Match the system of equations to their solutions

Final answer:

To find the height of the ball on the 10th bounce, an explicit formula that accounts for the initial conditions and the coefficient of restitution is needed. However, without specific information, we cannot calculate the exact height. Each bounce would generally be lower than the previous due to energy losses.

Explanation:

Finding the Height After 10th Bounce

The question of finding the height of a ball on the 10th bounce involves understanding the kinematic equations of motion. However, the question seems to be missing explicit information to compute the height, such as the initial height, velocity, and the coefficient of restitution that would tell us how the bounce height decreases with each bounce.

Assuming that a standard formula governing the bounce height is given, for example, height of nth bounce = initial height × (coefficient of restitution)ⁿ⁻¹ , we could insert the values to find the height of the 10th bounce.

Since we don't have the specifics needed, we cannot calculate the exact height. For an exact answer, we would need additional information on the initial conditions and the physical properties of the ball and the surface.

However, based on general physics principles, each bounce would be lower than the previous one because some energy is lost with each impact due to factors like air resistance and the inelastic collision with the floor.

Answer:

1/2

Step-by-step explanation:

We have [tex]\cos(\arccsc(\frac{2\sqrt{3}}{3}))[/tex].

Let [tex]u=\arccsc(\frac{2\sqrt{3}}{3})[/tex].

This implies [tex]\csc(u)=\frac{2\sqrt{3}}{3}[/tex].

Use that sine and cosecant are reciprocals.

[tex]\sin(u)=\frac{3}{2\sqrt{3}}[/tex]

Now I'm going to rationalize the denominator there by multiply numerator and denominator by [tex]\sqrt{3}[/tex]:

[tex]\sin(u)=\frac{3\sqrt{3}}{2(3)}[/tex]

[tex]\sin(u)=\frac{3\sqrt{3}}{6}[/tex]

Reduce the fraction:

[tex]\sin(u)=\frac{\sqrt{3}}{2}[/tex]

Now I'm going to use a Pythagorean Identity: [tex]\cos^2(u)+\sin^2(u)=1[/tex].

This will give me the value of cos(u) which would give me the answer to my question if it exists.

Replace [tex]\sin(u)[/tex] with [tex]\frac{\sqrt{3}}{2}[/tex] in:

[tex]\cos^2(u)+\sin^2(u)=1[/tex]

[tex]\cos^2(u)+(\frac{\sqrt{3}}{2})^2=1[/tex]

[tex]\cos^2(u)+\frac{3}{4}=1[/tex]

Subtract 3/4 on both sides:

[tex]\cos^2(u)=\frac{1}{4}[/tex]

Square root both sides:

[tex]\cos(u)=\pm \frac{1}{2}[/tex] (since 1/2*1/2=1/4 or -1/2*-1/2=1/4)

Now we must decide between the positive or the negative.

It depends where u lies.  What quadrant? Hopefully it lays between 0 and [tex]\pi[/tex].  Otherwise, it doesn't exist (unless you have a different definition for arc function).

So u led to this equation earlier:

[tex]\sin(u)=\frac{\sqrt{3}}{2}[/tex]

arcsin( ) only has outputs between [tex]\frac{-\pi}{2}[/tex] and [tex]\frac{\pi}{2}[/tex].

This would have to be in the first quadrant because we have only positive sine values there.

So this means cos(u)=1/2 and not -1/2 because we are using that u is in the 1st quadrant.

Remember u was [tex]\arccsc(\frac{2\sqrt{3}}{3})[/tex].

So we have actually evaluated

[tex]\cos(\arccsc(\frac{2\sqrt{3}}{3}))[/tex] without a calculator.

The value is 1/2.

Answer:

5/9

Step-by-step explanation:

F(-2)for f(x)=5•3^x

Let x = -2

f(-2)=5•3^(-2)

      = 5 * 1/3^2

     = 5 * 1/9

     = 5/9

Answer:

f(-2) = 5/9

Step-by-step explanation:

* lets explain the problem

∵ f(x) = 5(3)^x

- It is an exponential function

- (3) is the base of the function

- x is the exponent

- To find f(-2) means substitute x by -2

∵ [tex]f(x)=5(3)^{x}[/tex]

∵ x = -2

∴ [tex]f(-2)=5(3)^{-2}[/tex]

- If the power of the base is negative we can change its sign to

  positive by reciprocal the base

# Ex: [tex](a)^{-n}=(\frac{1}{a})^{n}[/tex]

- Lets do that withe the base 3 and power -2

∵ [tex](3)^{-2}=(\frac{1}{3})^{2}[/tex]

∴ [tex]f(-2)=5(\frac{1}{3})^{2}=5(\frac{1}{9})=\frac{5}{9}[/tex]

* f(-2) = 5/9

Answer:

This is an obtuse triangle

Step-by-step explanation:

Pythagoras theorem is used to determine if a triangle is right, acute or obtuse

If the sum of squares of two shorter lengths is greater than the square of third side then the triangle is an acute triangle.

If the sum of squares of two shorter lengths is less than the square of third side then the triangle is an obtuse triangle.

If the sum of squares of two shorter lengths is equal the square of third side then the triangle is a right triangle.

so,

[tex](41)^2 = (36)^2+(9)^2\\1681 = 1296+81\\1681>1377[/tex]

As 1681>1377, the given triangle is an obtuse triangle ..

Answer:

3/10

Step-by-step explanation:

Step 1 : Write the formula of calculating probability

Probability = Number of favourable outcomes/Number of total outcomes

Step 2 : Identify the favourable (requires) outcomes and the total outcomes

       12, 14, 18 = 3 cards

Step 3 : Substitute the values in the formula

Probability = Number of favourable outcomes/Number of total outcomes

Probability = 3/10

Therefore, the probability of picking a card with and even number is 3/10.

!!

Answer:

9:36 p.m.

Step-by-step explanation:

10 am + 12 hours = 10:00 pm

2 minutes every hour

12 hours x 2 = 24 minutes

12*2=24

10:00+12:00-0:24=21:36

The slope is the change in Y over the change in x:

Using the two points shown on the graph:

(1,7) and (0,0)

Slope = (7-0) / (1-0)

Slope = 7/1

Slope = 7

The number if front of the x would be the slope

The answer would be D. y = 7x

Answer:

9

Step-by-step explanation:

The distance between - 14 and -

5 on a number line is 9.

-14 - (-5) = 9

Answer:

starts wuth 46, and sells half, so she has 23. then when she buys 17 more she then has 40.

if you are asked to show work its 46/2=23, then 23+17=40!

Answer:

40 stamps

Step-by-step explanation:

She started with 46 stamps.

She sold half of them

46*1/2 = 23

She sold 23 46-23 = 23

So she has 23 stamps

Then she bought 17

23+17 = 40

She now has 40 stamps

Answer:

The equation is y=(-1/7)x+-1 or y=(-1/7)x-1.

Step-by-step explanation:

slope-intercept for is y=mx+b where m is the slope and b is the y-intercept.

You are given (7,-2) and (0,-1).

Line them up and subtract vertically then put 2nd difference over 1st difference:

( 7, -2)

-(0 , -1)

-----------

7      -1

So the slope is -1/7.

Now we know our equation is in the form

y=(-1/7)x+b.

Use one of the points you are given along with y=(-1/7)x+b to find b.

I'm choosing (0,-1) to plug in for (x,y):

-1=(-1/7)(0)+b

-1=0+b

-1=b

So the equation is y=(-1/7)x+-1 or y=(-1/7)x-1.

y = -1/7x - 1

In this question, we're trying to find the slope-intercept form with the information that is given.

Slope intercept form is represented as y = mx + b

In this case, we know that the points are at:

With the information above, we can solve the problem.

In order to find the slope, we would use the slope equation.

Slope equation:

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

To use this slope equation, we would need to plug in the points from the coordinates into the equation. Your equation should look like this:

[tex]m=\frac{-1--2}{0-7}[/tex]

Now, you solve to find the slope of the line.

[tex]m=\frac{-1--2}{0-7} =\frac{1}{-7} =-\frac{1}{7}[/tex]

When you're done solving, you should get -1/7.

This means that the slope of the line is -1/7. We would plug 9 in our "m" variable.

Your slope intercept form should look like this:

y = -1/7x + b

For our "b" variable, it's going to be the beginning point. When we look at our (0,-1) coordinate, we would know that -1 would be our beginning point, sicne thje 0 is on the x and there's a variable for the y.

Your slope intercept form should look like this:

y = -1/7x - 1

This means that the slope intercept form of the line is y = -1/7x - 1

Answer:

Step-by-step explanation:

Permutation is the ways in which a fixed group or members that can be arranged or ordered and combinations are used when a smaller group has to be chosen from a larger group.

Examples:

Permutation: How many unique combinations of the word MAHNOOR can be formed when the letters will be scrambled

2. In what order or arrangements five people can be seated in the front row?

The number of people are fixed, we have to find the order.

Combination: Choosing two cards from a deck of 52 cards. We are choosing a small group from a larger group of all cards.

2. Choosing 4 students from a class to take part in the competition.

The selection can be made in multiple ways ..

Answer:

Permutation - requires to arrange order with quantities

Combination - order is not required

Step-by-step explanation:

If a question demands you to select as well as arrange the given quantities then it means that it is a problem of permutation.

While on the other hand, a problem of combination would only ask you to select the quantities and not their order.

So with practice, you get to know how exactly it works.

Answer:

Slope = 2/3 Y intercept +3

Step-by-step explanation:

Plot the y-intercept (0,+3) in the xy axis. Remember, this point always lies on the vertical axis y.

Starting from the y-intercept, find another point using the slope. Slope contains the direction how you go from one point to another.

The numerator tells you how much steps to go up or down (rise) while the denominator tells you how many units to move left or right (run).

Connect the two points generated by the y-intercept and the slope using a straight edge (ruler) to reveal the graph of the line.

Answer:

Step-by-step explanation:

You find positive 3 and you put a dot on it, you begin to go up 2 then 3 to the right and put a point. Now connect those two dots and continue until you have no more space.

[tex](x-(-2))^2+(y-1)^2=3^2\\(x+2)^2+y^2-2y+1-9=0\\x^2+4x+4+y^2-2y-8=0\\x^2+y^2+4x-2y-4=0[/tex]

Answer:

x²+4x+y²−2y−4=0

Step-by-step explanation:

we can observe that,

1) the coordinates of the center of the circle (h,k) = (-2,1)

2) the radius, r = 3

using the standard form of the equation of circle:

(x−h)²+(y−k)²=r²  (substitute h= -2, k=1 and r=3), we get

(x+2)²+(y−1)²=9  ----> this is in "Standard" form, to get "General" form, simply expand the parenthesis reduce to simplest terms.

(x+2)²+(y−1)²=9

[x² + (2)(x)(2) + 2²] + [y² + (2)(y)(-1) + (-1)²] = 9 (expand and reduce)

x²+4x+y²−2y−4=0 (answer)

Answer: I believe 65 days will pass before be practices both the cello and piano again.

The number of days will pass before should be 65.

Mike practices the cello every 5 days and the piano every 13 days.

[tex]= 13 \times 5[/tex]

= 65 days

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Answer:

0.0268.

Step-by-step explanation:

This question can be solved using the binomial theorem.

Total attempts = n = 11

Required attempts = r = 5

Probability of success = p = 3/4 = 0.75

Binomial Theorem formula:

P(X=r) = nCr * p^r * (1-p)^(n-r).

Substituting the values:

P(X=5) = 11C5 * 0.75^5 * 0.25^6 = 0.0268 (to the nearest 3 significant figures).

So the probability that exactly 5 free throws are made out of 11 is 0.0268!!!

Answer:

The graph in the attached figure

Step-by-step explanation:

Let

x ----> the number of open acres

y ----> the number of developed acres

we know that

[tex]x+y \geq 4[/tex] -----> inequality A

The solution of the inequality A is the shaded area above the solid line [tex]x+y=4[/tex]

[tex]y-x\leq 1[/tex] -----> inequality B

The solution of the inequality B is the shaded area below the solid line [tex]y-x=1[/tex]

so

The graph in the attached figure

Answer:

Option B.

Step-by-step explanation:

Let x be the number of open acres and y be the number of developed acres.

It is given that a community must have at least 4 acres of developed and open space.

[tex]x+y\geq 4[/tex]

The difference between the number of developed acres, y, and the number of open acres, x, can be no more than 1.

[tex]y-x\leq 1[/tex]

The relative equations of both inequalities are

[tex]x+y=4[/tex]

[tex]y-x=1[/tex]

The table of values

For line 1                      For line 2

x      y                            x      y

0     4                            0      1

4     0                           -1      0

Plot these ordered pairs on a coordinate plane and draw both lines.

The related line of first inequality is a solid line and shaded area lies above the line because the sign of inequality is ≥.

The related line of second inequality is a solid line and shaded area lies below the line because the sign of inequality is ≤.

Therefore, the correct option is B.

[tex]\text{Use pythagorean theorem}[/tex]

[tex]$a^2 + b^2 = c^2 \longrightarrow a^2 +6^2=9^2 \longrightarrow a^2 =45\longrightarrow a =6.7082\longrightarrow a = 6.71$[/tex]

[tex]\textbf{a = 6.71}[/tex]

This is a right triangle and to solve this you must use Pythagorean theorem:

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

a and b are the legs (the sides that form a perpendicular/right angle)

c is the hypotenuse (the side opposite the right angle)

In this case...

a = 6

b = x

c = 9

^^^Plug these numbers into the theorem

[tex]6^{2} +x^{2} =9^{2}[/tex]

simplify

36 + [tex]x^{2}[/tex] = 81

Now bring 36 to the right side by subtracting 36 to both sides (what you do on one side you must do to the other). Since 36 is being added on the left side, subtraction (the opposite of addition) will cancel it out (make it zero) from the left side and bring it over to the right side.

36 - 36 + [tex]x^{2}[/tex] = 81 - 36

0 + [tex]x^{2}[/tex] = 45

[tex]x^{2}[/tex] = 45

To remove the square from x take the square root of both sides to get you...

x = √45

x ≈ 6.71

(choice A)

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

A (1,-5)

Step-by-step explanation:

the photo has the explanation

The solution of the system is x=1 and y= -5.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

3x+10y= -47..........(1)

5x-7y=40.............(2)

Solving equation (1) and (2) we get

15x + 50y = -235

15x - 35y = 200

____________

        85y = -435

         y = -5.11

and, 3x + 10(-5.11) = -47

3x -50 = -47

3x= 3

x= 1

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Answer: x=2/e^2

Step-by-step explanation:

Answer:

2a) [tex]g(x)=3x-6[/tex]

2b) This is a show kind of answer. The showing of this in the explanation.

2c) I provided the graph.  The lines should be a reflection through the y=x line.  The points (a,b) on y=(1/3)x+2 or swapped to get the points (b,a) on y=3x-6. That is for, example the point (-3,1) is on y=(1/3)x+2 while (1,-3) is on y=3x-6.

Step-by-step explanation:

2a) The inverse of a function is you just swapping x and y around. You also almost always asked to remake y the subject after that though.

So anyways we have this equation [tex]y=\frac{1}{3}x+2[/tex] to represent that function you have there.

We want to swap x and y:

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

Now we want to solve for y.

Subtract 2 on both sides:

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

Multiply both sides by 3:

[tex]3(x-2)=y[/tex]

[tex]y=3(x-2)[/tex]

Distribute:

[tex]y=3x-6[/tex]

So they want us to name the inverse g(x).

[tex]g(x)=3x-6[/tex]

2b) We want to show by composition that these functions are inverses.  That is we want to show f(g(x))=x and g(f(x))=x.

Let's do that:

f(g(x))

Replace g(x) with 3x-6 since g(x)=3x-6.

f(3x-6)

Replace the old input x with the new input (3x-6) in (1/3)x+2.

[tex]\frac{1}{3}(3x-6)+2[/tex]

Distribute:

[tex]\frac{3x}{3}-\frac{6}{3}+2[/tex]

Simplify:

[tex]x-2+2[/tex]

[tex]x-0[/tex]

[tex]x[/tex].

So we do have f(g(x))=x.

Now to show the other way:

g(f(x))

Replace f(x) with (1/3)x+2 since f(x)=(1/3)x+2.

g((1/3)x+2)

Replace the old input x with the new input (1/3)x+2 in 3x-6.

3((1/3)x+2)-6

Distribute:

[tex]3(\frac{1}{3})x+3(2)-6[/tex]

Simplify:

[tex]1x+6-6[/tex]

[tex]x+6-6[/tex]

[tex]x+0[/tex]

[tex]x[/tex]

So we do have g(f(x))=x.

We have confirmed that f and g are indeed inverses since f(g(x))=x and g(f(x))=x.

2c) Visually if two functions are inverses they should be reflections through the y=x line so that is what we should see since f and g are inverses.

I going to compare both equations to y=mx+b form to determine the y-intercept and the slope.

y=mx+b

y=(1/3)x+2 tells us the slope is 1/3 and the y-intercept is 2.

y=3x-6 tells us the slope is 3 and the y-intercept is -6.

I have color-coded the picture.

Answer:

16 x^6

Step-by-step explanation:

(4x^3)^2

Writing this expression out

(4x^3)(4x^3)

4 *4 x^3 *x^3

We know that a^b* a^b = a^(b+b)

16 x^(3+3)

16 x^6

Angle Q = 93° & angle R = 87°

The sum of any two adjacent angles of a parallelogram is equal to 180°, i.e. they are supplementary angles.

In parallelogram QRST,

∠Q= (9x-6)° & ∠R= (8x-1)°

Here, angle Q & angle R are adjacent angles of the parallelogram QRST.

∴ ∠Q+∠R = 180°

⇒ (9x-6)+(8x-1)= 180

⇒ 9x-6+8x-1 = 180

⇒ 17x-7= 180

⇒ 17x = 180+7

⇒ 17x = 187

⇒ x = 187/17

⇒ x = 11

So, ∠Q= (9x-6)° = {(9×11)-6}° = (99-6)° = 93°

& ∠R= (8x-1)° = {(8×11)-1}° = (88-1)°= 87°

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let's recall that a circular clock has 360° and 60 seconds per minute.

the seconds hand does 60 seconds in one go-around, and if it has covered 15 seconds on the clock, that gives us an angle of 90°, 15 is one quarter of 60, so in 15 seconds it has covered one quarter of the full circle or 90°.

we also know it covered 2cm, namely the arc's length on the 15 seconds is 2cm.

[tex]\bf \textit{arc's length}\\\\ s=\cfrac{\theta \pi r}{180}~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\ \cline{1-1} \theta =90\\ s=2 \end{cases}\implies 2=\cfrac{(90)\pi r}{180}\implies 2=\cfrac{\pi r}{2} \\\\\\ 4=\pi r\implies \cfrac{4}{\pi }=r\implies 1.27\approx r[/tex]

Answer:

B

Step-by-step explanation:

a root of a polynomial function is a value of the variable that makes the polynomial equal to zero.  

A root of a polynomial function is,

⇒ A value of the variable that makes the polynomial equal to zero.

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

A root of a polynomial function.

We know that;

The root of a polynomial function is satisfy the equation.

That's mean;

A value of the variable that makes the polynomial equal to zero is called root of the function.

Therefore, A root of a polynomial function is,

⇒ A value of the variable that makes the polynomial equal to zero.

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Answer:

B) 1,231 in3

Step-by-step explanation:

V = pi * r^2 * h

V = 3.14 * (14 in.)^2 * 2 in.

V = 1230.88 in.^3

Answer: B) 1,231 in3

Answer:

B) 1,231 in3  

Step-by-step explanation:

V = pi * r^2 * h

V = 3.14 * (14 in.)^2 * 2 in.

V = 1230.88 in.^3

= 1,231 in3  

i took geometry hope this helps

Answer:

B. See explanation

Step-by-step explanation:

Use the distance formula between two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2):[/tex]

[tex]d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}[/tex]

Find the lengths of all sides of quadrilateral DEFG:

[tex]DE=\sqrt{(-a-b-0)^2+(c-2c)^2}=\sqrt{(a+b)^2+c^2}\\ \\

EF=\sqrt{(a+-b-0)^2+(c-2c)^2}=\sqrt{(a+b)^2+c^2}\\ \\

FG=\sqrt{(a+b-0)^2+(c-0)^2}=\sqrt{(a+b)^2+c^2}\\ \\

GD=\sqrt{(-a-b-0)^2+(c-0)^2}=\sqrt{(a+b)^2+c^2}\\ \\[/tex]

All sides are of the same length. Now fond the slopes of all sides:

[tex]DE=\dfrac{2c-c}{0-(-a-b)}=\dfrac{c}{a+b}\\ \\EF=\dfrac{c-2c}{a+b-0}=-\dfrac{c}{a+b}\\ \\FG=\dfrac{c-0}{a+b-0}=\dfrac{c}{a+b}\\ \\GD=\dfrac{c-c}{-a-b-0}=-\dfrac{c}{a+b}\\ \\[/tex]

The slopes of the sides DE and FG are the same, so these sides are parallel. The slopes of the sides EF and GD are the same, so these sides are parallel.

Answer:

2/5

Step-by-step explanation:

probability of picking a quarter or a penny = (combined number of quarters and pennies)/(total number of coins)

combined number of quarters & pennies = 6 + 2 = 8

total number of coins = 20

probability of picking a quarter or a penny = 8/20 = 2/5

The required probability that Joseph selects a quarter or a penny from his pocket is 4/5

According to the problem,

There are 6 quarters and 2 pennies in his pocket which sum up to 8 coins.

This are actually the favorable cases.

So the required probability will be 8/20 = 4/5

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Answer:

3x²-10x+8

Step-by-step explanation:

To simplify the expression we need to multiply each term in the first bracket with the expression in the second bracket.

x(3x-4)-2(3x-4)

3x²-4x-6x+8

Executing the operations to give a quadratic expression we get:

3x²-10x+8

The simplified expression is therefore

3x²-10x+8

Answer:  [tex]3x^2-10x+8[/tex]

Step-by-step explanation:

Given the expression [tex](x-2)(3x-4)[/tex], you can apply Distributive property to simplify it. But first, you must remember the multiplication of signs:

[tex](+)(+)=+\\(-)(-)=+\\(-)(+)=-[/tex]

Then:

[tex](x-2)(3x-4)=(x)(3x)+(x)(-4)+(-2)(3x)+(-2)(-4)=3x^2-4x-6x+8[/tex]

The final step is to add the like terms. Therefore, you get:

[tex]3x^2-10x+8[/tex]

Answer:

C. 3.131131113…

Step-by-step explanation:

Irrational numbers cannot be written as a ratio of two integers and irrational numbers can have decimal that goes on forever without repeating

C. 3.131131113…

It's a decimal which is not repeating, therefore it's an irrational number.