A particle starts from rest at a fixed point A and moves in a straight line with an acceleration which, t seconds after leaving A, is given by a = 4t. After 2 seconds the particle reaches a point B and the acceleration then ceases. Find:
i) the velocity when the particle reaches B
ii) the distance AB

The particle moves on immediately with acceleration given by -3t, where t seconds is the time after the particle leaves A, until it comes to rest at a point C. Find:
iii) the value of t when the particle reaches C
(iv) the distance AC​

The periodic time, t, of a pendulum range directly as the square root of its length, l. t = 6 when l = 9. If l = 25 then the periodic time, T exists at 10.

The period, or periodic time, of a periodic variation of a quantity, exists described as the time interval between two consecutive repetitions.

Given: The periodic time, t, of a pendulum goes directly as the square root of its length, l if t = 6 when l = 9.

If [tex]T \alpha\ l^2[/tex] then [tex]T^2[/tex] α [tex]\sqrt{l}[/tex]

[tex]T^2 = l[/tex]

Let, [tex]T^2 = kl,[/tex] where k exists constant

t = 6 when l = 9.

So, [tex]6^2 = k*9[/tex]

[tex]k = 6^2/9 = 4[/tex]

[tex]T^2 = 4*l[/tex]

If l = 25

[tex]T^2 = 4 * 25 = 100[/tex]

[tex]T = \sqrt{100}[/tex]

T = 10

Therefore, the periodic time, T exists 10.

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The correctly matched quadratic functions are given as follows:

Quadratics are polynomial equations of the second degree, which means that they contain at least one squared term.

Quadratic equations are another name for it. The quadratic equation has the following general form: ax2 + bx + c = 0.

Recall that:
f(x) - a (x -c) (x - d)

where

Part A) First graph

We are given to know

The solutions or zeros of the first graph are

x=-1 and x=3

The parabola open up, so the leading coefficient a is positive, the function therefore, is equal to

The vertex is equal to the point (1, -4)

Substitute and solve for a

-4 = a(1-3)(1+1)

-4 = a(-2)(2)

a = 1

Hence, the function is equal to f(x) = (x-3) (x + 1)

Part B) Second Graph

We are also given to know that

The solutions or zeros of the first graph are

x=-3 and x=1

The parabola open down, so the leading coefficient a is negative

The function is equal to f(x) = a (x-1)(x+3)

Find the value of the coefficient a

The vertex is equal to the point (-1,8)

to solve for a, we must substitute:

8 = a(-1-1) (-1+3)

8 = a(-2)(2)

a = -2

Hence, the function is equal to: f(x) = -2 (x - 1) ( x + 3)

Part C)

We are given to know that the solutions or zeros of the first graph are

x=-2 and x=6

The parabola open up, so the leading coefficient a is positive

The function is equal to f(x)  = a(x-6) (x+2).

Find the value of the coefficient a

The vertex is equal to the point (2,-8)

We substitute and solve for a:

-8 = a(2-6) (2+2)

-8 = a(-4)(4)

a = 0.5

Hence, the function is equal to f(x) = 0.5(x - 6) (x + 2)

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Answer:

RT = 16[tex]\sqrt{3}[/tex]

Step-by-step explanation:

using the sine ratio in right triangle QRT and the exact value

sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then

sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{RT}{QR}[/tex] = [tex]\frac{RT}{32}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

2 RT = 32[tex]\sqrt{3}[/tex] ( divide both sides by 2 )

RT = 16[tex]\sqrt{3}[/tex]

Answer:

  3

Step-by-step explanation:

The quadratic formula is used to solve a quadratic equation in standard form, based on the values of the coefficients.

The standard-form quadratic ax² +bx +c = 0 has solutions given by the quadratic formula:

  [tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

For the quadratic 3x² +4x -2 = 0, we have a=3, b=4, c=-2 and the formula gives ...

  [tex]x=\dfrac{-4\pm\sqrt{4^2-4(3)(-2)}}{2(3)}=\dfrac{-4\pm\sqrt{16+24}}{6}\\\\x=\dfrac{-4\pm 2\sqrt{10}}{6}=\dfrac{-2\pm\sqrt{10}}{3}[/tex]

The denominator in the solution is 3.

Answer:

The answer is x=(-2±√10)/3

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

|-80 - 15| = 95

The distance between -80 and 15 is 95

80 to 0 plus 15.

Answer:

The actual prime factors of 60 are 2, 3, and 5.

Step-by-step explanation:

Greetings !

Use a factor tree to express 60 as a product of prime factors. So the prime factorization of 60 is 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.

The sum of each of the 4 polynomials and their answers are respectively;

12x² + 3x + 6 + (-7x²) – 4x – 2  = 5x² - x + 4

(2x² - x) + (-x – 2x² – 2) = -2x - 2

(x³ + x² + 2) + (x² – 2 - x³) = 2x²

(x² + x) + (x² + 8x - 2) = 2x²+ 9x - 2

1) We want to find the sum of the polynomials (12x² + 3x + 6) and (-7x² – 4x – 2). Thus, we have;

12x² + 3x + 6 + (-7x²) – 4x – 2

= 5x² - x + 4

2)  We want to find the sum of the polynomials (2x² - x) and (-x – 2x² – 2). Thus, we have;

(2x² - x) + (-x – 2x² – 2)

= -2x - 2

3) We want to find the sum of the polynomials (x³ + x² + 2) and (x² – 2 - x³). Thus, we have;

(x³ + x² + 2) + (x² – 2 - x³)

= 2x²

4) We want to find the sum of the polynomials (x² + x) and (x² + 8x - 2). Thus, we have;

(x² + x) + (x² + 8x - 2) = 2x²+ 9x - 2

The sum of each of the 4 polynomials and their answers are respectively;

12x² + 3x + 6 + (-7x²) – 4x – 2  = 5x² - x + 4

(2x² - x) + (-x – 2x² – 2) = -2x - 2

(x³ + x² + 2) + (x² – 2 - x³) = 2x²

(x² + x) + (x² + 8x - 2) = 2x²+ 9x - 2

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- 1.3 is the number we get when we subtract the sum of -5/6 and -1 3/5 from the sum of 2 2/3 and -6 2/5. This can be obtained by finding sum separately and then subtracting them.

Here in the question it is given that,

subtract the sum of -5/6 and -1 3/5 from the sum of 2 2/3 and -6 2/5

By separating them as two parts

⇒ sum of -5/6 and -1 3/5  

- 5/6 + - 1 3/5 = - 5/6 + - 8/5  (∵ a b/c = (ac+b)/c(5+3)/5 = 8/5)

                      = (- 25 - 48)/30  (LCM = 30)

                      = - 73/30

⇒ sum of 2 2/3 and -6 2/5

2 2/3 + -6 2/5 = 8/3 + -32/5

                       = (40 - 96)/15  (LCM = 15)

                       = - 56/15

subtract the sum of -5/6 and -1 3/5 from the sum of 2 2/3 and -6 2/5

= (sum of 2 2/3 and -6 2/5) - (sum of -5/6 and -1 3/5)

= (- 56/15) - (- 73/30 )  

= - 56/15 + 73/30

= - 112/30 + 73/30   (LCM = 30)

= - 39/30

= - 13/10

= - 1.3

Hence - 1.3 is the number we get when we subtract the sum of -5/6 and -1 3/5 from the sum of 2 2/3 and -6 2/5.

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The perimeter of the triangle is 56.4 units.

The perimeter of the triangle can be found as follows:

The perimeter of a figure is the sum of the whole sides. Therefore, the perimeter of a triangle is the sum of the three sides.

perimeter of a triangle = x + y + z

where

Therefore,

perimeter of the triangle = 19.8 + (9.8 + 8.4) + (8.4 + (19.8 - 9.8))

perimeter of the triangle = 19.8 + 18.2 + 18.4

perimeter of the triangle = 56.4

perimeter of the triangle = 56.4 units

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The graph of the function f(x) = -(x+1)^2 shows that the domain of the function f(x) = -(x+1)² is: -∞ < x < ∞. The range of the function is f(x) ≤ 0.

The graph of a function is the arrangement of all ordered pairs of the function. Typically, they are expressed as points in a cartesian coordinate system. The graph of f is the collection of all ordered pairings (x, f(x)) such that x lies inside the domain of f.

The graph of a function might similarly be defined as the graph of the equation y = f(x). As a result, the graph of a function is a subset of the graph of an equation.

From the given information: the graph of the function f(x) = -(x+1)² can be determined if the domain, the range, and the vertex of the function are known.

Since the parabola curve from the graph shows that the graph is facing down, then the function is negative and decreasing.

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1.) maximum value

2.) for no values of x

3,) when x > -1

4.) all real numbers

5.) all numbers less than or equal to 0

The value of x from the given diagram is 20

According to theorem, the right triangle altitude theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.

Using the theorem above;

RT^2 = 9 * 16

RT^2 = 144

R = 12 units

Determine the value of x using the Pythagoras theorem;

x² =12² + 16²

x² = 144 + 256

x² = 400

Take the square root of both sides

x = √400

x = 20

Hence the value of x from the given diagram is 20

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Answer:

Okay, so:

sinA = 22.5/25.5

<A = 62* (correct)

cos C = 22.5/25.5 (correct)

tanA = 22.5/12.0 = 1.875 (correct)

sinC = 12.0/25.5

tan C = 12.0/22.5 = 0.53 (correct)

Step-by-step explanation:

I'll explain the wrong ones in the comments hold on-

The first integer is 5.

The second integer  is 10.

If an integer is higher than zero, it is positive; if it is lower than zero, it is negative. Zero can be either positive or negative. Since a b and c d, then a + c b + d, the ordering of integers is consistent with algebraic operations.

One integer is twice the other

so,

If one integer is x then

The other will be 2x.

Their product :

2x * x = 50

2x² = 50

x² = 50/2

x² = 25

x = √25

x = 5

the first integer is 5.

the second integer is 2x = 2 x 5

                                        = 10.

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Answer:

b/a

Step-by-step explanation:

Perpendicular lines have slopes that are opposite sign and reciprocals (flipped over).

In the equation

ax + by = c,

the slope of the line is

-a/b

If you haven't memorized this pattern yet, you can calculate it by solving ax+by=c for y.

by = -ax +c

y = -a/b x + c/b

The slope is -a/b

So a perpendicular line would be opposite sign and flipped, b/a

7.2 + c = 19

c = 19 - 7.2

c = 11.8

Answer:

Your answer is 5.

Step-by-step explanation:

7.2 + c = 19

or, 14 + c = 19

or, c = 19 - 14

or, c = 5       ans.

Hope its helpful :-)

Answer:

y-6= -5/3(x+2)

Step-by-step explanation:

the general way to write it is y-y1 = m(x-x1), so i plugged in the values

y1 = 6, x1 = -2

m = (y2-y1)/(x2-x1) = -5/3

Answer:

Step-by-step explanation:

First, you have to multiply. 15x4=60. All you need to do is add a zero to that product. You will get 600. Product means the answer to a multiplication problem. That is why you would have to multiply to get your answer. Your answer to that part is 600. 15*40= 600. Then, you would do 45+15= 60. 600/60= 10. Your answer is 10.

Answer:

  (d)  (4, -10)

Step-by-step explanation:

We are asked to identify the point that falls on a line defined by parametric equations.

The parametric equation can be written in a number of forms One of these is a vector form, in which some multiple of a vector is added to a known point.

  (x(t), y(t)) = (1/2t +4, 2t -10) = (4, -10) +(1/2t, 2t)

  (x, y) = (4, -10) +(t/2)(1, 4)

Clearly, when t=0, one point on the line will be (4, -10) — the last of the offered choices.

Another form of the equation can be had by solving the x(t) and y(t) equations for t, then equating those values.

  x = 1/2t +4   ⇒   2(x -4) = t

  y = 2t -10   ⇒   (y +10)/2 = t

Equating these gives ...

  2(x -4) = (y +10)/2 . . . . t = t

  4x -16 = y +10 . . . . . . . multiply by 2

  4x -y = 26 . . . . . . . . . add 16-y to put in standard form

Dividing by 26 gives intercept form:

  x/6.5 +y/(-26) = 1 . . . . . x-intercept: 6.5, y-intercept: -26.

This tells you the value of y will be negative for all x-values less than 6.5. All of the offered points have x-values less than that, so the only viable answer choice is (4, -10).

Answer:

(4.-10)

Step-by-step explanation:

When t = 0:

x(t)=1/2t+4

x(0)=1/2(0)+4

x(0) = 4

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

y(t)=2t−10

y(0)=2(0)−10

y(0) = -10

(x,y) at t = 0 is (4, -10)

The behavior sees the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

The behavior of the graph of the polynomial function f(x) as the variable x approaches either positive or negative infinity represents the end behavior of the function. The final behavior of a polynomial function's graph is determined by the degree of the function as well as the leading coefficient.

Generally, the equation for is zeros  mathematically given as

y=x^4−2x^3−11x^2+12x+36

Therefore

(x+2)^2(x-3)^2=0

x=-2

x=3

since the graph attached has a positive leading coefficient and a positive degree

In conclusion, If the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

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Angles are said to be produced whenever two or more straight lines intersect. Some types of angles are right angle, obtuse angle, acute angle, etc. Thus, the required proof is stated below:

Prove: <A > <C

           <B > <D

An obtuse angle is a given angle that is greater than [tex]90^{o}[/tex] but less than [tex]180^{o}[/tex]. While an acute angle is a given angle that is less than [tex]90^{o}[/tex].

Thus extend AB to F, such that BF = CD.

Draw a perpendicular from a to intersect CD at point E.

So that;

<BAD = <BAE + <DAE (addition property of an angle)

Such that;

<A is an obtuse angle, while <C is an acute angle. Thu,

<A > <C (different types of angles)

Also,

<ADC = <CDF - <ADF (subtraction property of angles)

But both angles B and D are acute angles.

So that;

<B > <C (comparison of measures of two angles)

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The measure of the side SQ from the given diagram is 19/6

Similar shapes are shapes that has equal length and equal side measures and angles.

From the given diagram, the measure of the sides TSis congruent to SP and the measure of MS is congruent to SQ.

Using the expression below to determine the value of x

MS/ST = SQ/SP

Given the following parameters

MS =30

ST = 10

SQ = 6x-1

SP =  3 - 2x

Substitute the given parameters into the formula to have:

30/10 = 6x-1/3-2x

Cross multiply

10(6x-1) = 30(3-2x)

Expand

60x - 10 = 90 - 60x

60x + 60x = 90 + 10

120x = 100

x = 10/12

x = 5/6

Determine the measure of SQ

SQ = 6x - 1

SQ = 5(5/6) - 1

SQ = 25/6  - 1

SQ = 19/6

Hence the measure of the side SQ from the given diagram is 19/6

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Answer: 42 and 31

Step-by-step explanation:

       Let x be one number and y be the other.

       We will write mathematical equations for these two numbers.

The sum of 2 numbers is 73.

x + y = 73

Their difference is 11.

x - y = 11

       See attached for the graph.

       The point of intersection is our solution.

Answer:

2.1 litres

Step-by-step explanation:

the 5 part of the ratio refers to 3.5 litres of red paint

divide this amount by 5 to find the value of one part of the ratio

3.5 litres ÷ 5 = 0.7 litres ← value of 1 part of the ratio , then

3 parts = 3 × 0.7 litres = 2.1 litres ← amount of blue paint used

With convolution theorem the equation is proved.

According to the statement

we have given that the equation and we have to evaluate with the convolution theorem.

Then for this purpose, we know that the

A convolution integral is an integral that expresses the amount of overlap of one function as it is shifted over another function.

And the given equation is solved with this given integral.

So, According to this theorem the equation becomes the

[tex]\mathscr{L} \left( \int_{0}^{t} e^{-\tau} \cos \tau d \tau \right) = \frac{ \mathscr{L} (e^{-\tau} \cos \tau ) }{s} \\\mathscr{L} \left( \int_{0}^{t} e^{-\tau} \cos \tau d \tau \right) = \frac{\frac{s+1}{(s+1)^2+1}}{s} \\\mathscr{L} \left( \int_{0}^{t} e^{-\tau} \cos \tau d \tau \right) = \frac{1}{s}\left (\frac{s+1}{(s+1)^2+1} \right).[/tex]

Then after solving, it become and with theorem it says that the

[tex]\mathscr{L} \left( \int_{0}^{t} f(\tau) d\tau \right) = \frac{\mathscr{L} ( f(\tau))}{s} .[/tex]

Hence by this way the given equation with convolution theorem is proved.

So, With convolution theorem the equation is proved.

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The number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16

The equation is given as:

a+b+c+d+e+f = 2006

In the above equation, we have:

Result = 2006

Variables = 6

This means that

n = 2006

r = 6

The number of solutions is then calculated as:

(n + r - 1)Cr

This gives

(2006 + 6 - 1)C6

Evaluate the sum and difference

2011C6

Apply the combination formula:

2011C6 = 2011!/((2011-6)! * 6!)

Evaluate the difference

2011C6 = 2011!/(2005! * 6!)

Expand the expression

2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006 * 2005!/(2005! * 6!)

Cancel out the common factors

2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/6!

Expand the denominator

2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/720

Evaluate the quotient

2011C6 = 9.12 * 10^16

Hence, the number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16

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The answer is 48π units³ or 150.72 units³.

To find the volume of the cone, use the formula : 1/3 × πr²h

We are given that r = 6 and h = 4.

Solving :

Answer:

Step-by-step explanation:

Convert 2y=-3x+2 into y=mx+b form

[tex]\frac{2y}{2} =\frac{-3x+2}{2}[/tex]

[tex]y=\frac{-3x}{2} +1[/tex]

Y intercept is 1, slope is -3/2

Graph looks like this:

Answer:

23 customers on payment plan B

13 customers on plan text B

Probability: 13/23

Step-by-step explanation:

Answer: 3.2

Step-by-step explanation: 3.2 is the answer because the number 7 is higher than 5. so the number 1 goes up by one, which that equals 3.2