rm a_1x+a_2x^2+a_3x^3+a_4x^4..........a_nx^n[/tex]

We have a polynomial function:

f(x) = x⁷ – 2x⁴ + 7x² + 2x – 2

[tex]\rm f(x) =\left(x+1\right)\left(x^6-x^5+x^4-3x^3+3x^2+4x-2\right)[/tex]

Using the zero product property:

[tex]\rm \left x+1\right = 0 \ \ \ or \left x^6-x^5+x^4-3x^3+3x^2+4x-2\right = 0[/tex]

After solving:

x = -1, x = -0.853, and x = 0.418 (using the graph method)

Thus, there are two negative roots to the provided polynomial function

f(x) = x⁷ – 2x⁴ + 7x² + 2x – 2

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Using the normal distribution, considering it's symmetry, it is found that the value of C is of 2.19.

The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The standard normal distribution has mean and standard deviation given, respectively, by:

[tex]\mu = 0, \sigma = 1[/tex].

Hence, considering the symmetry of the normal distribution, the value of c is Z with a p-value of (1 + 0.9715)/2 = 0.98575, hence c = Z = 2.19.

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Answer:The answer is 8 sides

Step-by-step explanation:

Determinants are defined only for square matrices, so I assume you are given the 3×3 matrix

[tex]A = \begin{bmatrix}-2 & -4 & -3 \\ -1 & -9 & -8 \\ -15 & -5 & 6 \end{bmatrix}[/tex]

Let's compute the determinant by taking the cofactor expansion along the first column:

[tex]\det A = -2 \det\begin{bmatrix}-9 & -8 \\ -5 & 6\end{bmatrix} + \det\begin{bmatrix}-4 & -3 \\ -5 & 6\end{bmatrix} - 15 \det \begin{bmatrix}-4 & -3 \\ -9 & -8\end{bmatrix}[/tex]

[tex]\det A = -2 ((-9)\times6-(-8)\times(-5)) + ((-4)\times6 - (-3)\times(-5)) \\ ~~~~~~~~~~~~~~~- 15 ((-4)\times(-8)-(-3)\times(-9))[/tex]

[tex]\det A = -2(-54-40) + (-24 - 15) - 15 (32 - 27)[/tex]

[tex]\det A = \boxed{74}[/tex]

The differentiation of the function g(t) = 7/t⁴ will be equal to g¹(t)=-28/t⁵

The method of determining the derivative, or rate of change, of a function in mathematics.is termed as the differentiation.

Given that:-

[tex]g(t)= \dfrac{7}{t^4}[/tex]

The derivative will be calculated as:-

[tex]g'(t)=7 \dfrac{d}{dt}(t^{-4})\\\\\\g'(t)=7\times (-4t^{-5})\\\\\\g'(t)=\dfrac{-28}{t^5}[/tex]

Therefore the differentiation of the function g(t) = 7/t⁴ will be equal to g¹(t)=-28/t⁵

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Step-by-step explanation:

similar shapes means that they have the same angles, and there is one scale factor that applies to all sides and other lines in the shapes (like heights).

therefore the ratios between 2 correlated lines are all the same.

so,

2/7 = 5/height

2×height / 7 = 5

2×height = 7×5 = 35

height = 35/2 = 17.5 in