During the first four months of the year, Jack earned $1270, $1150, $870 and $1450 If Jack must have an average salary of at least $1150 in order to earn retirement benefits, what must Jack earn in the fifth month in order to qualify for benefits?

[tex](g\circ f)(x)=(2x+8)^4\\\\(g\circ f)(-3)=(2\cdot(-3)+8)^4=2^4=16[/tex]

Final answer:

To find (g°f)(-3), we first calculate f(-3) = 2, and then g(2) which equals 16. Therefore, (g°f)(-3) is 16.

Explanation:

The question provided is asking us to compute the composition of two functions, denoted as (g°f)(-3), where f(x) and g(x) are both defined algebraically.

The composition of functions refers to applying one function to the results of another.

To find (g°f)(-3), we first evaluate f(-3) and then use that result as the input for g(x).

Firstly, let's evaluate f(-3):
f(x) = 2x + 8f(-3) = 2(-3) + 8 = -6 + 8 = 2

Now, we evaluate g(2) using the result from f(-3):
g(x) = x⁴g(2) = 2⁴ = 16

Therefore, (g°f)(-3) = g(f(-3)) = g(2) = 16.

Answer:

309

Step-by-step explanation:

(2x-)^5

step 1

1 5 10 10 5 1

step 2

1(2x¹-3^0) 5(2x²-3^5) 10(2x³-3^4)

10(2x^4-3³) 5(2x^5-3²)

1(2x^0-3¹)

step 3

(2x) (10x²-243) (20x³-81) (20x^4-27)

(10x^5-9) (2-3)

step 4

2 -243-61-7 +1 -1=309

The coefficient of [tex]x^3[/tex] in the expansion of [tex](2x-3)^5[/tex] is 720

The correct answer is an option (E)

"It refers to a number or quantity placed with a variable."

"The alphabetic character that expresses a numerical value."

"It is a mathematical expression which consists of numbers, variables and mathematical operations."

" [tex](a-b)^3=a^3-3a^2b+3ab^2- b^3[/tex] "

" [tex](a-b)^2=a^2-2ab+b^2[/tex] "

For given question,

We have been given an expression [tex](2x-3)^5[/tex]

We need to expand given expression.

[tex](2x-3)^5\\\\= (2x-3)^3\times (2x-3)^2\\\\=[(2x)^3-3(2x)^23+3(2x)(3)^2- 3^3]\times ((2x)^2-2(2x)(3)+3^2)\\\\=[8x^3-36x^2+54x-27]\times [4x^2-12x+9]\\\\=-243+810x-1080x^2 +720x^3-240x^4+32x^5\\\\=32x^5-240x^4+720x^3-1080x^2+810x-243[/tex]

Therefore, the coefficient of [tex]x^3[/tex] in the expansion of [tex](2x-3)^5[/tex] is 720

The correct answer is an option (E)

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Ella can make 21 complete envelopes with her available red, white, and blue pages, since the number of complete sets is limited by the blue pages.

Ella has 48 red pages, 25 white pages, and 84 blue pages. She needs to put 2 red pages, 1 white page, and 4 blue pages in every envelope. To find out how many complete envelopes she can make, we need to calculate the number of complete sets of pages she can create based on the quantity of each color she has and the number she needs for each envelope.

For the red pages: 48 red pages / 2 pages per envelope = 24 envelopes

For the white pages: 25 white pages / 1 page per envelope = 25 envelopes

For the blue pages: 84 blue pages / 4 pages per envelope = 21 envelopes

Since the number of envelopes is limited by the color with the least amount of complete sets, Ella can make 21 complete envelopes because she will run out of blue pages after the 21st envelope.

Answer:

$384.00

Step-by-step explanation:

Since $8..0 is 1 hour, just multiply 48 x 8! Hope this helps!

Answer:

320

Step-by-step explanation:

The reason that you  48* 8 is because the 48 has overtime included in it so you would only 40*8.

Answer:

The product is 25

Step-by-step explanation:

we know that

The complex number z1 is equal to

z1=(-4-3i)

we know that

To find the complex conjugate of (-4 - 3i) we change the sign of the imaginary part

so

The conjugate is equal to (-4+3i)

therefore

[tex](-4-3i)(-4+3i)=16-9(-1)=25[/tex]

Answer:

[tex](-4-3i) (- 4 + 3i) = 25[/tex]

Step-by-step explanation:

Notice in the graph that z1 has a real component of -4 and an imaginary component of -3.

Then we know that:

[tex]z_1 = -4-3i[/tex]

By definition for an imaginary number of the form [tex]a-bi[/tex] its conjugate will always be the number [tex]a + bi[/tex]

So the conjugate of [tex]z_1[/tex] is:

[tex]-4 + 3i[/tex]

The product of both numbers is:

[tex](-4-3i) (- 4 + 3i) = 16-12i + 12i-9i ^ 2\\\\(-4-3i) (- 4 + 3i) = 16-9 (-1)\\\\(-4-3i) (- 4 + 3i) = 16 + 9\\\\(-4-3i) (- 4 + 3i) = 25[/tex]

Answer:

see explanation

Step-by-step explanation:

Co terminal angles are angle ± 2π, that is

[tex]\frac{11\pi }{8}[/tex] + 2π

= [tex]\frac{11\pi }{8}[/tex] + [tex]\frac{16\pi }{8}[/tex] = [tex]\frac{27\pi }{8}[/tex]

and

[tex]\frac{11\pi }{8}[/tex] - 2π

= [tex]\frac{11\pi }{8}[/tex] - [tex]\frac{16\pi }{8}[/tex] = - [tex]\frac{5\pi }{8}[/tex]

Answer:

x = - 1

Step-by-step explanation:

The equation of the axis of symmetry for a parabola in standard form

y = ax² + bx + c : a ≠ 0 is found using

x = - [tex]\frac{b}{2a}[/tex]

y = - x² - 2x - 5 ← is in standard form

with a = - 1 and b = - 2, thus equation of axis of symmetry is

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

Equation of axis of symmetry is x = - 1

(10х — 4х2) - (7х + ?) = 3х – 6х2

10x - 7x = 3x which is given,

Now you have -4x^2 - ? needs to equal -6x^2

write it as an equation:

-4x^2 - ? = -6x^2

To solve for ? add 4x^2 to each side:

-? = -2x^2

Multiply each side by -1:

? = 2x^2

Answer:

C

Step-by-step explanation:

Given

- 2 > 5x - 37 ( add 37 to both sides )

35 > 5x ( divide both sides by 5 )

7 > x ⇒ x < 7 → C

Answer:

[tex]\large\boxed{\dfrac{(2x^3y)^3}{(4xy^2)^2(xy^3)}=\dfrac{x^6}{2y^4}}[/tex]

Step-by-step explanation:

[tex]\dfrac{(2x^3y)^3}{(4xy^2)^2(xy^3)}\qquad\text{use}\ (ab)^n\ \text{and}\ (a^n)^m=a^{nm}\\\\=\dfrac{2^3(x^3)^3y^3}{4^2x^2(y^2)^2xy^3}\qquad\text{cancel}\ y^3\\\\=\dfrac{8x^{(3)(3)}y^3\!\!\!\!\!\diagup}{16x^2y^{(2)(2)}xy^3\!\!\!\!\!\diagup}\\\\=\dfrac{8\!\!\!\!\diagup^1x^9}{16\!\!\!\!\!\diagup_2x^2y^4x}\qquad\text{use}\ a^na^m=a^{n+m}\\\\=\dfrac{x^9}{2x^{2+1}y^4}\\\\=\dfrac{x^9}{2x^3y^4}\qquad\text{use}\ \dfrac{a^n}{a^m}=a^{n-m}\\\\=\dfrac{x^{9-3}}{2y^4}\\\\=\dfrac{x^6}{2y^4}[/tex]

An outside angle is equal to the sum of the two opposite inside angles.

The triangle has an outside angle of 98, it's two opposite inside angles are given as 32 and X.

To find X, subtract 32 from 98.

X = 98 - 32 = 66

The answer is B.

Answer:

[tex]y=\frac{1}{2}x-6[/tex]

Step-by-step explanation:

The general equation of a line: [tex]y = ax+b[/tex]

In our case a is given

[tex]a=\frac{1}{2}[/tex]

Now we plug in for a and we get:

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

Now we plug in our coordinates into this equation and we solve for b

[tex]-3=\frac{1}{2}*6+b\\b = -6[/tex]

So the equation of the line is [tex]y=\frac{1}{2}x-6[/tex]

Answer:

[tex]y=\frac{1}{2}x-6[/tex]

Step-by-step explanation:

Since the Inclination of the line was already given to us. Let us use Inclination of line formula:

[tex]m=\frac{y-y_0}{x-x_0}[/tex]

Plugging it in those values then we have the equation of the line:

[tex]\frac{1}{2}=\frac{y+3}{x-6}[/tex]

Rewriting, multiplying both sides

2y+6=x-6

2y=x-12 Rearranging

[tex]y=\frac{1}{2}x-6[/tex]

Answer:

(4,0)

Step-by-step explanation:

So we have the system:

-x+5y=-4

4x+4y=16

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

We are asked to solve this for elimination.

We aren't totally setup for elimination though.

Both equations are in the same form, the form being ax+by=c form, so we are good there.

Now the other requirement is one the columns that contain variables need to be opposites (you will add) or same (you will subtract).

I'm going to multiply the first equation by 4.  The reason I'm going to do this is because the first column will contain -4x and 4x.  Those are opposites.  When you add opposites, you get 0 (they cancel).

So applying the multiplication of 4 to first equation gives:

-4x+20y=-16

4x+   4y=16

----------------- Add the equations now.

0x+24y=0    Again this is called elimination, because we made it to where a                         variable is canceled when combining the equations.

0x+24y=0

0+ 24y=0

     24y=0

Divide both sides by 24:

         y=0

So using one of the equations (you only need one of them) along with the y=0, let's find x.

I'm going to go with -x+5y=-4 with y=0.

Plugging in the 0 for y gives us:

-x+5y  =-4

-x+5(0)=-4

-x+  0  =-4

-x        =-4

Multiply both sides by -1:

x        =4

The solution is (x,y)=(4,0).

ANSWER

The ordered pair is

(4,0)

EXPLANATION

To solve a simultaneous equation using the elimination method means making the coefficient of one of the variables the same. We then add or subtract to eliminate that variable.

The given system has equations:

[tex] - x + 5y = - 4...(1)[/tex]

[tex]4x + 4y = 16...(2)[/tex]

Divide the second equation by 4

[tex]x + y = 4...(3)[/tex]

Add equation (1) and (3) to eliminate x.

[tex] \implies \: 5y + y = - 4 + 4[/tex]

[tex] \implies \: 6y = 0[/tex]

[tex] \implies \: y = \frac{0}{6} = 0[/tex]

Put

[tex]y = 0[/tex]

into the first equation to get:

[tex] - x + 5 \times 0 = - 4[/tex]

[tex] - x = - 4[/tex]

[tex] \therefore \: x = 4[/tex]

The ordered pair is

(4,0)

Answer:

[tex]\frac{1215}{32}[/tex]

Step-by-step explanation:

The n th term of a geometric sequence is

[tex]a_{n}[/tex] = a₁ [tex](r)^{n-1}[/tex], hence

[tex]a_{6}[/tex] = 5 × [tex](\frac{3}{2 )} ^{5}[/tex] = 5 × [tex]\frac{243}{32}[/tex] = [tex]\frac{1215}{32}[/tex]

The sixth term of the geometric sequence is 1215/32 given that the first term a₁=5 and common ratio r=3/2. This can be obtained by using formula for nth term of the geometric sequence.

Sequence is s collection of objects in a particular order and repetitions are allowed.

   Geometric Sequence:

a, ar, ar¹, ..., arⁿ⁻¹ is a geometric sequence, where a is the first term, r is the common ratio and arⁿ⁻¹ is the nth term.

From the definition, nth term of a geometric sequence is arⁿ⁻¹.

Given that, a₁=5 and r=3/2​

To find sixth term, put n=6 ⇒ arⁿ⁻¹=(5)(3/2)⁶⁻¹

                                                       =(5)(3/2)⁵

                                                       =5(3⁵/2⁵)

                                                       =5×(243/32)

                                                       =1215/32

Hence the sixth term of the geometric sequence is 1215/32 given that the first term a₁=5 and common ratio r=3/2.

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Step-by-step explanation and answer:

The given equations P=2T=3Q can be split up into

P=2T..................(1)

P=3Q................(2)

since they are all equal.

Rewrite the given equation

3Q + 4T = 26  .............(3)

as

3Q + 2(2T) = 26  ...........(4)

and substitute (1) and (2) into (4)

P + 2(P) = 26 .................(5)

Multiply (5) by two to get

6P = 2*26 = 52   ..............(6)

as given.

Answer:

see explanation

Step-by-step explanation:

Given P = 2T = 3Q, then

3Q + 4T = 26 can be expressed as

P + 2P = 26

3P = 26 ( multiply both sides by 2 )

6P = 52 ← as required

4x + 3y = -6

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

The angle on the circumference is half the angle subtended at the centre by the arc CB, that is

∠CAB = 0.5 × ∠COB = 0.5 × 96° = 48° → B

m∠CAB is 48° given that point O is the center of the circle and ∠COB is 96°. This can be obtained by using the circle theorem.

Given that, ∠COB = 96°

By applying Circle theorem, we can say that

                                 ∠COB = 2∠CAB

                                      96° = 2∠CAB

                                 ∠CAB = 96°/2 = 48°

                                 ∠CAB = 48°

Hence m∠CAB is 48° given that point O is the center of the circle and ∠COB is 96°.

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Answer: 5

times are properly represented on the chart, the other represents other possible colored teams

Step-by-step explanation:

There are 6 colors in the key, indicating there are at least 6 teams. But we only have a formal data set of 5 of those colors, therefore representing 5 teams.

The number of teams represented by each color in the pie chart are as follows:

Black: 24%, Navy blue: 21%, White: 19%, Gray: 16%, Maroon: 8%, Other: 12%

The pie chart shows that the dominant uniform color in the dataset is black, with 24% of teams represented by that color. This is followed by navy blue (21%), white (19%), gray (16%), and maroon (8%). The remaining 12% of teams are represented by a variety of other colors.

Here is a table showing the number of teams represented by each color:

Color | Number of teams

Black 24%

Navy blue 21%

White | 19%

Gray | 16%

Maroon | 8%

Other | 12%

Please note that the pie chart does not specify the total number of teams in the dataset, so it is not possible to say exactly how many teams are represented by each color. However, we can estimate that approximately 24% of the teams are black, 21% are navy blue, 19% are white, 16% are gray, 8% are maroon, and 12% are represented by a variety of other colors.

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Find the ratio of the known sides and calculate X.

1. The smaller triangle is half the size of the lager one:

6/12 = 1/2

10/12 = 1/2

This means x is half the length of 16

x = 16/2 = 8

2.  18/30  = 3/5

     9/15 = 3/5

This means x is 3/5 of 25

x = 25 * 3/5 = 15

3. 32/24 = 1 1/3

   16 /12 = 1 1/3

X is 1 1/3 times the length of 21

x = 21 x 1 1/3

x = 28

4.  12/15 = 4/5

    20/25 = 4/5

x is 4/5 the length of 40

x = 40 * 4/5

x = 32

Answer:

the answer is D

Step-by-step explanation:

a is not it has a negative

a b is not it has an equation in it

so that leaves it to see and d c does not work

because it has to have two variables per one number

For this case we have the following equation:

[tex]2x + 5 = -x + (- 1)[/tex]

Below are the correct steps to solve:

We eliminate the parenthesis taking into account that [tex]+ * - = -[/tex]

[tex]2x + 5 = -x-1[/tex]

We add "x" to both sides of the equation:

[tex]2x + x + 5 = -x + x-1\\3x + 5 = -1[/tex]

We subtract 5 from both sides of the equation:

[tex]3x + 5-5 = -1-5\\3x = -6[/tex]

We divide by 3 on both sides of the equation:

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

ANswer:

[tex]x = -2[/tex]

For this case we have the following function:

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

We look for the points of intersection with the x axis, doing y = 0

[tex]0 = 5-x\\x = 5[/tex]

We look for the points of intersection with the y axis, doing x = 0

[tex]y = 5-0\\y = 5[/tex]

We can also observe that the slope is -1.

[tex]y = mx + b\\y = -x + 5[/tex]

Answer:

See attached image

Answer:

[tex]x^{4}[/tex]

Step-by-step explanation:

[tex]\frac{x^{5}y^{2}  }{xy^{2} }[/tex]

= [tex]x^{4}[/tex]

Question 1:

For this case we have the following expression:

[tex]\frac {x ^ 5y ^ 2} {xy ^ 2}[/tex]

We eliminate similar terms in the numerator and denominator:

[tex]\frac {x ^ 5} {x}[/tex]

By definition of division of powers of the same base we have to place the same base and subtract the exponents:

[tex]\frac {x ^ 5} {x} = x ^ {5-1} = x ^ 4[/tex]

ANswer:

Option D

Question 2:

For this case we must indicate which expression is not equal to 125.

We tested option A:

[tex]5 (\frac {5 ^ 3} {\frac {2} {5}}) ^ 2 =[/tex]

Simplifying we have:

[tex]5 (\frac {5 ^ 3 * 5} {2}) ^ 2 =\\5 (\frac {5 ^ 8} {4}) =\\\frac {5 ^ 9} {4}[/tex]

The expression is not equal to 125.

Answer:

option A

Answer:

0.5

Step-by-step explanation:

Integers will not include decimals.

Integers are numbers you count with or the opposite of the counting numbers and also 0.

.5 is not a counting number (or the opposite of one or 0) so this is the one that isn't an integer.

Final answer:

Among the options provided, 0.5 is not an integer because it is a decimal number, whereas -22, 75, and 0 are all whole numbers and thus integers.

Explanation:

The question is asking to identify which of the given numbers is not an integer. An integer is defined as any whole number, including negatives, zero, and positive whole numbers. Therefore, out of the options given (0.5, -22, 75, 0), the number 0.5 is not an integer as it is not a whole number but a decimal.

Answer:

5/7

Step-by-step explanation:

Simplify the radical by breaking the radical up into a product of known factors.

Answer:

[tex]\frac{5}{7}[/tex]

Step-by-step explanation:

You can rewrite the expression as [tex]\frac{\sqrt{25}}{\sqrt{49}}[/tex].

Then you just need to simplify the numerator and denominator. the square root of 25 is 5, and the square root of 49 is 7, therefore the answer is [tex]\frac{5}{7}[/tex]

Answer:

(5,-3) if the system is

y=x-8

2x+3y=1

Step-by-step explanation:

I think the system is to read:

y=x-8

2x+3y=1.

Please correct me if I'm wrong.

I'm going to plug 1st equation into 2nd equation giving me:

2x+3(x-8)=1      ->I replaced y with (x-8).

Distribute:

2x+3x-24=1

Combine like terms:

5x-24=1

Add 24 on both sides:

5x=25

Divide both sides by 5:

x=5

If y=x-8 and x=5, then y=5-8=-3.

The solution is (5,-3).

(5, −3) is the solution of given system of equations.

"It is a mathematical statement which consists of equal symbol between two algebraic expressions."

"It is set of equations for which we find a common solution."

"It is a method of solving system of equation we substitute the value of a variable found by one equation in the second equation."

For given question,

We have been given a system of equations.

y = x - 8                ..............(i)

2x + 3y = 1                 ...............(ii)

We use substitution method to solve given system of equations.

Substitute the value of y from (i) to equation (ii).

⇒ x + 3(x - 8) = 1

⇒ 2x + 3x - 24 = 1

⇒ 5x = 1 + 24

⇒ 5x = 25

⇒ x = 5

Substitute above value of x in an equation (i)

⇒ y = 5 - 8

⇒ y = -3

So, the solution of given system of equations is x = 5 and y = -3

Therefore (5, -3) is the solution of given system of equations.

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Answer:

4

Step-by-step explanation:

6/7 divided by 3/14​ is 4.

Find common denominators.

6/7 would be changed into 12/14.

3/14 would stay the same because its denominator is already 14.

12/3 = 4

Therefore, 6/7 divided by 3/14​ is 4.

Answer:

The answer is 4

Step number one use the butterfly method:

6 14

-- ×---- =84/21

7 3

Step 2: Divide since it's improper

84÷21=4

4 21

-- × --- =4

1 21

Answer:

x^4+100x^2-5x^2-500=0

x^2(x^2+100)-5(x^2+100)=0

(x^2+100)(x^2-5)=0

from first expression

x^2+100=0

x^2=-100

x= root under -100

Therefore x=-10

from next expression

x^2-5=0

x^2=5

Or x=2.23

Step-by-step explanation:

Answer:

x = ± sqrt(5)

x = ±10i

Step-by-step explanation:

x^4+ 95x^2-500=0

What two numbers multiply to -500 and add to 95

-5 * 100 = -500

-5 +100 = 95

(x^2 - 5) (x^2 + 100) = 0

Using the zero product property

x^2 -5 =0          x^2 +100 =0

x^2 -5+5 =0      x^2 +100-100 = 0-100

x^2 =5               x^2 = -100

Take the square root of each side

x = ± sqrt(5)       x = imaginary numbers

                           x = ±10i

Answer:

(0.618,4.236) and (-1.618,-0.236)

Step-by-step explanation:

To find the intersection, we are looking for a common point between the curves.

We are solving the system:

[tex]y=x^2+3x+2[/tex]

[tex]y=2x+3[/tex].

I'm going to do this by substitution:

[tex]x^2+3x+2=2x+3[/tex]

Subtract 2x and 3 on both sides:

[tex]x^2+1x-1=0[/tex]

[tex]x^2+x-1=0[/tex]

To solve this equation I'm going to use the quadratic formula:

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

To find [tex]a,b,\text{ and }c[/tex], you must compare [tex]x^2+x-1=0[/tex]

to [tex]ax^2+bx+c=0[/tex].

[tex]a=1,b=1,c=-1[/tex].

Now inputting the values into the quadratic formula gives us:

[tex]x=\frac{-1\pm\sqrt{(1)^2-4(1)(-1)}}{2(1)}[/tex]

[tex]x=\frac{-1\pm\sqrt{1+4}}{2}[/tex]

[tex]x=\frac{-1\pm\sqrt{5}}{2}[/tex]

This means you have two solutions:

[tex]x=\frac{-1+\sqrt{5}}{2} \text{ or } x=\frac{-1-\sqrt{5}}{2}[/tex]

It does say approximately.

So I'm going to put both of these in my calculator and I guess round to the nearest thousandths.

[tex]x=0.618 \text{ or } x=-1.618[/tex]

Now to find the corresponding y coordinates, I need to use one the equations along with each x.

I choose the linear equation: y=2x+3.

y=2x+3 when x=0.618

y=2(0.618)+3=4.236

So one approximate point is (0.618,4.236).

y=2x+3 when x=-1.618

y=2(-1.618)+3=-0.236

So another approximate point is (-1.618,-0.236).

Answer:

20.6

Step-by-step explanation:

28.6×20.6=589.16

Answer:

20.6

Step-by-step explanation:

Area of a rectangle = base times height

A = bh

Substitute in what we know

589.16 = 28.6 * h

Divide each side by 28.6

589.16/28.6 = 28.6h/28.6

20.6=h