A theme park guest takes an innertube into the wave pool. The innertube moves up and down in a way such that the height h of the innertube, in meters, above the floor of the pool can be modeled by the function h = asinbx + 5. When the wave pool is turned on, the innertube's height fluctuates between 2 meters and 8 meters, with an interval of 12 seconds between maximums. Based on the context of the problem, what are the values of a and b?

The net force on the cart is 62N

and the vertical component =36.64 N

the horizontal component= 50N

What is revolution of vector?

The process of splitting a vector into its components is called resolution of the vector. We resolve a vector into two components which are. component along the x-axis called horizontal component. component along the y-axis called vertical component

30N tension on the cart is westward which mean the vertical component will be 0 and the horizontal component will be 30N.

40N tension isN 30° E which means

the vertical component= 40 sin60=36.64N

Horizontal component= 40 cos 60= 20N

sum of vertical = 36.64N +0= 36.64N

sum of horizontal= 30N+20N= 50N

60° was used for tension 40N because the angle of vector must always lie on the horizontal axis

therefore the net force F is obtained by using Pythagoras theorem

F= √(50^2+36.64^2)

= 62N to the nearest whole number.

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The value of the variable 'x' in logarithmic expression is: 2

What is logarithm?

In mathematics, the term logarithmic expression can be explained as the exponent or the power which is raised on the base value of the logarithmic expression.

According to the question, the given logarithmic expression are as follows:

[tex]log_{1/2}x = -1[/tex]

Using standard logarithmic property, we get

x = (1/2)^-1

⇒x = 2

Hence, the logarithmic value of the variable 'x' is: 2

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Recall that the "discriminat" of a quadratic equation of the form:

a x^2 + b x + c = 0

is given by the formula: b^2 - 4 a c

So for our equation:

x^2 - 5 x = -2, we start by adding 2 to both sides, in order to get the appropriate form equal to zero:

x^2 - 5 x + 2 = 0

Now we recognize:

a = 1

b = - 5

c = 2

and write the discriminant

b^2 - 4 a c = (-5)^2 - 4 (1) (2) = 25 - 8 = 17

Then, the discriminat of the equation is: 17.

Equation of the line in slope-intercept form = y = 8x − 1

How to calculate the equation of the line?

Given:

Co-ordinates of the first line (x, y)=(0,-1), (1,7), (2,15), (3,23),(4,31)

We know that,

[tex]\text{slope }=m=\frac{y_2-y_1}{x_2-x_1}\\\\= > \frac{7+1}{1-0}\\\\=8[/tex]

The slope - intercept form of a line is given by,

y = mx + b    ---(1)

Let us substitute ( x, y )= (0, -1) in (1)

-1 =(8)*0+b

-1= b

b = - 1 ----(2)

Let us substitute m and b in the slope-intercept equation (1)

y = 8x − 1

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we have the following formula for the area of a sector of a circle:

In this case, we have the following information:

then, using the formula and solving for r, we get the following:

therefore, the measure of the radius is r = 30

If ∆ABC is congruent to ∆PQR, find the length of QR.

Given that If ∆ABC is congruent to ∆PQR:

The simplest way to prove that triangles are congruent is to prove that all three sides of the triangle are congruent.

When all the sides of two triangles are congruent, the angles of those triangles must also be congruent.

From the triangle ABC, the side BC = 4,

Hence the PQR, the side QR = 4 since they are congruent

Answer:21 to each of his 3 friends

Step-by-step explanation:

Answer

Option B is correct.

The latitude and longitude of the point X is 42.5°N, 77.5°W

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The total invested = $7000

Part of the money at 7% and the reminder at 15%

Let the part at 7% = x

So, the part at 15% = 7000 - x

Given the annual interest = $510

so,

7% of x + 15% of ( 7000 - x ) = 510

0.07 * x + 0.15 * ( 7000 - x ) = 510

0.07 x + 0.15 * 7000 - 0.15 x = 510

0.07 x + 1050 - 0.15x = 510

0.07 x - 0.15 x = 510 - 1050

-0.08 x = - 540

divide both sides by -0.08

x = -540/-0.08 = 6,750

7000 - x = 7000 - 6750 = 250

So, the part that is invested at a rate of 7% = $6,750

And the part that invested at a rate of 15% = $250

The augmented matrix is [tex]\begin{bmatrix}0&9&9&|&5\\7&3&-1&|&7\\5&7&0&|&5\end{bmatrix}[/tex].

What is a matrix?

A rectangular array or table containing numbers, symbols, or phrases that are arranged in rows and columns is known as a matrix.

The given system of equations is:

9y - 5 +9z = 0

7x = -3y +7 + z

5x + 7y = 5

Rewrite the equations as:

0x + 9y + 9z =  5

7x +3y -1z = 7

5x + 7y + 0z= 5

Therefore, the augmented matrix for the system of equations is:

[tex]\begin{bmatrix}0&9&9&|&5\\7&3&-1&|&7\\5&7&0&|&5\end{bmatrix}[/tex]

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We can show that one third + one fourth is not equal to 3/7 by following two ways.

We are given to prove that the one third and one forth combination of any object/value does not equals to the 3/7 of that same value/object.

let's Consider we have value as 1.

∴ one third of value will be [tex] \frac{1}{3} [/tex]

∴ one forth of value will be [tex] \frac{1}{4} [/tex]

∴ three of seven of value will be [tex] \frac{3}{7} [/tex]

Now we can do the mathematic calculation on one third and one forth of the value.

As given we have to calculate the addition of  one third and one forth of the value

∴ [tex] \frac{1}{3} + \frac{1}{4} [/tex]

= [tex] \frac{4}{12} + \frac{3}{12} [/tex] --(make base same )

= [tex] \frac{4\ + 3}{12}[/tex]

= [tex] \frac{7}{12}[/tex]

∴ By the calculation we clearly got that the addition of  one third and one forth of the value is equal to the seven of twelve of the value.

We can also prove it by analysis as follows ;

Consider a number say 120.

∴ 1/3 of 120 will be 40

∴ 1/4 of 120 will be 30

∴ 3/7 of 120 will be 51.42

Now Addition of 1/3 and 1/4 of 120 will be 40 + 30 = 70

Comparing it with 3/7 i.e 51.42 ≠ 70

Hence proved 1/3 + 1/4 ≠ 3/7 .

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If 2+[tex]\sqrt{3}[/tex] is a polynomial root, then the another root of the polynomial will be 2 -[tex]\sqrt{3}[/tex]

The root of the polynomial =  2 +[tex]\sqrt{3}[/tex]

We know by using the quadratic equation we can find the solution of the polynomial

The quadratic equation of the polynomial

= [tex]\frac{-b+/-\sqrt{b^2-4ac} }{2a}[/tex]

Then the root of the polynomial is [tex]\frac{-b+\sqrt{b^2-4ac} }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^2-4ac} }{2a}[/tex]

Here one root is given that 2 + [tex]\sqrt{3}[/tex]

Then using the quadratic equation of the polynomial, the other root of  the polynomial will be the conjugate of the given root.

The other root of the polynomial = 2 - [tex]\sqrt{3}[/tex]

Hence, If 2 + [tex]\sqrt{3}[/tex] is a polynomial root, then the another root of the polynomial will be 2 - [tex]\sqrt{3}[/tex]

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Given:

The functions

Required:

Determine changes made in g(x) to get f(x).

Explanation:

The steps are:

So. fourth option is describing the f(x).

The graph

Answer:

Completed the answer.

Answer:

Sample space = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C}

Event that the number chosen is 2 = {2A, 2B, 2C}

Explanation:

The sample space is the set with all the possible outcomes, if she can select any number from 1, 2, or 3 and she can select any letter of A, B, or C, we get that the sample space is

Sample space = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C}

Then, the outcome for the event that the select number is 2 is

Event that the number chosen is 2 = {2A, 2B, 2C}

Therefore, the answers are:

Sample space = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C}

Event that the number chosen is 2 = {2A, 2B, 2C}

Answer:

Triangle UVT and TSR are isosceles triangles (two equal sides).

Angle WUV = Angle WTV (base angles of isosceles triangle)

Angle RTS = Angle TRS (base angles of isosceles triangle)

Line VW is perpendicular to line UT.

Angle WTV = Angle RTS (vertically opposite angles)

Angle WTR = Angle VTS (vertically opposite angles)

The probability of the first sock being blue is 1/3 and the second sock being white is 1/4.

So,

Total number of socks: 10 + 8 + 6 = 24

Formula: P = favourable events/Total number of events

Probability of picking a blue sock first:

Probability of picking a white sock at second:

Therefore, the probability of the first sock being blue is 1/3 and the second sock being white is 1/4.

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The total loss after 4 years is $47840.

From the information, we want to calculate the total loss after 4 years of 299000 is reduced by 4% per annum.

It should be noted that this simply means the calculation using the interest formula. In this case, the total loss will be gotten by knowing the interest.

In this case, the interest formula can be illustrated as:

Interest = PRT / 100

where P = Principal = 299000

R = Rate = 4%

T = Time = 4 years

Interest = (299000 × 4 × 4) / 100

Interest = 4784000 / 100

Interest = 47840

In this case, the calculation of the interest is the loss that the person gets.

Therefore, the loss is $47840.

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Number..................Difference1........... Difference 2.....Difference 3

6...........................................................................................

1................................ -5......................................................

0.............................. -1....................................4..............

10.............................. 10..................................11......................7

38.............................. 28................................18..................... 7

91.............................. 53................................25................... 7

176............................85..............................32

Answer:

Next number in sequence 176

Solution

For the solid line we can select two points:

(0,2) and (1,0)

We can calculate the slope and we have:

Then we can find the intercept and we got:

0= -2*1+ b

b =2

Then the equation is:

y= -2x +2

For the dashed line we can select (0,1) and (1,2) then the slope is:

The intercept is:

2= 1*1 +b

b=1

Then the equation is:

y= x +1

Part 1

The y-intercept is when x=0, and [tex]f(0)=(0-7)(0+6)(0-5)=210[/tex]. So, the y-intercept is at (0, 210).

Part 2

The x-intercept(s) are when y=0, so:

[tex]f(x)=0\\\\(x-7)(x+6)(x-5)=0\\\\x-7=0, x+6=0, x-5=0\\\\x=-6, 5, 7[/tex]

So, the x-intercepts are: (-6, 0), (5, 0), (7, 0)

Using our roots from Part 2, we know that:

Part 3: [tex](-6, 5) \cup (7, \infty)[/tex]

Part 4: [tex](-\infty, -6) \cup (5, 7)[/tex]

Hello there. To solve this question, we'll have to remember some properties about the pythagorean theorem and right triangles.

Given a triangle with side lengths 10 m, 24 m and 26 m, we have to determine whether this is a right triangle or not.

First, let's assume these are the sides lengths of a right triangle, as follows:

Notice in this case we choose the largest side to be the hypotenuse. We know that the largest side of a triangle has to be less than the sum of the legs and any combination would do, but since we'll apply the Pythagorean theorem, we already know that the hypotenuse has to be the largest side of the triangle.

For a right triangle with legs a, b and hypotenuse c, the Pythagorean theorem says that the sum of the squares of the legs is equal to the square of the hypotenuse, that is

Hence we plug a = 10, b = 24 and c = 26.

We have to see if this equality will hold for the values;

On the right hand side, square the number

On the left hand side, square and add the numbers

Notice we got the same answer, then we say that these values satisfy the Pythagorean theorem.

Therefore, this is a right triangle.

1) 1/6

3) -7/6

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With the use of factorization, we can state that s and r have values of -2 and 3, respectively.

The given values are

Solving with the help of factotisation.

Dividing the eqn by 6

hence,from above solving we can say that values of s and r are: 2,3

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Answer:

no

Step-by-step explanation:

because 3*2 isnot equal to13

Determine the amount paid by the Tony overall.

At first ,we should know that :

The relation between x and y will be a function when each value of x appeared one time

We will check the options:

A. for the figure of option A, each value of x connected one time to a value of y

B . there are 2 arrows from 0 and 1

C. the values of x = -1 , connected four times to the value of y

So, the answer is option A

⇒g(f(x)) means that in the function g(x) insert/plug in what f(x) is equal to in the place of x. f(x)= 3x+2 meaning in the place of x in g(x) insert 3x+2 and simplify.

[tex]g(f(x))=-2(3x+2)+4\\g(f(x))=-6x-4+4\\g(f(x))=-6x[/tex]

Attached is the solution of what g(f(x)) is equal to.

Explanation

Step 1

the distance between 2 points P1 and P2 is given by:

Step 2

let

P1=M(4,0)

P2=N(-2,-3)

replace

I hope this helps you