If a polynomial function, f(x), with rational coefficients has roots 3 and startroot 7 endroot, what must also be a root of f(x)?

The velocity vector of the given path  r(t) = (9cos2(t), 3t - t^3, 2t) is [tex]v = 9sint\hat{i}+(3-3t^{2} )\hat{j} +2\hat{k}[/tex].

According to the given question.

We have a path

r(t) = (9cos2(t), 3t - t^3, 2t)

So, the vector form of the above vector form can be written as

[tex]r(t) = 9cos2(t)\hat{i}+ (3t - t^{3} )\hat{j} + 2t\hat{k}[/tex]

As, we know that the rate of change of position of an object is called velocity vector.

Therefore, the velocity vector of the given path r(t) = (9cos2(t), 3t - t^3, 2t) is given by

[tex]v = \frac{d(r(t))}{dt}[/tex]

[tex]\implies v = \frac{d(9cost\hat{i}+(3t-t^{3})\hat{j}+2t\hat{k} }{dt}[/tex]

[tex]\implies v = \frac{d(9cost\hat{i})}{dt} +\frac{d(3t-t^{3})\hat{j} }{dt} +\frac{d(2t)}{d(t)}[/tex]

[tex]\implies v = 9sint\hat{i}+(3-3t^{2} )\hat{j} +2\hat{k}[/tex]

Hence, the velocity vector of the given path  r(t) = (9cos2(t), 3t - t^3, 2t) is [tex]v = 9sint\hat{i}+(3-3t^{2} )\hat{j} +2\hat{k}[/tex].

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Answer:

3x + 16 + 30/x - 13

Step-by-step explanation:

The vertex form of the equation y = -x^2 + 4x - 1 is y = -(x - 2)^2 + 3

The quadratic equation is given as:

y = -x^2 + 4x - 1

Differentiate the above quadratic equation.

This is done with respect to x by first derivative

So, we have:

y' = -2x + 4

Set the derivative to 0

-2x + 4 = 0

Subtract 4 from both sides of the equation

-2x + 4 - 4 = 0 - 4

Evaluate the difference in the above equation

-2x = -4

Divide both sides of the above equation by -2

x = 2

Rewrite as

h = 2

Substitute 2 for x in the equation y = -x^2 + 4x - 1

y = -2^2 + 4 *2 - 1

Evaluate the equation

y = 3

Rewrite as:

k = 3

A quadratic equation in vertex form is represented as:

y = a(x - h)^2 + k

So, we have:

y = a(x - 2)^2 + 3

In the equation y = -x^2 + 4x - 1, a = -1

So, we have:

y = -(x - 2)^2 + 3

Hence, the vertex form of the equation y = -x^2 + 4x - 1 is y = -(x - 2)^2 + 3

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The values of b, h and k are (c) b = 2, h = -2 and k = -9

The logarithmic function is given as:

f(x) = log₂(x + 2) - 9

A logarithmic function is represented as:

f(x) = logb(x - h) + k

By comparing both equations, we have

b = 2

h = -2

k = -9

Hence, the values of b, h and k are (c) b = 2, h = -2 and k = -9

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The true statement relating to a property of the function y =sin x is that the maximum and minimum values of the function are 1 and -1 respectively. Option B

The following are the properties of the sin trigonometric ratio of the function;

From the above listed deductions, we can see that the true statement about the function y = sin x is that the range which is always known as the maximum and minimum values of the function are 1 and - 1 respectively.

Thus, the true statement relating to a property of the function y =sin x is that the maximum and minimum values of the function are 1 and -1 respectively. Option B

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In order to solve this IVP using Laplace transforms, we must first write f(t) in terms of the Heaviside function.

f(t)=0*(u(t)-u(t-Pi))+1*(u(t-Pi)-u(t-2Pi))+0*(u(t-2Pi))

f(t)=u(t-π)-u(t-2π)

So, the rewritten IVP is

y'' +y = u(t-π)-u(t-2π)y(0)=0, y'(0)=1

Taking the Laplace transform of both sides of the equation, we get:

s2L{y}-sy(0)-y'(0)+L{y}=(1/s)*e-πs-(1/s)*e-2πs

s2L{y}-1+L{y}=(1/s)*e-πs-(1/s)*e-2πs

(s2+1)L{y}=1+(1/s)*e-πs-(1/s)*e-2πs

L{y}=1/(s2+1)+(1/s(s2+1))e-πs-(1/s(s2+1))*e-2πs

Now, we must take the inverse transform of both sides to solve for y.

The first inverse transform is easy enough. By definition, it is sin(t).

The second two inverse transforms will be a little tougher, we will have to use partial fraction decomposition to break them down into terms that are easier to compute.

A/s+(Bs+C)/(s^2+1)=1/(s(s^2+1))

A(s^2+1)+(Bs+C)(s)=1

As^2+A+Bs^2+Cs=1

Rewriting this system in matrix form, we get:

1  1  0    A     0

0  0  1  * B  = 0

1  0  0    C     1

Using row-reduction we find that A=1, B=-1, and C=0. So, our reduced inverse transforms are:

L-1{(e-πs)(1/s-s/(s2+1))}

and

L-1{(e-2πs)(1/s-s/(s2+1))}

Using the first and second shifting properties, these inverse transforms can be computed as.

L-1{(e-πs)(1/s-s/(s2+1))}=u(t-π)-cos(t-π)u(t-π)

L-1{(e-2πs)(1/s-s/(s2+1))}=u(t-2π)-cos(t-2π)u(t-2π)

Combining all of our inverses transforms, we get the solution the IVP as:

y=sin(t)+u(t-π)-cos(t-π)u(t-π)+u(t-2π)-cos(t-2π)u(t-2π)

In mathematics, the Laplace transform, named after its discoverer Pierre Simon Laplace (/ləˈplɑːs/), transforms a function of real variables (usually in the time domain) into a function of complex variables (in the time domain). is the integral transform that Complex frequency domain, also called S-area or S-plane).

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The average value of the given function is [tex]f_{avg} = \frac{-2a^{2} + 6a + 6 }{6}[/tex].

According to the statement

we have given that the function f(x) and we have to find the average value of that function.

So, For this purpose, we know that the

The given function f(x) is

[tex]f(x) = -x^{2} + 2x +1[/tex]

And now integrate this function with the limit 0 to a then

[tex]f_{avg} = \frac{1}{b - a} \int\limits^a_0 {f(x)} \, dx = -x^{2} + 2x +1[/tex]

Now integrate this then

[tex]f_{avg} = \frac{1}{a} \int\limits^a_0 {-x^{2} + 2x +1} \, dx[/tex]

Then the value becomes according to the integration rules is:

[tex]f_{avg} = \frac{1}{a} \int\limits^a_0 {-\frac{x^{3} }{3} + \frac{2x^{2} }{2} +x} \,[/tex]

Now put the limits then answer will become as output is:

[tex]f_{avg} = \frac{1}{a} [ {-\frac{a^{3} }{3} + \frac{2a^{2} }{2} +a} \,][/tex]

Now solve this equation then

[tex]f_{avg} = [ {-\frac{a^{2} }{3} + \frac{2a }{2} +1} \,][/tex]

Now

[tex]f_{avg} = \frac{-2a^{2} + 6a + 6 }{6}[/tex]

This is the value which represent the average of the given function in the statement.

So, The average value of the given function is [tex]f_{avg} = \frac{-2a^{2} + 6a + 6 }{6}[/tex].

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If you know the slope of the lines, you can also use these methods:

- see if the slopes multiplied by each other are -1 (should be if they're perpendicular)

- test if the slopes are opposite reciprocals (if they are, they are perpendicular)

Answer: 243

Step-by-step explanation:

If you divide the 18 inch by two to get the r, you will get 9. So you should multiple the 3 and the 9 square. So it should be 3x81=243

The area of the shaded region in the given image is: 1,111.84 ft²

To find the area of the shaded region in the image given, find the areas of the unshaded region, then subtract it from the total area of the whole figure.

Area of the total figure = area of two semicircles + area of the rectangle.

Dimensions of the rectangle is given as, 18ft x 22ft.

Area of the rectangle = 18ft × 22ft = 396 ft²

The diameter of each of the semicircle = 7 + 7 + 18 = 32 ft

The radius of each of the semicircle = 1/2(32) = 16 ft

Area of the two semicircles = 2(1/2 × πr²) = 2(1/2 × 3.14 × 16²)

Area of the two semicircles = 803.84 ft²

Area of the rectangle = 32 × 22 = 704 ft²

The total area = 704 + 803.84

The total area = 1,507.84 ft²

The area of the shaded region = 1,507.84 - 396

The area of the shaded region = 1,111.84 ft²

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Considering the given graph, we have that:

The slope of a function, given two points, is given by the change in y divided by change in x.

When x = -3, y = -3, and while x = -2, y = -2, hence when x changes by 1, y also changes, hence the slope of the graph of the function is equal to 1 for x between x = -3 and x = -2.

When x = 3, y = 4, and when x = 4, y = 4, hence the slope of the graph is equal to 0 for x between x = 3 and x = 4.

Looking at the vertical axis:

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The most general antiderivative of the given function g(t) is (8t + t³/3 + t²/2 + c).

The antiderivative of a function is the inverse function of a derivative.

This inverse function of the derivative is called integration.

Here the given function is: g(t) = 8 + t² + t

Therefore, the antiderivative of the given function is

∫g(t) dt

= ∫(8 + t² + t) dt

= ∫8 dt + ∫t² dt + ∫t dt

= [8t⁽⁰⁺¹⁾/(0+1) + t⁽²⁺¹⁾/(2+1) + t⁽¹⁺¹⁾/(1+1) + c]

= (8t + t³/3 + t²/2 + c)

Here 'c' is the constant.

Again, differentiating the result, we get:

d/dt(8t + t³/3 + t²/2 + c)

= [8 ˣ 1 ˣ t⁽¹⁻¹⁾ + 3 ˣ t⁽³⁻¹⁾/3 + 2 ˣ t⁽²⁻¹⁾/2 + 0]

= 8 + t² + t

= g(t)

The antiderivative of the given function g(t)is (8t + t³/3 + t²/2 + c).

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Answer:

  p(x) = x² +30x -375

Step-by-step explanation:

When a quadratic has real rational coefficients, any irrational or complex roots come in conjugate pairs.

A root of p means (x -p) is a factor of the polynomial. Here, we have roots of -15+10√6 and -15-10√6, so the factored form can be written ...

  p(x) = (x -(-15 +10√6))(x -(-15 -10√6))

Using the factoring of the difference of squares, we can write this as ...

  p(x) = (x +15)² -(10√6)²

Expanding the factored form, we can write the polynomial as ...

  p(x) = x² +30x +225 -600

  p(x) = x² +30x -375

Answer:

2/9 - 2/15

Solution

LCM = 45

45 ÷9 ×2 = 10

45÷ 15 ×2 = 6

10 - 6 = 4

ANSWER = 4/45

Answer:

l

Step-by-step explanation:

take the lcm of 9 and 15. it will be 45 . than continue

Answer: line ZX is congruent to line VX (option 4)

Step-by-step explanation:

We already know <X is congruent to <X, we also know that line YX is congruent to line XW. Now all we need is one more line adjacent to X which is going to be ZX ad VX

Answer:

1.0954.

Step-by-step explanation:

Standard error  =  std dev / √n

=  6 / √30

= 1.0954.

The tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.

First, let's find the derivative of the given function y = 1 + 60x³ − 2x⁵ using the power rule for differentiation:

dy/dx = 0 + 3(60)x² - 5(2)x⁴

= 180x² - 10x⁴

To find the critical points, we set the derivative equal to zero and solve for x:

180x² - 10x⁴ = 0

Factoring out common terms, we get:

10x²(18 - x²) = 0

Setting each factor equal to zero, we have:

10x² = 0 or 18 - x² = 0

From the first equation, we find x = 0.

From the second equation, we have:

18 - x² = 0

x² = 18

Taking the square root, we get:

x = ±√18

= ±3√2

So the critical points are x = 0, x = 3√2, and x = -3√2.

Now we need to evaluate the slope at these critical points. We can do this by plugging each x-value into the derivative:

When x = 0:

dy/dx = 180(0)² - 10(0)⁴ = 0

When x = 3√2:

dy/dx = 180(3√2)² - 10(3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080

When x = -3√2:

dy/dx = 180(-3√2)² - 10(-3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080

The slope is 0 when x = 0 and 1080 when x = 3√2 or x = -3√2.

Therefore, the tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.

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The complete question is as follows:

At which points on the curve y = 1 + 60x³ − 2x5⁵ does the tangent line have the largest slope?

[tex]x = \frac{\sqrt{5} + 1}{2} \approx 1.618033989[/tex]

Answer:

  (c)  f(g(3)) > g(f(3))

Step-by-step explanation:

The relationship between the function values can be found by evaluating the functions.

The order of operations tells us we must first evaluate g(3).

  g(x) = -3x +15

  g(3) = -3(3) +15 = 6

Then we can evaluate f(x) for x=6:

  f(x) = 7x

  f(6) = 7(6) = 42

So, the composition is ...

  f(g(3)) = 42

As above, we must first evaluate f(3):

  f(3) = 7(3) = 21

Then we can evaluate g(x) for x=21:

  g(21) = -3(21) +15 = -63 +15 = -48

That means the composition is ...

  g(f(3)) = -48

The first is greater than the second.

  42 > -48

  f(g(3)) > g(f(3))

The distance the player is from the right post is = 5.1 meters

The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.

Formula for the Pythagorean theorem =

a²= b²+c²

From the diagram given,

The hypotenuse (a) = x²

b= 1.82²

c = 4.8²

x² = 1.82² + 4.8²

x²= 3.3124 + 23.04

x²= 26.3524

a= √26.3524

a= 5.1 meters.

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Answer:

x=2

Step-by-step explanation:

2^x *2^4=2^3x

using law of indices

2^x+4=2^3x

x+4=3x

4=3x-x

4=2x

2=x

          Both of these problems will be solved in a similar way, but with different numbers. First, we set up an equation with the values given. Then, we solve. Lastly, we plug into the original expressions to solve for the angles.

[23] ABD = 42°, DBC = 35°

(4x - 2) + (3x + 2) = 77°

4x+ 3x + 2 - 2  = 77°

4x+ 3x= 77°

7x= 77°

x= 11°

-

ABD = (4x - 2) = (4(11°) - 2) = 44° - 2 = 42°

DBC = (3x + 2) = (3(11°) + 2) = 33° + 2 = 35°

[24] ABD = 62°, DBC = 78°

(4x - 8) + (4x + 8) = 140°

4x + 4x + 8 - 8 = 140°

4x + 4x = 140°

8x = 140°

8x = 140°

x = 17.5°

-

ABD = (4x - 8) = (4(17.5°) - 8) = 70° - 8° = 62°

DBC =(4x + 8) = (4(17.5°) + 8) = 70° + 8° = 78°

The number set(s) that - 10 belong to are rational numbers, integers and real numbers. Option C, D and E

A rational number can be defined as a number expressed as the ratio of two integers, where the denominator is not be equal to zer0

- 10 can be written as = 1/ 10

Integers are whole number that could be positive, negative and even zero

- 10 is a negative whole number

Real numbers are numbers with continuous quantity that can represent  distance along a number line

-10 can represent distance along a number line.

Thus, the number set(s) that - 10 belong to are rational numbers, integers and real numbers. Option C, D and E

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[tex]\sum^{\infty}_{n=1} (a/b)^n=5 \\ \\ =\frac{a/b}{1-\frac{a}{b}}=5 \\ \\ \frac{a}{b-a} =5 \\ \\ \frac{a}{b}=\frac{5}{6}[/tex]

So, we need to find

[tex]\sum^{\infty}_{n=1} n(5/6)^n

[/tex]

Let this sum be S.

Then,

[tex]S=(5/6)+2(5/6)^2 +3(5/6)^3+\cdots \\ \\ \frac{5}{6}S=(5/6)^2 + 2(5/6)^3+\cdots \\ \\ \implies \frac{1}{6}S=(5/6)+(5/6)^2+(5/6)^3+\cdots=5 \\ \\ \implies S=\boxed{30}[/tex]

Answer:7 21/100

Step-by-step explanation: We will covert 7.21 into a fraction. .21 = 21/100. Since 21 can only be divided by 3 and 7, the simplest form of 21/100 is 21/100. So, the mixed number is 7 and 21/100

Answer:

7 and 21/100

Step-by-step explanation:

Since .01 is 1/100, 7.21 will be 7 and 21/100. There is no way to further simplify this, therefore, it is our final answer. Hope this helps! :)

Answer:

20 cookies

Step-by-step explanation:

8 in 1.5 minutes

so we want to find how many in 3.75 minutes since 3 + 45/60 = 3.75

so then its 1.5*2 = 3 so 8*2 = 16 to get that 16 cookies in 3 minutes

then we still have .75 left so then divide 8/2 to get 4 cookies in 0.75 minutes

16+4 = 20

you can also just find how many in 0.25 minutes (15 seconds) you get 6/8

multiply that by 3.75/0.25 = 15 you get 15*(8/6) = 20

The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).

According to the statement, we must find in what conditions a given expression may be a solution of an ordinary differential equation. Then, first and second derivatives of the equation are:

y' = α · cos x - β · sin x                 (1)

y'' = - α · sin x - β · cos x              (2)

Then, we substitute on the ordinary differential equation:

(- α · sin x - β · cos x) + (α · cos x - β · sin x) = 0

And by algebraic handling we simplify the resulting expression:

- (α + β) · sin x + (α - β) · cos x = 0

Where each coefficient represents a constant of a linear combination:

α + β = 0

α - β = 0

Then, the solution of the system of linear equations is (α, β) = (0, 0). The expression y = α · sin x + β · cos x is a solution of the ordinary differential equations if and only if (α, β) = (0, 0).

The statement is incomplete and complete form cannot be found, then we decided to create a new statement:

Please prove that y = α · sin x + β · cos x is the solution of the differential equation d²y / dx² + dy /dx = 0 where the following condition is observed, if and only if α = β = 0.

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Type of model best fits the given situation linear.  Since it has a constant of +3 everyday.

What is a linear equation in math?

the membership of the local pta increases by 3 members a day for each day during the month of September.

Since it has a constant of +3 everyday.

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Answer: 45

Step-by-step explanation: Given the way the formula is formatted, the first term is 1. The common difference can be found by subtracting a number from the number that follows (ex. 3-2 or 4-3), therefore it's 1. The desired term is what you're trying to find so 44-1=43. When you put it all together, the formula should be 2+1(44-1) which equals 45 when you follow the rules of PEMDAS.