simplify e^ln3x please​

Answer:

the answer is negative infinity to infinity because it's all real numbers.

The domain of g(x) is the set of all real numbers.

A real number is a value of a continuous quantity that can represent a distance along a line.

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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:

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:

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:

[tex]\large\boxed{a(V)=\sqrt{\dfrac{3V}{8}}}[/tex]

Step-by-step explanation:

The formula of a volume of a square pyramid:

[tex]V=\dfrac{1}{3}a^2h[/tex]

a - base edge

h - height of a pyramid

We have H = 8in.

Substitute and solve for a:

[tex]\dfrac{1}{3}a^2(8)=V\\\\\dfrac{8}{3}a^2=V\qquad\text{multiply both sides by}\ \dfrac{3}{8}\\\\\dfrac{3\!\!\!\!\diagup^1}{8\!\!\!\!\diagup_1}\cdot\dfrac{8\!\!\!\!\diagup^1}{3\!\!\!\!\diagup_1}a^2=\dfrac{3}{8}V\\\\a^2=\dfrac{3V}{8}\Rightarrow a=\sqrt{\dfrac{3V}{8}}[/tex]

Answer:

Answer:

Step-by-step explanation:

The formula of a volume of a square pyramid:

a - base edge

h - height of a pyramid

We have H = 8in.

Substitute and solve for a:

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Step-by-step explanation:

Answer: 54 bags; equation is 51=213-3x

Step-by-step explanation:

If you divid 213 by 3, you see it’s divisible and equals 71. 51 divided by 3 is 17. 71 minus 17 is 54.

Answer:

x < 54 (bags)

Step-by-step explanation:

213 - 3x < 51, or  

162 - 3x < 0, or

162 > 3x, or, finally,

162

----- > x, or x < 54 (bags)

 3

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:

Volume: 45000cm^3

Surface area: 6600cm^2

Step-by-step explanation:

The volume is all the side lengths multiplied, so 60*25*30 = 45,000 cm^3.

Now let's calculate the surface area.

The bottom is the width * the length, or 60*25 cm^2.

One of the sides is the length * the height, or 60*30 cm^2. This is multiplied by 2 because there are 2 of those sides.

The other pair of sides is the width * the height, or 25*30 cm^2, again multiplied by 2.

In total, the surface area is

60*25+2*60*30+2*25*30 = 6600cm^2

To fill the aquarium to the top with water, it will take 7800 cubic centimeters (cm³). To make the aquarium, it will take 7800 square centimeters (cm²) of glass.

Find the volume of the aquarium: Since there is no top to the aquarium, we will consider the volume as the combined volume of the five faces of the rectangular prism (4 faces with dimensions Length x Width and 1 face with dimensions Length x Height).

Volume = 4 * (Length * Width) + (Length * Height)

Volume = 4 * (60 cm * 25 cm) + (60 cm * 30 cm)

Volume = 4 * 1500 cm² + 1800 cm²

Volume = 6000 cm² + 1800 cm²

Volume = 7800 cm³

So, it will take 7800 cubic centimeters (cm³) of water to fill the aquarium to the top.

Find the surface area of the aquarium: To find the surface area, we will consider the area of each face and add them together.

Area of the 4 faces with dimensions Length x Width:

4 * (60 cm * 25 cm) = 4 * 1500 cm² = 6000 cm²

Area of the face with dimensions Length x Height:

(60 cm * 30 cm) = 1800 cm²

Total Surface Area = 6000 cm² + 1800 cm² = 7800 cm²

It will take 7800 square centimeters (cm²) of glass to make the aquarium.

Answer:

49.6mL

Step-by-step explanation:

Given parameters:

Volume of solution = 386mL

Percentage composition of acid = 12.86%

Solution

The solution is made up of water and acid in their respective proportions.

We have been given the percentage composition of acid to be 12.86%, so we can easily obtain the exact volume of the acid in the compositon.

Volume of acid in solution = perenctage of acid x volume of solution

Volume of acid = [tex]\frac{12.86}{100}[/tex] x 386mL = 49.6396mL

Rounding to the nearest tenth gives: 49.6mL

Answer:

It increased by 94 percent or 94%.

Answer:

162% ( to the nearest percent )

Step-by-step explanation:

The percentage change is calculated as

[tex]\frac{change}{original}[/tex] × 100%

Change = 152 - 58 = 94, so

percent change = [tex]\frac{94}{58}[/tex] × 100% = [tex]\frac{94(100)}{58}[/tex] = 162%

Answer:

6^ (1/12)

Step-by-step explanation:

6 ^ 1/3

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

6 ^ 1/4

x^a / x^b = x^ (a-b)

6^(1/3-1/4)

Getting a common denominator)

6^(4/12-3/12)

6^1/12

Answer:

6^1/12

Step-by-step explanation:

You can rewrite the numbers as 6^1/3 and 6^1/4. Now we need common denominators on the powers. You can make the 1/3 into 4/12 and the 1/4 into 3/12. When you divide powers, you just subtract the numerators for each fraction. So you do 4-3. This leaves you with 6^1/12. Hope this helps :)

[tex]8v-6j-10v+8j =2j-2v[/tex]

Answer:

A given point on a line graph represents a data point with the value of the responding variable as the height of the point at the given value of the manipulated variable on the horizontal axis.

Step-by-step explanation:

A given point in a line graph signifies a specific value of an ordered pair of variables. The x-coordinate represents one variable, and the y-coordinate another. It essentially reflects the relationship between these two variables at a particular time.

In mathematics, specifically in the field of graphs and data visualization, a given point on a line graph represents a specific value for an ordered pair of variables. The horizontal coordinate (x-value) stands for one variable, while the vertical coordinate (y-value) represents another variable.

For example, if we're graphing a student's test scores over time, the time period will be represented on the x-axis, and the test score would be represented on the y-axis. If we have a point at coordinates (3, 85), then that means the student scored 85 on their test in the third time period.

This single point, therefore, is a representation of the interaction or relation between these two variables at a particular interval.

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Answer:

see below

Step-by-step explanation:

Y=x^2

x=-3            -2           -1       0       1       2        3

y=(-3)^2    (-2)^2    (-1)^2    0^2    1^2    2^2   3^2

       9           4           1         0        1         4       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:

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:

B

Step-by-step explanation:

The question has no division answer. That makes C and D incorrect.

A for some reason is backwards. It makes no sense to use that. You are left with B. The reason A or B  is correct is that the burn of calories must get larger with an increase in hours.

Answer:

[tex]7\frac{15}{16}[/tex]

Step-by-step explanation:

[tex]8-\frac{1}{16}=\frac{8}{1}-\frac{1}{16}=\frac{128}{16}-\frac{1}{16}=\frac{127}{16}=7\frac{15}{16}[/tex]

Hi there!

Simplfly 8:

[tex]8= 7 \frac{16}{16}[/tex]

Now subtract:

[tex]7-0=7[/tex]

[tex]16-1=15[/tex]

Denominator stays the same so the answer would be 7 15/16.

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:

The value of the discriminate is -27 and there are 2 complex roots

Step-by-step explanation:

* Lets explain what is discriminant

- The form of the quadratic equation is y= ax² + bx + c

- The roots of the equation is the values of x when y = 0

- There are three types of roots:

# Two different real roots

# One real root

# No real roots or two complex roots

- We can know the types of roots of the equation without solve it by

 using the discriminant which depends on the value of a , b , c

- The discriminant = b² - 4ac, where a is the coefficient of x² , b is the

  coefficient of x and c is the numerical term

# If b² - 4ac > 0, then there are two different real roots

# If b² - 4ac = 0, then there is one real root

# If b² - 4ac < 0, then there is no real root (2 complex roots)

* Lets solve the problem

∵ x² + x + 7 = 0

∴ a = 1 , b = 1 , c = 7

∵ The discriminant = b² - 4ac

∴ The discriminant = (1) - 4(1)(7) = 1 - 28 = -27

∵ -27 < 0

∴ There is no real solution there are two complex roots

* The value of the discriminate is -27 and there are 2 complex roots

Answer:

The number of roots are 2 and type of roots is complex

Step-by-step explanation:

Points to remember

Discriminant of a quadratic equation ax² + bx + c = 0

x = b² - 4ac

To find the discriminant of the given equation

Here quadratic equation be x² + x + 7 = 0

a = 1, b = 1 and c = 7

discriminant =  b² - 4ac

 =  1² - (4 * 1 * 7)

 = 1 - 28

 = -27

To find number and type of roots

Here discriminant  is negative

Therefore the number of roots are 2 and type of roots is complex

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

Answer:

[tex]b=6.8\ units[/tex]

Step-by-step explanation:

we know that

Applying the law of sines in the triangle ABC

[tex]\frac{AB}{sin(C)}=\frac{AC}{sin(B)}[/tex]

substitute the given values

[tex]\frac{15}{sin(140\°)}=\frac{b}{sin(17\°)}[/tex]

Solve for b

[tex]b=(sin(17\°))\frac{15}{sin(140\°)}[/tex]

[tex]b=6.8\ units[/tex]

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:

m=22.5

Step-by-step explanation:

m is proportional to n means there is some constant k such that:

m=kn

If m=5 when n=4 then we have the following equation to solve for our constant k:

5=k(4)

5=4k

Divide both sides by 4:

5/4 =k

So k=5/4 no matter what (m,n) pair they give you where the equation is m=kn.

m=(5/4)n

What is m when n=18?

m=(5/4)(18)

m=(5/2)(9)

m=45/2

m=22.5

Final answer:

To find the value of m when n=18 for a proportional relationship where m is 5 when n is 4, calculate the constant of proportionality (k=5/4) and multiply by 18 to get m=22.5.

Explanation:

The question asks to find the value of m when n=18 given that m is proportional to n and m=5 when n=4. To solve this, we first establish the relationship between m and n using the provided values:

m = kn
(where k is the constant of proportionality)

Given that m=5 when n=4:

5 = k × 4
k = 5/4

Now we can determine the value of m when n=18 with the constant k:

m = (5/4) × 18
m = 22.5

The value of m when n is 18 is 22.5.

Answer:

The radius is 5

Step-by-step explanation:

The equation for Area of a sphere is

4pi*r^2

Since you have the area and not the radius you do the opposite of the equation so instead of multiplying you divide

100/4 = 25

Then find the square root of 25, which is 5

So your radius is 5.

Hope This Helps!!! :}

Answer:

Hi there!

The answer to this question is: A

y= [x] + 5

Answer:

D

Step-by-step explanation:

Answer:

[tex]y_Q=\dfrac{21}{5}=4.2[/tex]

Step-by-step explanation:

If the point Q that is located two over three the distance from point P to point R, then PQ:QR=2:3.

Use formula to find the coordinates of the point Q:

[tex]x_Q=\dfrac{3x_P+2x_R}{3+2}\\ \\y_Q=\dfrac{3y_P+2y_R}{3+2}[/tex]

In your case, P(-2,7) and R(1,0), then

[tex]x_Q=\dfrac{3\cdot (-2)+2\cdot 1}{3+2}=\dfrac{-4}{5}\\ \\y_Q=\dfrac{3\cdot 7+2\cdot 0}{3+2}=\dfrac{21}{5}[/tex]

Answer: x=2/e^2

Step-by-step explanation:

Answer:

The area of the pattern is 43.6 inches²

Step-by-step explanation:

* Lets explain how to solve the problem

- A decorative pillow can be modeled by Δ ABC

- In Δ ABC: AB = 9 inches , BC = 15 inches , AC = 10 inches

- To find the area of the triangle we can use the rule:

  A = 1/2 × (AB) × (BC) × sin∠B

- We will use the cosine rule to find the measure of angle B

∵ [tex]cos(B)=\frac{(AB)^{2}+(BC)^{2}-(AC)^{2}}{2(AB)(BC)}[/tex]

∵ AB = 9 , BC = 15 , AC = 10

∴ [tex]cos(B)=\frac{9^{2}+15^{2}-10^{2}}{2(9)(15)}=\frac{81+225-100}{270}=\frac{206}{270}=\frac{103}{135}[/tex]

∴ m∠B = [tex]cos^{-1}\frac{103}{135}=40.27[/tex]°

* Lets find the area of the triangle

∴ The area = 1/2 × (9) × (15) × sin(40.27) = 43.6 inches²

* The area of the pattern is 43.6 inches²

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.

!!