Select the property of equality used to arrive at the conclusion.

If x = 3, then x^2 = 3x

a. the multiplication property of equality

b. the division property of equality

c. the addition property of equality

d. the subtraction property of equality

(a) Wild guess:

[tex]f(x)=\dfrac1{(6+x)^2[/tex]

Recall the power series

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

With some manipulation, we can write

[tex]\displaystyle\frac1{6+x}=\frac16\frac1{1-\left(-\frac x6\right)}=\frac16\sum_{n=0}^\infty\left(-\frac x6\right)^n=\sum_{n=0}^\infty\frac{(-x)^n}{6^{n+1}}[/tex]

Take the derivative and we get

[tex]\displaystyle-\frac1{(6+x)^2}=-\sum_{n=0}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=1}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

so we have

[tex]\displaystyle\frac1{(6+x)^2}=\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

By the ratio test, this series converges if

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+2)(-x)^{n+1}}{6^{n+3}}}{\frac{(n+1)(-x)^n}{6^{n+2}}}\right|=\left|\frac x6\right|\lim_{n\to\infty}\frac{n+2}{n+1}=\left|\frac x6\right|<1[/tex]

or [tex]|x|<6[/tex], so that the radius of convergence is [tex]R=6[/tex].

(b). If we take the second derivative, we get

[tex]\displaystyle\frac2{(6+x)^3}=\sum_{n=0}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=1}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

[tex]\displaystyle\frac1{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

Apply the ratio test again and we get [tex]R=6[/tex].

(c) Multiply the previous series by [tex]x^2[/tex] and we get

[tex]\displaystyle\frac{x^2}{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^nx^2}{6^{n+3}}[/tex]

[tex]\displaystyle=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-1)^nx^{n+2}}{6^{n+3}}[/tex]

The ratio test yet again tells us [tex]R=6[/tex].

Hello there!

To find which tuna is a better deal, divide the cost by the number of ounces of tuna you are getting to get the cost per ounce.

$0.90/5  = $0.18 per ounce

$2.40/12 = $0.20 per ounce

Since the 5-ounce can of tuna has a cheaper unit rate price, meaning you are getting a better value, makes this the best option. I hope this was helpful and have a great day! :)

The 5 once one hope this helps

The Ice-Cream Palace needed to buy 34 more pints of milk to reach their requirement for the daily special, as they only had 14 pints and required 48 pints in total.

To solve this question, we need to convert all the quantities to the same unit. In this case, we'll use pints, since it's the smallest unit given. We know that 1 gallon equals 4 quarts, and 1 quart equals 2 pints. Therefore, 1 gallon equals 8 pints.

Ice-Cream Palace needed 6 gallons of milk, which is 6 * 8 = 48 pints. They already had 6 1/2 quarts of skim milk and 1 pint of whole milk. The 6 1/2 quarts equals 6.5 * 2 = 13 pints.

So, in total, Ice-Cream Palace had 13 pints + 1 pint = 14 pints of milk. This means they still needed to buy 48 - 14 = 34 pints of milk for their daily special.

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The Ice-Cream Palace does not need to buy any more pints of milk because they already have more milk than they need.

To find the number of pints of milk the Ice-Cream Palace still needed to buy, we first need to convert the given quantities to the same unit. Since we need to find the number of pints, we'll convert the 6 gallons and 6 1/2 quarts to pints.

1 gallon = 4 quarts = 8 pints

So, 6 gallons = 6 x 8 = 48 pints

1 quart = 2 pints

So, 6 1/2 quarts = 6 x 2 + 1 x 2 = 12 + 2 =14 pints

1 pint is already given as 1 pint.

Now, to find the number of pints of milk still needed to buy, we subtract the total amount of milk they already have (48 pints + 14 pints + 1 pint) from the amount they need (6 gallons = 48 pints).

48 pints (needed) - (48 pints + 14 pints + 1 pint) (already have) = 48 pints - 63 pints = -15 pints

Since the result is negative, it means they have more milk than they need. So, they don't need to buy any more pints of milk.

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There are 1000 meters in 1 kilometer so you can set up a proportion to find the number of meters. 1000m / 1km = x / 40075 km cross multiply and you get the answer of 40,075,000 meters. To change to scientific notation, move the decimal point next to the 4 (becuase 4.0075 is less than 10) and you get the answer of 4.0075 x 10^7 meters.

Final answer:

After solving the given system of equations, the value of y is determined to be 6.4, but this answer does not match any of the provided choices. There appears to be an error in the options given or the original equations.

Explanation:

To find the value of y in the solution to the system of equations, we first need to solve the system. We have two equations:

x + y = 1

2x – 3y = –30

From the first equation, we can express x in terms of y:

x = 1 - y

Next, we substitute this expression for x into the second equation:

2(1 - y) - 3y = -30

Expanding this equation:

2 - 2y - 3y = -30

Combining like terms:

-5y = -32

Divide both sides by -5 to find the value of y:

y = 32 / 5

Now, we can see that none of the options provided in the question (-8, -3, 3, 8) match the correct value we found, which is 6.4. It seems there might be an error in the options provided. However, if the question intended for y to be an integer, then we should re-examine the arithmetic or consider that there might be a typo in the original equations or options given.

Answer: option c.

Step-by-step explanation:

To solve the exercise you must apply the formula given in the problem, which is the following:

[tex]y=33,430e^{0.0397t}[/tex]

The problem asks for the projected population of Bakersfield in 2010.

Therefore, keeping on mind that  t is th number of years since 1950, you have that:

[tex]t=2010-1950\\t=60[/tex]

Substitute the value of t into the formula.

Therefore, you obtain:

 [tex]y=33,430e^{0.0397(60)}=361,931[/tex]

x=8

y=2

8/2=4

the value of the x is 4

Answer:

[tex](x',y')-->(x-3,y-2)[/tex]

Step-by-step explanation:

Notice that we can get from the x-coordinate of A, 1, to the x-coordinte of A', -2, by subtracting 3 from the x-coordinate of A. More formaly:

[tex]1+a=-2[/tex]

[tex]a=-2-1[/tex]

[tex]a=-3[/tex]

Similarly, we can get from the y-coordinate of A, 5, to the y-coordinate of A', 3, by subtracting 2 from the y-coordinte of A. More formaly:

[tex]5+b=3[/tex]

[tex]b=3-5[/tex]

[tex]b=-2[/tex]

Now we now that to get to A' from A, we need to subtract 3 to the x-coordinate of A and subtract 2 to the y-coordinate. Knowing this, we can create the expression to translate any point of the polygon ABCD to create the polygon A'B'C'D':

[tex](x',y')-->(x-3,y-2)[/tex]

75÷12=6.25

which means Steve needs 7 bags of apples total

Answer:

Answer:2.4

Step-by-step explanation:

The correct answer will be 2.4

Hope this helped

Answer:

Step-by-step explanation:

A graphing program can help a lot in this case. Even without the program, it seems clear from a graph that only the first three points will lie on the same line.

The slope of the line can be found from the first two points as ...

  ∆y/∆x = (7/6 -1/6)/(-2 -1) = (6/6)/-3 = -1/3

Then the point-slope form of the equation can be written as

  y -1/6 = -1/3(x -1)

Multiplying by 6 gives ...

  6y -1 = -2x +2

Adding 2x+1 puts the equation into standard form:

  2x + 6y = 3

Answer:

6

Step-by-step explanation:

48 tiles divided by 8 ft is 6 ft

Answer:

B

Step-by-step explanation:

C and D dont make since to the problem

so i was left was A and B and i can't realy how i got my answer but i came down B

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this one you can solve easily.

take each equation and put number of miles in that , calculate cost and compare the table.

for example

taking first equation

c = 0.15m + 5.75

noe.from table take m = 10, after putting this we should get cost equal to 5.75 as mentioned in table.

c = 0.15×10 + 5.75 = 7.25 ≠ 5.75

so ignore this and take next equation

c = 0.15m + 4.25

m =10

c = 0.15×10 + 4.25 = 5.75

which matches with our table. putting other values of m from table gives us the right cost mentioned in table . so answer is option B

you can verify that no other equation satisfies the table data.

Answer:18.375

Step-by-step explanation:8x/7+9=30

We simplify the equation to the form, which is simple to understand  

8x/7+9=30

Simplifying:

+1.14285714286x+9=30

We move all terms containing x to the left and all other terms to the right.  

+1.14285714286x=+30-9

We simplify left and right side of the equation.  

+1.14285714286x=+21

We divide both sides of the equation by 1.14285714286 to get x.  

x=18.375

Brainliest please :)

Answer: x=18 3/8

Step-by-step explanation:

I got it right on odyssey ware

Let width = X

Then length would = 3x

The perimeter is 2 times the length + 2 times the width, so half the perimeter would be the length plus the width.

L + W = 1/2 perimeter

L +w = 48

Replace L and W withe the variables from above:

3x + x = 48

Simplify:

4x = 48

Divide both sides by 4:

x = 48/4

x = 12

Now you have a value for x, replace x with 12 in the variables:

Width = x = 12 cm

Length = 3x = 3*12 = 36 cm

The median of a set is defined as the middle element of the sorted set.

So, the only way to add two elements without changing the median is to add an element before it, and one element after it.

Think about it: if you are the middle element in a queue, and we add something before you and something after you, you're still in the middle:

[tex] \{2,\ 3,\ 4\} \mapsto \{1,\ 2,\ 3,\ 4,\ 5\}[/tex]

In both cases, the middle element is 3.

If, instead, we add two elements before 3,

[tex] \{2,\ 3,\ 4\} \mapsto \{0,\ 1,\ 2,\ 3,\ 4\}[/tex]

the median has changed, and similarly if we add two elements after 3

[tex] \{2,\ 3,\ 4\} \mapsto \{2,\ 3,\ 4,\ 5,\ 6\}[/tex]

the median has changed again.

So, if we sort our set, we have

[tex]\{27,\ 28,\ 35,\ 37,\ 43,\ 47\}[/tex]

And the middle elements are 35 and 37. The only option that adds something bigger than then and something smaller than them is option a.

Answer:

Part 1) The dimensions are

Length is [tex]20\ ft[/tex], Width is [tex]12\ ft[/tex]

Part 2) The perimeter is [tex]64\ ft[/tex]

Step-by-step explanation:

Part 1) Determine both dimensions

Let

x-----> the length of the pool

y----> the wide of the pool

we know that

The area of the rectangle (pool) is equal to

[tex]A=xy[/tex]

we have

[tex]A=240\ ft^{2}[/tex]

so

[tex]240=xy[/tex] -----> equation A

[tex]x=y+8[/tex] ----> equation B

substitute equation B in equation A and solve for y

[tex]240=(y+8)y[/tex]

[tex]240=y^{2}+8y[/tex]

[tex]y^{2}+8y-240=0[/tex]

using a graphing tool to solve the quadratic equation

the solution is

[tex]y=12\ ft[/tex]

see the attached figure

Find the value of x

[tex]x=y+8[/tex] ------> [tex]x=12+8=20\ ft[/tex]

Part 2) Find the perimeter

The perimeter of rectangle (pool) is equal to

[tex]P=2(x+y)[/tex]

we have

[tex]x=20\ ft, y=12\ ft[/tex]

substitute

[tex]P=2(20+12)=64\ ft[/tex]

The dimensions of the pool is 12ft by 20ft and the perimeter of the pool is 64feet

The perimeter of a rectangle is calculated as;

P= 2(l+w)

where

l is the length

w is the width

If the pool is to be 8ft longer than it is wide and cover 240 square feet then the equation becomes;

240 = w(w+8)

240 = w² + 8w

w² + 8w - 240 = 0

w² +20w -12w -240 = 0

w(w+20) -12(w+20) = 0

w = 12feet

Determine the length of the rectangle. Recall that:

l = 8 + w

l = 8+12

l = 20feet

Hence the dimensions of the pool is 12ft by 20ft

Perimeter = 2(12+20)

P = 2(32)

P = 64ft

Hence the perimeter of the pool is 64feet

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Answer:

see explanation

Step-by-step explanation:

Using the trigonometric identities

• 1 + cot² x = csc²x and csc x = [tex]\frac{1}{sinx}[/tex]

• sin²x + cos²x = 1 ⇒ sin²x = 1 - cos²x

Consider the left side

sin²Θ( 1 + cot²Θ )

= sin²Θ × csc²Θ

= sin²Θ × 1 / sin²Θ = 1 = right side ⇔ verified

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

Consider the left side

cos²Θ - sin²Θ

= cos²Θ - (1 - cos²Θ)

= cos²Θ - 1 + cos²Θ

= 2cos²Θ - 1 = right side ⇒ verified

Answer:

4.780 °

144''

Step-by-step explanation:

Given that,

door's level is 12 feet off the ground

the ramp can not have an incline surpassing a ratio of 1:12

An incline surpassing a ratio of 1:12 , means that every 1" of vertical rise requires at least 12" of ramp length

So,

1' rise = 12' length

12' = 12'x12

   = 144''

now we know the length and height of ramp so we can use trigonometry identities to find the angle

sinФ = height / length

sinФ = 12 / 144

sinФ =  sin^-1(12/144)

Ф       =  4.780 °

Answer: option a.

Step-by-step explanation:

 To solve the given exercise and write the expression as a single logarithm, you must keep on mind the following properties:

[tex]log(a)+log(b)=log(ab)\\m*log(a)=log(a)^m[/tex]

Therefore, by applying the properties shown above, you can rewrite the expression given, as following:

[tex]3logx+4log(x-2)=logx^3+log(x-2)^4=logx^3(x-2)^4[/tex]

Then, the answer is the option a.

Answer:

6 cm

Step-by-step explanation:

A = 1/2(b + 12)(4)

36 = 1/2(b + 12)(4)

72 = (b + 12)(4)

18 = b + 12

18 - 12 = b

6 = b

or

b = 6 cm

The calculated missing base of the trapezoid is 6 cm

From the question, we have the following parameters that can be used in our computation:

The trapezoid

The area of a trapezoid is calculated as

Area = 1/2 * Sum of parallel base * Height

Using the above as a guide, we have the following:

1/2 * (b + 12) * 4 = 36

So, we have

b + 12 = 18

Evaluate

b = 6

Hence, the missing base of the trapezoid is 6 cm

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Answer:

the difference between bobby and the pool deck is 35 feet

Step-by-step explanation:

The difference between the pool deck and Bobby's position is 35 feet.

To find the difference between the pool deck and Bobby's position, we subtract Bobby's depth from the pool deck's height. The pool deck is 15 feet above sea level, while Bobby is diving 50 feet below sea level. We can compare these two numbers by subtracting 50 from 15.

15-50=35

The difference between the pool deck and Bobby's position is 35 feet.

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Answer:

Step-by-step explanation:

56 ft is the answer

Answer:

[tex]A1=28.3\ m^{2}[/tex],[tex]A2=78.5\ m^{2}[/tex],[tex]A3=132.7\ m^{2}[/tex]

Step-by-step explanation:

we know that

The area of a circle is equal to

[tex]A=\pi r^{2}[/tex]

Part 1) Find the area of the circle with diameter 6 m

we have

[tex]r=6/2=3\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A1=(3.14)(3)^{2}=28.3\ m^{2}[/tex]

Part 2) Find the area of the circle with diameter 10 m

we have

[tex]r=10/2=5\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A2=(3.14)(5)^{2}=78.5\ m^{2}[/tex]

Part 3) Find the area of the circle with diameter 13 m

we have

[tex]r=13/2=6.5\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A3=(3.14)(6.5)^{2}=132.7\ m^{2}[/tex]

The areas of the circles with diameters 6m, 10m, and 13m are approximately A₁=28.3 m², A₂ =78.5 m², and A₃=132.7 m², respectively.

To determine the area of each circle, we use the formula for the area of a circle: A = πr², where A is the area and r is the radius of the circle. First, we need to find the radii of the circles, which are half of their diameters.

Therefore, the areas of the circles are approximately 28.3 m², 78.5 m², and 132.7 m² respectively.

Answer:

K

Step-by-step explanation:

The factor that will change in the expression will be K.

n = number of walls

K = units of $/ft²

L = length of each wall in feet

h = height of each wall in feet

If the customer will request a more expensive brand of paint, the only factor that will change is K, because K holds the price of the paint over how many square feet.

By using more expensive brand of paint, the cost "K" would change.

Given:

painter's fee = nKLh

length= L

h= Height

n = number of walls.

The constant "K" represents the cost per square foot of the paint.

By using a more expensive brand of paint, the cost per square foot would increase.

Therefore, the value of "K" in the expression would change to reflect the new cost associated with the more expensive paint.

The other factors in the expression, "n" (number of walls), "L" (length of each wall), and "h" (height of each wall) remains same.

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Answer:

I think the answer would be B.

Answer:

Step-by-step explanation:

Volume = length * width * height.

Volume = 20 * 10 * 5

The tub can hold 1000 cubic feet of water.

The amount of water tub hold is 28316.8 liters.

Space is used by every three-dimensional object. The volume of this space is used as a measurement. The volume of an object in three-dimensional space is the amount of space it occupies within its boundaries. The capacity of the object is another name for it.

An object's volume can help us figure out how much water is needed to fill it, like how much water is needed to fill a bottle, aquarium, or water tank.

Given shape of bath tub is rectangular prism

which is  20 feet long, 10 feet wide, and 5 feet deep

length = 20 feet

width = 10 feet

height = 5 feet

to find the capacity of tub we need to calculate volume of tub

volume for rectangular prism = l x b x h

V = 20 x 10 x 5 = 1000 cubic feet

and 1 cubic feet = 28.3168 liter

1000 cubic feet = 28316.8 liter

Hence, the capacity of tub is 28316.8 liters.

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Answer:

11.3cm = RC

Step-by-step explanation:

We can find the radius by using Pythagorean theorem and finding RC

a^2 + b^2 = c^2

RQ^2 + QC^2 = RC^2

We are given QC = 8

QC is the perpendicular bisector of PR so QR is 1/2 PR

QR = 1/2 (16) = 8

RQ^2 + QC^2 = RC^2

8^2 + 8^2 = RC^2

64+64 = RC^2

128 = RC^2

Take the square root of each side

sqrt(128) = sqrt(RC^2)

11.3137 = RC

Rounding to the nearest tenth

11.3cm = RC

Answer:

[tex]f(x)=5x-7[/tex]

Step-by-step explanation:

The given line passes through the points;

[tex](x_1,y_1)=(3,8)[/tex]

and

[tex](x_2,y_2)=(4,13)[/tex]

Let us find the slope using;

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{13-8}{4-3}[/tex]

[tex]m=\frac{5}{1}=5[/tex]

The equation is given by;

[tex]y-y_1=m(x-x_1)[/tex]

We substitute the values into the formula to obtain;

[tex]y-8=5(x-3)[/tex]

[tex]y=5x-15+8[/tex]

[tex]y=5x-7[/tex]

There the equation of the line is

[tex]f(x)=5x-7[/tex]

To find the equation of the line passing through the points (3,8) and (4,13), use the slope-intercept form, which is y = mx + b.

To find the equation of the line passing through the points (3,8) and (4,13), we can use the slope-intercept form which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). Substituting the coordinates of the points, we have m = (13 - 8) / (4 - 3) = 5 / 1 = 5.

Next, we can choose any of the two points to substitute into the equation to find the y-intercept (b). Using the point (3,8), we have 8 = 5(3) + b. Solving for b, we get b = -7.

Therefore, the equation of the line passing through the points (3,8) and (4,13) is y = 5x - 7.

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