Bev earns $2 a week for taking out her neighbor’s trash cans. Complete the table of values. Then state the domain and range.

Number of Weeks | Money Earned
1 |
2 |
3 |
4 |

Answer:

1176

Step-by-step explanation:

A room with dimensions 10 ft by 5 ft has an area of 50 ft².

A room with dimension 7 ft by 21 ft has an area of 147 ft².

Writing a proportion:

400 tiles / 50 ft² = x / 147 ft²

x = 1176 tiles

Answer:

1176

Step-by-step explanation:

A room with dimensions 10 ft by 5 ft require 400 tiles. There would be 1176 tiles needed for a room that measures 7 ft by 21 ft.

Answer:

x = 6

Step-by-step explanation:

Given

6x² - 2x + 36 = 5x² + 10x ( subtract 5x² + 10x from both sides )

x² - 12x + 36 = 0 ← in standard form

This is a perfect square of the form

(x - a)² = x² - 2ax + a²

36 = 6² ⇒ a = 6 and 2ax = (2 × 6)x = 12x, hence

(x - 6)² = 0

x - 6 = 0 ⇒ x = 6

Answer:

The surface area of the triangular prism is 468 square centimeters. Therefore the correct option is 4.

Step-by-step explanation:

It is given that the bases are right triangles with perpendicular legs measuring 9 centimeters and 12 centimeters. Using Pythagoras theorem, the third side of the base is

[tex]hypotenuse^2=leg_1^2+leg_2^2[/tex]

[tex]hypotenuse^2=(9)^2+(12)^2[/tex]

[tex]hypotenuse^2=225[/tex]

[tex]hypotenuse=\sqrt{225}[/tex]

[tex]hypotenuse=15[/tex]

The area of a triangle is

[tex]A=\frac{1}{2}\times base \times height[/tex]

Area of the base is

[tex]A_1=\frac{1}{2}\times 9\times 12=54[/tex]

The curved surface area of triangular prism is

[tex]A_2=\text{perimeter of base}\times height[/tex]

[tex]A_2=(9+12+15)\times 10[/tex]

[tex]A_2=9\times 10+12\times 10+15\times 10[/tex]

[tex]A_2=360[/tex]

The surface area of the triangular prism is

[tex]A=2A_1+A_2[/tex]

[tex]A=2(54)+360[/tex]

[tex]A=108+360[/tex]

[tex]A=468[/tex]

The surface area of the triangular prism is 468 square centimeters. Therefore the correct option is 4.

Answer:

468 square centimeters

Step-by-step explanation:

Median: The middle number when all the numbers are listed in order.

First, we would put our numbers in order from least to greatest.

0 - 0 - 1 - 1 - 1 - 2 - 2 - 2 - 4 - 4

Next, we need to find the middle number. When we cross of one number from the left and one number from the right and keep doing this, we come across two numbers that are in the middle. The two numbers that are in the middle are 1 and 2. We find the median by adding 2 and 1 together to get a sum of 3 and then divide it by 2 to get an answer of 1.5

The median of this set of numbers is 1.5

Answer:

1.5

Step-by-step explanation:

the median if even if the two middle added then divided like.

0,0,1,1,(1,2,)2,2,4,4

1+2=3 ÷2 = 1.5

Answer:

r=5.15

d=10.31

Step-by-step explanation:

Given:

circumference of circle=32.4 mm

radius=?

diameter=?

formula of circumference of circle is given as:

C=2πr

Putting value we get

32.4=2πr

r=32.4/2π

r=5.15

d=2r

d=2(5.15)

d=10.31 !

For this case we have that by definition, the circumference of a circle is given by:

[tex]C = \pi * d[/tex]

Where:

d: It is the diameter of the circle

We have as data that:

[tex]C = 32.4 \ mm[/tex]

Taking[tex]\pi = 3.14[/tex]

We have:

[tex]32.4 = 3.14 * d\\d = \frac {32.4} {3.14}\\d = 10.32 \ mm[/tex]

The radius is given by:

[tex]\frac {10.32} {2} = 5.16 \ mm[/tex]

Answer:

[tex]Diameter: 10.32 \ mm\\Radius: 5.16 \ mm[/tex]

Answer:

Step-by-step explanation:

We know:

The root with an odd degree is exist for any real number.

The root with an even degree is exist for any non-negative real number.

The domain:

[tex]\sqrt[n]{a}[/tex]

is all real number if n is odd number.

is alle non-negative real number if n is even number.

Answer:

b) −8x+6

Step-by-step explanation:

-3x+4 - (5x-2)

Distribute the minus sign

-3x+4 -5x+2

Combine like terms

-3x-5x +4+2

-8x+6

Answer:

the answer is b -8x+6

Step-by-step explanation:

Answer: The quotient of powers was used because 25=5^2 which means that 5^4/25 is the same as 5^4/5^2. 5^4/5^2= 5^2. You can check your answer by simplifying 5^4 which is 625 and 5^2 which is 25, then divide the two which is 625/25 which equals 25 (or 5^2)

Step-by-step explanation:

[tex]\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{5^4}{25}\implies \cfrac{5^4}{5^2}\implies \cfrac{5^4}{1}\cdot \cfrac{1}{5^2}\implies 5^4\cdot 5^{-2}\implies 5^{4-2}\implies 5^2[/tex]

Answer:

1,779.33 ft³

Step-by-step explanation:

volume of cone = 1/3(pi)r²h (r=radius, h=height)

= 1/3 x 3.14 x (10)² x 17,

= 1/3 x 314 x 17

= 1/3 x 5338

= 1779.33 ft³

Answer:

just answering so this guy can get brainiest

Rewrite the given number as

[tex]\dfrac{a+ib}{c+id}=\dfrac{(a+ib)(c-id)}{(c+id)(c-id)}=\dfrac{ac+bd+i(bc-ad)}{c^2+d^2}[/tex]

If it's purely real, then the complex part should be 0, so that

[tex]\dfrac{bc-ad}{c^2+d^2}=0\implies bc-ad=0\implies\boxed{ad-bc}[/tex]

as required.

Answer:

Probability = 5/6

Step-by-step explanation:

Step 1: Write the formula of probability

Probability = number of possible outcomes/total number of outcomes

There is only one 28 so the chance of getting a 28 is 1/6.

Not getting a 28 would mean getting any one number from 23, 24, 25, 26 and 27.

Step 2: Apply the probability formula

Probability = number of possible outcomes/total number of outcomes

Probability of not getting 28 = 5/6

!!

Answer:

7/8

Step-by-step explanation:

Answer:

Step-by-step explanation:

sin=y/r

cos=x/r

sin=21/29

cos=x/29

x^2+y^2=r^2

x^2+21^2=29^2

x^2+441=841

x=sqrt(841-441)

x=20

cos=20/29

  Only              |

   sin +             |     All Positive

---------------------|-------------------

           only      |    Only cos +

            tan +    |

Answer:

C

Step-by-step explanation:

Using the trigonometric identity

sin²x + cos²x = 1 ⇒ cosx = ± [tex]\sqrt{1-sin^2x}[/tex]

Given

sinx = [tex]\frac{21}{29}[/tex], then

cosx = [tex]\sqrt{1-(\frac{21}{29})^2 }[/tex] ( positive since 0 < x < 90 )

       = [tex]\sqrt{1-\frac{441}{841} }[/tex]

       = [tex]\sqrt{\frac{400}{841} }[/tex] = [tex]\frac{20}{29}[/tex]

The length of the edge of the cube whose net area is 24 in sq is calculated as: 2 inches.

What is the length of a cube?

The net of a cube consists of six identical square faces. Let's denote the length of one side of the square as s.

The total surface area of the cube is the sum of the areas of its six faces. Since each face has an area of s², the total surface area (A) is given by:

[tex]\[ A = 6s^2 \][/tex]

You mentioned that the net has an area of 24 square inches. Therefore, we can set up the equation:

6s² = 24

Now, solve for s:

[tex]s^2 = \frac{24}{6}[/tex]

s² = 4

Take the square root of both sides:

[tex]s = \sqrt{4}[/tex]

s = 2

So, the length of each edge of the cube is 2 inches.

Answer:

26

Step-by-step explanation:

The formula is a half the positive difference of the measurements of the intercepted arcs.

That is you do .5(66-14) here.

I'm going to distribute first and instead of find the difference first.

.5(66)-.5(14)

   33-  7

   26

Or.... you could do the difference first which gives us .5(66-14)=.5(52)=26.

So that angle is 26 degrees.

Answer:

x = 26°

Step-by-step explanation:

A secant- secant angle is an angle whose vertex is outside the circle and whose sides are 2 secants of the circle. It's measure is

x = 0.5(66 - 14)° = 0.5 × 52° = 26°

Answer:

m=-2/3  (slope)

b=3        (y-intercept)

Step-by-step explanation:

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

You have 3y+2x=9.

We need to solve this for y to get it into y=mx+b form.

3y+2x=9

Subtract 2x on both sides:

3y     =-2x+9

Divide both sides by 3:

[tex]y=\frac{-2}{3}x+\frac{9}{3}[/tex]

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

Now compare this to:

y=mx+b

m=-2/3  

b=3

Answer:

y=(x-3)^2 -2

Step-by-step explanation:

when the number to the power of two is positive the graph will aslo go up, (both ends go up as shown in the graph. the parabula of the graph is -2.

Answer:

$311.20

Step-by-step explanation:

Here we are required to use the Compound interest formula for finding the Amount at the end of 9th year

The formula is given as

[tex]A=P(1+\frac{r}{n})^{tn}[/tex]

Where ,

A is the final amount

P is the initial amount = $200

r is the rate of interest = 5% annual = 0.05

n is the frequency of compounding in a year ( Here it is compounding monthly) = 12

t is the time period = 9

Now we substitute all these values in the formula and solve for A

[tex]A=200(1+\frac{0.05}{12})^{9\times 12}[/tex]

[tex]A=200(1+0.00416)^{108}[/tex]

[tex]A=200(1.00416)^{108}[/tex]

[tex]A=200 \times 1.556[/tex]

[tex]A=311.20[/tex]

Hence the amount after 9 years will be $311.20

Answer:

Step-by-step explanation:

The first choice describes the right way to find the length of an arc in a circle.

The length of an arc is defined as

[tex]L=2\pi r (\frac{\theta}{360\°} )[/tex]

Where [tex]2\pi r[/tex] represents the circumference of the circle and [tex]\theta[/tex] represents the arc's degree measure.

So, as you can observe through this formula, we need to divide the arc's degree measure by 360°, and then multiply this result with the circumference of the circle, that's the right way based on the definition of arc length.

Therefore, the right answer is A.

The length of an arc in the circle is L = Divide the arc's degree measure by 360°, then multiply by the circumference of the circle

The central angle is an angle with two arms and a vertex in the middle of a circle. The two arms of the circle's two radii intersect the circle's arc at two separate locations. It is an angle whose vertex is the center of a circle with the two radii lines as its arms, that intersect at two different points on the circle.

The central angle of a circle formula is as follows.

Central Angle = ( s x 360° ) / 2πr

where s is the length of the arc

r is the radius of the circle

Central Angle = 2 x Angle in other segment

Given data ,

Let the length of the arc of the circle be L

Now , Central Angle = ( L x 360° ) / 2πr

where s is the length of the arc

On simplifying the equation , we get

Divide by 360° on both sides , we get

L / 2πr = Central Angle θ / 360°

Multiply by 2πr ( circumference ) on both sides , we get

L = ( θ / 360° ) x 2πr

Hence , the length of an arc is L = ( θ / 360° ) x 2πr

To learn more about central angle click :

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Step-by-step explanation:

Length of arc = (Central Angle/360) × 2

[tex]\pi[/tex]

r

= 21/360 × 2 × 3.14 × 3

Length = 1.099 feet

Please mark Brainliest if this helps!

Answer:

Your answer is [tex]\frac{7\pi}{20}[/tex].

If you prefer an answer rounded to nearest hundredths you would have 1.10 or just 1.1.

Step-by-step explanation:

The formula for finding the arc length s is given by:

[tex]s=r \cdot \frac{\theta \pi}{180^\circ}[/tex]

where [tex]\theta[/tex] is in degrees.

Plug in 3 for r and 21 for [tex]theta[/tex]:

[tex]s=3 \cdot \frac{21 \pi}{180}[/tex]

I'm going to reduce 21/180 by dividing top and bottom by 3:

[tex]s=3 \cdot \frac{7 \pi}{60}{/tex]

I'm going to multiply 3 and 7:

[tex]s=\frac{21 \pi}{60}[/tex]

I'm going to reduce 21/60 by dividing top and bottom by 3:

[tex]s=\frac{7\pi}{20}[/tex]

Your answer is [tex]\frac{7\pi}{20}[/tex].

If you prefer an answer rounded to nearest hundredths you would have 1.10 or just 1.1.

Answer:

-a^2 +2ab + b^2

Step-by-step explanation:

b (a+b) - a (a-b)

Distribute

ab +b^2 -a^2 +ab

Combine like terms

b^2 -a^2 +2ab

-a^2 +2ab + b^2

The value of z in the table is 40.

To find the value of z in the table, we can set up an equation using the concentrations and amounts of the solutions. Let's denote the amount of the 4% peroxide solution as x and the amount of the 10% peroxide solution as y. We can then set up the equation:

0.04x + 0.1y = 0.08(100)

Simplifying this equation, we have:

0.04x + 0.1y = 8

Now, let's refer to the table to find the values of x and y. Since the sum of the amounts is 100 L, we have:

x + y = 100

From the information in the table, we can see that the value of x is 60. This means that y must be 40, as the sum of the amounts is 100. Now we can substitute the values of x and y into the equation:

0.04x + 0.1y = 8

0.04(60) + 0.1(40) = 8

2.4 + 4 = 8

The equation holds true, so the value of z in the table is 40.

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Answer:

The correct answer is second option

78°

Step-by-step explanation:

Points to remember

The measure of arc BC = 2 * measure of angle BDC

To find the measure of arc BC

From the figure we can see the BD is the diameter of the given circle.

Therefore the ΔBDC is right angled triangle. m<C = 90°

m<CBD = 51°   (given)

m<CBD + m<BDC = 90

m<BDC = 90 - m<CBD

 = 90 - 51 = 39

Therefore measure of arc BC = 2 *m<BDC

  = 2 * 39

 = 78°

The correct answer is second option

78°

Answer:

30

Step-by-step explanation:

350/12=29.6

(29.6, six is neverending)

Round the decimal

Answer is 30

Mrs. Yamato must bake 30 pans of muffins to have enough for 350 people, as each pan makes 12 muffins and she cannot bake a fraction of a pan.

Mrs. Yamato needs to calculate how many pans of muffins to bake to serve 350 people, given each pan holds 12 muffins. To find the number of pans, we divide the total number of muffins needed by the number of muffins each pan can hold.

The calculation would be 350 muffins  7 12 muffins/pan = 29.17 pans.

Since she cannot bake a fraction of a pan, Mrs. Yamato will need to round up to the nearest whole number, which means she must bake 30 pans of muffins to ensure there are enough muffins to serve 350 people.

Answer:

788.13  to the nearest hundredth.

Step-by-step explanation:

Let A be the total amount in the account after 3 years.

The formula is A = P(1 + x/100)^t .

Here P = 5000, x = 5 %  and the time t = 3. years.  

Amount after 3 years = 5000(1 + 5/100)^3

=  5788.13

So the Interest is 5788.13 - 5000

= 788.13.

Answer:

Compound interest = 788.125

Step-by-step explanation:

Points to remember

Compound interest

A = P[1 + R/100]^N

Were A - Amount

P - Principle

R - Rate of interest

N - Number of years

To find the compound interest

Here P - 5,000

R = 5%

N - 3 years

A = P[1 + R/100]^N

 = 5000[1 + 5/100]^3

 = 5000[1 + 0.05]^3

 = 5788.125

Compound interest  = A - P

 = 5788.125 - 5000

 = 788.125

Answer:

-12 <p <12

Step-by-step explanation:

|p|<12

We can write this without the absolute values

Take the equation with the positive value on the right hand side  and take the equation with a negative value on the right side remembering to flip the inequality.  Since this is less than we use and in between

p < 12 and p >-12

-12 <p <12

Step-by-step explanation:

[tex]For\ a>0\\\\|x|<a\Rightarrow x<a\ \wedge\ x>-a\\\\|x|>a\Rightarrow x>a\ \wedge\ x<-a\\\\===============================\\\\|p|<12\Rightarrow p<-12\ \wedge\ p>-12\Rightarrow-12<p<12[/tex]

Answer:

Hi there!

The answer to this question is: The lines are perpendicular.

Step-by-step explanation:

If you take the slope of the first equation its 3. To find its perpendicular slope you take the negative reciprocal of it. You flip the number into a fraction and make it negative, this case you get -1/3 which is the slope of the second equation therefore they are perpendicular

Answer:

The lines are perpendicular.

Step-by-step explanation:

If two lines are graphed on the coordinate plane showing y equals 3 x plus 1 and y equals negative one third x minus 2, we can conclude that the lines are perpendicular.

Slope = 3

Slope of second equation: -1/3

Therefore, thee slopes are perpendicular.

Answer:

See below in bold.

Step-by-step explanation:

The x intercepts occur when f(x) = 0.

1.  logx  - 1 = 0

logx = 1

By the definition of a log ( to the base 10):

x  = 10^1 = 10

So the x-intercept is  c (10,0).

2. - (logx - 2) = 0

logx - 2 = 0

log x = 2

so x = 100.

So it is d (100,0).

3 .   log(-x - 2)  = 0

-x - 2 = 10^0 = 1

-x = 3

x = -3

So it is  b (-3, 0).

4.  f(x) = -log -(x - 1)

log - (x - 1) = 0

log 1 = 0

so -(x - 1) = 1

- x + 1 = 1

x = 1-1 = 0

So  it is a. (0,0).

     Function                               x-intercept

[tex]f(x)=\log x-1[/tex]                              [tex](10,0)[/tex]

[tex]f(x)=-(\log x-2)[/tex]                        [tex](100,0)[/tex]

[tex]f(x)=\log (-x-2)[/tex]                         [tex](-3,0)[/tex]

[tex]f(x)=-\log -(x-1)[/tex]                     [tex](0,0)[/tex]

We know that the x-intercept of a function is the point where the function value is zero.

i.e. the x where f(x)=0

1)

[tex]f(x)=\log x-1[/tex]

when [tex]f(x)=0[/tex] we have:

[tex]\log x-1=0\\\\i.e.\\\\\log x=1\\\\i.e.\\\\\log x=\log 10[/tex]

Hence, taking the exponential function on both the sides of the equation we have:

[tex]x=10[/tex]

The x-intercept is: (10,0)

2)

[tex]f(x)=-(\log x-2)[/tex]

when, [tex]f(x)=0[/tex]

we have:

[tex]-(\log x-2)=0\\\\i.e.\\\\\log x-2=0\\\\i.e.\\\\\log x=2\\\\i.e.\\\\\log x=2\cdot 1\\\\i.e.\\\\\log x=2\cdot \log 10\\\\i.e.\\\\\log x=\log (10)^2[/tex]

Since,

[tex]m\log n=\log n^m[/tex]

Hence, we have:

[tex]\log x=\log 100[/tex]

Taking anti logarithm on both side we get:

[tex]x=100[/tex]

Hence, the x-intercept is:

(100,0)

3)

[tex]f(x)=\log (-x-2)[/tex]

when

[tex]f(x)=0[/tex]

we have:

[tex]\log (-x-2)=0\\\\i.e.\\\\\log (-x-2)=\log 1[/tex]

On taking anti logarithm on both the side of the equation we get:

[tex]-x-2=1\\\\i.e.\\\\x=-2-1\\\\i.e.\\\\x=-3[/tex]

Hence, the x-intercept is: (-3,0)

4)

[tex]f(x)=-\log -(x-1)[/tex]

when,

[tex]f(x)=0\ we\ have:[/tex]

[tex]-\log -(x-1)=0\\\\i.e.\\\\\log -(x-1)=0\\\\i.e.\\\\\log -(x-1)=\log 1\\\\i.e.\\\\-(x-1)=1\\\\i.e.\\\\x-1=-1\\\\i.e.\\\\x=-1+1\\\\i.e.\\\\x=0[/tex]

Hence, the x-intercept is:  (0,0)

Answer:

"The maximum number of solutions is one."

Step-by-step explanation:

Hopefully the drawing helps visualize the problem.

The circle has a radius of 9 because the vertex is 9 units above the center of the circle.

The circle the parabola intersect only once and cannot intercept more than once.  

The solution is "The maximum number of solutions is one."

Let's see if we can find an algebraic way:

The equation for the circle given as we know from the problem without further analysis is so far [tex]x^2+y^2=r^2[/tex].

The equation for the parabola without further analysis is [tex]y=ax^2+9[/tex].

We are going to plug [tex]ax^2+9[/tex] into [tex]x^2+y^2=r^2[/tex] for [tex]y[/tex].

[tex]x^2+y^2=r^2[/tex]

[tex]x^2+(ax^2+9)^2=r^2[/tex]

To expand [tex](ax^2+9)^2[/tex], I'm going to use the following formula:

[tex](u+v)^2=u^2+2uv+v^2[/tex].

[tex](ax^2+9)^2=a^2x^4+18ax^2+81[/tex].

[tex]x^2+y^2=r^2[/tex]

[tex]x^2+(ax^2+9)^2=r^2[/tex]

[tex]x^2+a^2x^4+18ax^2+81=r^2[/tex]

So this is a quadratic in terms of [tex]x^2[/tex]

Let's put everything to one side.

Subtract [tex]r^2[/tex] on both sides.

[tex]x^2+a^2x^4+18ax^2+81-r^2=0[/tex]

Reorder in standard form in terms of x:

[tex]a^2x^4+(18a+1)x^2+(81-r^2)=0[/tex]

The discriminant of the left hand side will tell us how many solutions we will have to the equation in terms of [tex]x^2[/tex].

The discriminant is [tex]B^2-4AC[/tex].

If you compare our equation to [tex]Au^2+Bu+C[/tex], you should determine [tex]A=a^2[/tex]

[tex]B=(18a+1)[/tex]

[tex]C=(81-r^2)[/tex]

The discriminant is

[tex]B^2-4AC[/tex]

[tex](18a+1)^2-4(a^2)(81-r^2)[/tex]

Multiply the (18a+1)^2 out using the formula I mentioned earlier which was:

[tex](u+v)^2=u^2+2uv+v^2[/tex]

[tex](324a^2+36a+1)-4a^2(81-r^2)[/tex]

Distribute the 4a^2 to the terms in the ( ) next to it:

[tex]324a^2+36a+1-324a^2+4a^2r^2[/tex]

[tex]36a+1+4a^2r^2[/tex]

We know that [tex]a>0[/tex] because the parabola is open up.

We know that [tex]r>0[/tex] because in order it to be a circle a radius has to exist.

So our discriminat is positive which means we have two solutions for [tex]x^2[/tex].

But how many do we have for just [tex]x[/tex].

We have to go further to see.

So the quadratic formula is:

[tex]\frac{-B \pm \sqrt{B^2-4AC}}{2A}[/tex]

We already have [tex]B^2-4AC}[/tex]

[tex]\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}[/tex]

This is t he solution for [tex]x^2[/tex].

To find [tex]x[/tex] we must square root both sides.

[tex]x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]

So there is only that one real solution (it actually includes 2 because of the plus or minus outside) here for x since the other one is square root of a negative number.

That is,

[tex]x=\pm \sqrt{\frac{-(18a+1) \pm \sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]

means you have:

[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]

or

[tex]x=\pm \sqrt{\frac{-(18a+1)-\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex].

The second one is definitely includes a negative result in the square root.

18a+1 is positive since a is positive so -(18a+1) is negative

2a^2 is positive (a is not 0).

So you have (negative number-positive number)/positive which is a negative since the top is negative and you are dividing by a positive.

We have confirmed are max of one solution algebraically. (It is definitely not 3 solutions.)

If r=9, then there is one solution.

If r>9, then there is two solutions as this shows:

[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}}[/tex]

r=9 since our circle intersects the parabola at (0,9).

Also if (0,9) is intersection, then

[tex]0^2+9^2=r^2[/tex] which implies r=9.

Plugging in 9 for r we get:

[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+4a^2(9)^2}}{2a^2}}[/tex]

[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{36a+1+324a^2}}{2a^2}}[/tex]

[tex]x=\pm \sqrt{\frac{-(18a+1)+\sqrt{(18a+1)^2}}{2a^2}}[/tex]

[tex]x=\pm \sqrt{\frac{-(18a+1)+18a+1}{2a^2}}[/tex]

[tex]x=\pm \sqrt{\frac{0}{2a^2}}[/tex]

[tex]x=\pm 0[/tex]

[tex]x=0[/tex]

The equations intersect at x=0. Plugging into [tex]y=ax^2+9[/tex] we do get [tex]y=a(0)^2+9=9[/tex].  

After this confirmation it would be interesting to see what happens with assume algebraically the solution should be (0,9).

This means we should have got x=0.

[tex]0=\frac{-(18a+1)+\sqrt{36a+1+4a^2r^2}}{2a^2}[/tex]

A fraction is only 0 when it's top is 0.

[tex]0=-(18a+1)+\sqrt{36a+1+4a^2r^2}[/tex]

Add 18a+1 on both sides:

[tex]18a+1=\sqrt{36a+1+4a^2r^2[/tex]

Square both sides:

[tex]324a^2+36a+1=36a+1+4a^2r^2[/tex]

Subtract 36a and 1 on both sides:

[tex]324a^2=4a^2r^2[/tex]

Divide both sides by [tex]4a^2[/tex]:

[tex]81=r^2[/tex]

Square root both sides:

[tex]9=r[/tex]

The radius is 9 as we stated earlier.

Let's go through the radius choices.

If the radius of the circle with center (0,0) is less than 9 then the circle wouldn't intersect the parabola.  So It definitely couldn't be the last two choices.

Answer:

Option A.

Step-by-step explanation:

A circle was drawn with the center at origin (0, 0) and a point (0, 9) on the circle.

So the radius will be r = [tex]\sqrt{(0-0)+(0-9)^{2}}=9[/tex]

Equation of this circle will be in the form of x² + y² = r²

Here r represents radius.

So the equation of the circle will be x² + y² = 9²

Or x² + y² = 81

Now we will form the equation of the parabola having vertex at (0, 9)

y² = (x - h)² + k

where (h, k) is the vertex.

Equation of the parabola will be y² = (x - 0)² + 9

y² = x² + 9

Now we will replace the value of y² from this equation in the equation of circle to get the solution of this system of the equations.

x² + x² + 9 = 81

2x² = 81 - 9

2x² = 72

x²= 36

x = ±√36

x = ±6

Since circle and parabola both touch on a single point (0, 9) therefore, there will be only one solution that is x = 6.

For x = 6,

6² + y² = 9²

36 + y² = 81

y² = 81 - 36 = 45

y = √45 = 3√5  

Option A. will be the answer.

Click here for the step by step: https://brainly.com/question/2301848

15.9

Answer: 15.9 centimeters.

Step-by-step explanation:

The formula for calculate the circumference of a circle is this one:

[tex]C=2\pi r[/tex]

Where "C" is the circumference of the circle and "r" is the radius.

We know that:

[tex[C=50cm\\\pi=3.14[/tex]

Then, substituting these values into the formula and solving for "r", we get:

[tex]50cm=2(3.14)r\\\\r=\frac{50cm}{6.28}\\\\r=7.96cm[/tex]

Since the diameter of a circle is twice the radius, we can multiply the radius by 2 to get the diameter of this circle. Then, rounded to the nearest tenth of a centimeters, this is:

[tex]D=2r\\\\D=2(7.96cm)\\\\D=15.9cm[/tex]

Answer:

The first step would to be use quotient rule.

3

Step-by-step explanation:

ln(x-1)=ln(6)-ln(x)

The first step would to be use quotient rule there on the right hand side:

ln(x-1)=ln(6/x)

*Quotient rule says ln(a/b)=ln(a)-ln(b).

Now that since we have ln(c)=ln(d) then c must equal d, that is c=d.

ln(x-1)=ln(6/x)

implies

x-1=6/x

So you want to shove a 1 underneath the (x-1) and just cross multiply that might be easier.

[tex]\frac{x-1}{1}=\frac{6}{x}[/tex]

Cross multiplying:

[tex]x(x-1)=1(6)[/tex]

Multiplying/distribute[/tex]

[tex]x^2-x=6[/tex]

Subtract 6 on both sides:

[tex]x^2-x-6=0[/tex]

Now this is not too bad to factor since the coefficient of x^2 is 1.  All you have to do is find two numbers that multiply to be -6  and add up to be -1.

These numbers are -3 and 2 since -3(2)=-6 and -3+2=-1.

So the factored form of our equation is

[tex](x-3)(x+2)=0[/tex]

This implies that x-3=0 or x+2=0.

So solving x-3=0 gives us x=3 (just added 3 on both sides).

So solve x+2=0 gives us x=-2 (just subtracted 2 on both sides).

We need to see if these are actually the solutions by plugging them in.

Just a heads up: You can't do log(negative number).

Checking x=3:

ln(3-1)=ln(6)-ln(3)

ln(2)=ln(6/3)

ln(2)=ln(2)

This is true.

Checking x=-2:

ln(-2-1)=ln(6)-ln(-2)

ln(-3)=ln(6)-ln(-2)

We don't need to go further -2 makes the inside of our logarithms negative above.

The only solution is 3.

Step-by-step explanation:

[tex]\ln(x-1)=\ln6-\ln x\\\\D:\ x-1>0\ \wedge\ x>0\\\\x>1\ \wedge\ x>0\Rightarrow x>1[/tex]

[tex]\ln(x-1)=\ln6-\ln x\qquad\text{use}\ \log_ab-\log_ac=\log_a\dfrac{b}{c}\\\\\ln(x-1)=\ln\dfrac{6}{x}\iff x-1=\dfrac{6}{x}\\\\\dfrac{x-1}{1}=\dfrac{6}{x}\qquad\text{cross multiply}\\\\x(x-1)=(1)(6)\qquad\text{use the distributive property}\ a(b+c)=ab+ac\\\\(x)(x)+(x)(-1)=6\\\\x^2-x=6\qquad\text{subtract 6 from both sides}\\\\x^2-x-6=0\\\\x^2+2x-3x-6=0\\\\x(x+2)-3(x+2)=0\\\\(x+2)(x-3)=0\iff x+2=0\ \vee\ x-3=0\\\\x+2=0\qquad\text{subtract 2 from both sides}\\x=-2 \notin D\\\\x-3=0\qquad\text{add 3 to both sides}\\x=3\in D[/tex]

Solution: