What is the maximum Number of different sizes of faces that a rectangular Prism can have.

The real solutions are -5, -4, 4, 5. There are no complex solutions.

The equation ...

... x^4 -41x^2 +400 = 0

can be factored as ...

... (x^2 -16)(x^2 -25) = 0

... (x -4)(x +4)(x -5)(x +5) = 0

So, all roots are real and are ...

... x ∈ {-5, -4, 4, 5}

_____

These are the values of x that make the factors zero.

Final answer:

The real solutions of the polynomial equation x^4 - 41x^2 = -400 are x = 4, -4, 5, and -5. There are no complex solutions since the equation can be factored into real numbers without the need for complex terms.

Explanation:

To find the real and complex solutions of the polynomial equation x^4 - 41x^2 = -400, we can begin by rewriting the equation in a more familiar quadratic form. Adding 400 to both sides gives us:

x^4 - 41x^2 + 400 = 0

We can set y = x^2, which turns our equation into:

y^2 - 41y + 400 = 0

Factoring this quadratic equation, we get:

(y - 16)(y - 25) = 0

So, y = 16 or y = 25. Since y = x^2, we solve for x:

x^2 = 16 → x = ±4

x^2 = 25 → x = ±5

Therefore, the real solutions are x = 4, -4, 5, -5. There are no complex solutions in this case because all values of x are real numbers.

2x^2 -5x -2

See the attached for the polynomial long division.

Answer:

rational: √81, √121

irrational: √89, √131

Step-by-step explanation:

You know your squares, so you know that 81 = 9² is a perfect square. Its square root is 9, a rational number.

And you know that 121 = 11², another perfect square. Its square root is 11, a rational number.

The remaining numbers are not the roots of squares of integers, so will be irrational.

Answer:

at least 43 1/4 minutes (or 44 rounded)

Step-by-step explanation:

I set up an equation:

175+6x=10x

Isolate x:

175=4x

x=43.25

is this a college question? (just curious)

Answer:

x=2, y=0

Step-by-step explanation:

x − 2y = 2

3x + y = 6

I am going to use elimination to solve this system.

Multiply the second equation by 2

2(3x+ y) =2* 6

6x +2y =12

Now add the first equation.  This will eliminate y

6x +2y =12

x − 2y = 2

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

7x = 14

Divide by 7

7x/7 =14/7

x=2

We still need to find y

3x + y = 6

3x+y=6

We know x so substitute it in

3(2) +y =6

6+y=6

Subtract 6 from each side

6+y-6 = 6-6

y=0

Answer:

60°

Step-by-step explanation:

The triangle PTX is equilateral, so all of its angles are 60°.

___

Adding up the lengths that make PX, you find they're the same as the lengths that make PT. The sides of ΔPQU are all the same length, so all the triangles are similar and equilateral.

Answer:

r(b) = 4b +3

Step-by-step explanation:

There are only two variables and the one that is not b is r. So, this amounts to solving for r. Divide the given equation by the coefficient of r:

... 4b = r -3

Add the constant:

... 4b +3 = r

Express in function notation.

... r(b) = 4b +3

Answer:

To see if a table of values represents a linear function, check to see if there's a constant rate of change. If there is, you're looking at a linear function!

Step-by-step explanation:

Answer:

Step-by-step explanation:

If it has an exponent, it's non-linear. (There are other ways it can be non-linear, too, but this is the one applicable here.)

The side length of the original square is calculated by associating the area of the triangle with the square's side length. It is determined to be 1.2 units.

To find the side length of the original square, we need to first understand how the area of a triangle is related to the square. When a square is cut on the diagonal, it forms two triangles. The area of a triangle is given by the formula 1/2 * base * height. In the case of these triangles, the base and height are the same, and equal to the side length of the square, which we'll call 'a'.

The area of one triangle is given as 0.72 square units. So, 0.72 = 1/2 * a * a. Simplifying this, we get a² = 2*0.72 = 1.44. Taking the square root of both sides, we find that a = √1.44 or a = 1.2 units.

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Answer:

an = 65 + 5[tex]_{n - 1}[/tex]

a1 = 65

Step-by-step explanation:

If the begining value is 65, then a1 has to be itL

a1 = 65

Then, if this amount increased every minute then 5 has to be added to this value each time:

an = 65 + 5

But this means that 5 is added to 65 every time with no increase, so the five must me multiplied by the n value:

an = 65 + 5[tex]_{n}[/tex]

a2 = 65 + 10 = 75

But this value should be 70 because it is the first point after 65 so it should be +5 not +10

To do this we can simply subtract one from the n value that is being multiplied by 5:

an = 65 + 5[tex]_{n - 1}[/tex]

Which is the recursive rule!

Answer:

2 cis (7/6 pi)

Step-by-step explanation:

r = sqrt( a^2 + b^2)

r = sqrt (-sqrt(3) )^2 + (-1)^2)

   = sqrt(3 +1)

    = sqrt(4)

    = 2

theta  = arctan (b/a)

theta = arctan (-1/-sqrt(3))

theta = 30

but this is in the third quardrant  -a and -b

so add 180

theta = 210 degrees

convert this to radians

210 * pi/180 = 210/180 * pi = 21/18 * pi = 7/6 * pi

r cis (theta)

2 cis (7/6 pi)

Answer:

Step-by-step explanation:

Given that total amount available is 90$

x represent the number of pens, and  y represent the number of pencils.

cost of each pen is 0.75 and cost of each pencil is 0.15

Then we have total cost = cost of pens + cost of pencils

= [tex]0.75x+0.15y[/tex]=90

Hence option B is right

For finding out the greatest number of each type of writing tool, we can make the other 0 and find out

Hence maximum x = [tex]\frac{90}{0.75} =120[/tex]

maximum y =[tex]\frac{90}{0.15} \\=600[/tex]

Part A

The equation that describes the number of club pens and pencils that the glee club can buy is 0.75x + 0.15y = 90

Part B

Greatest number of pens that the club can buy = 120

Greatest number of pencils that the club can buy = 600

Cost of each pen = $0.75

Number of pens = x

Cost of each pencil = $0.15

Number of pencils = y

Total cost on pens and pencils = $90

(Cost of each pen) x (Number of pens)  +  (Cost of each pencil)x(Number of pencils)  = Total cost

The equation that describes the number of club pens and pencils is:

0.75x  +  0.15y    =   90

The club buys the greatest number of pens when:

0.75x  = 90

x  =  90/0.75

x  =  120

Greatest number of pens that the club can buy = 120

The club buys the greatest number of pens when:

0.15x  = 90

x  =  90/0.15

x  =  600

Greatest number of pencils that the club can buy = 600

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Answer:

a = 0, b = 1/2

Step-by-step explanation:

Multiplication and division of complex numbers in polar form is pretty simple.

... a·Cis(α) × b·Cis(β) = ab·Cis(α+β) . . . . . multiply magnitudes; add angles

... a·Cis(α) / (b·Cis(β)) = (a/b)·Cis(α-β) . . . divide magnitudes, subtract angles

In your case, you have ...

... 2·Cis(120°) / (4·Cis(30°)) = (2/4)·Cis(120° -30°) = (1/2)·Cis(90°)

Of course, Cis(90°) = cos(90°) +i·sin(90°) = 0 +i.

... (1/2)·Cis(90°) = 1/2·(0 +i) = 0 + 0.5i

_____

Comment on the use of a calculator

It helps to be aware of how your calculator handles complex numbers. This one can express complex numbers in exponential format, equivalent to a polar form, but requiring specific notation. In this case, the variable D has been assigned the value π/180 so multiplying by it converts degrees to radians. The result of the division is (1/2)i, not 1/(2i).

Answer:

4.32%

Step-by-step explanation:

Step 1:

To solve this, we must know how to find sales tax first. Let's use this equation:

t = p × r

Let t stand for the total amount of sales tax

Let c stand for the purchase

Let r stand for the sales tax rate.

Step 2:

Now, let us plug in what is given. We know that the purchase made cost $2,500:

t = $2,500 x r

Annddd, we also know that the sales tax is $108:

$108 = $2,500 x r

Therefore, our equation is:

$108 = $2,500 x r

Step 3:

We can simplify this to:

$108 = $2,500r

Step 4:

All we need to do is divide each side by 2500 because the goal is get r all by itself

[tex]\frac{108}{2500} =\frac{2500r}{2500}[/tex]

Step 5:

This gives us:

r = 0.0432

Step 6:

We're not done! Since we are dealing with a percentage, we would multiply .0432 by 100% and that gives us our final answer of

4.32% is our sales tax

Answer:

It is irrational and equal to 4 + 5 square root of 2.

Step-by-step explanation:

[tex]\sqrt{2}(5+\sqrt{8})=5\sqrt{2}+\sqrt{2\cdot 8}=5\sqrt{2}+\sqrt{16}\\\\=4+5\sqrt{2}[/tex]

The square root of 2 in the result makes the result irrational.

Answer:

It is irrational since the square root sign in the answer is present.

Step-by-step explanation:

We are given with two factors: square root of 2 and 5 + square root of 8. We use a scientific calculator that displays irrational answers. The product of the two factors is 4 + 5 square root of 2. The answer is irrational since the square root sign in the answer is present.

[tex]\bf \textit{Logarithm of exponentials} \\\\ log_a\left( x^b \right)\implies b\cdot log_a(x) \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\bf 5\cdot 21^{5x}=16\implies 21^{5x}=\cfrac{16}{5}\implies \stackrel{\textit{taking \underline{log } to both sides}}{\log\left( 21^{5x} \right)=\log\left( \cfrac{16}{5} \right)} \\\\\\ 5x\log(21)=\log\left( \cfrac{16}{5} \right)\implies 5x=\cfrac{~~\log\left( \frac{16}{5}\right)~~}{\log(21)}\implies x=\cfrac{~~\log\left( \frac{16}{5}\right)~~}{5\log(21)} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill x\approx 0.0764094095937~\hfill[/tex]

Answer:

X=log 3.2/5 log 21

Step-by-step explanation:

X=0.0764094

Answer:

y=14x+75 where y is equal to 14x+75. Hope this helps.

Step-by-step explanation:

Answer:

y=14x+75

Step-by-step explanation:

y=mx+b is the slope intercept form which you will use for equations like these.

the M represents what it recurring, and in this case it is 14 dollars per month

The b represents your initial value and in this case it is 75 dollars

The 14x represents that it is 14 per, and the 75 represents the initial cost.

The area of the label for a cylindrical coffee can with a height of 14 cm and a radius of 5 cm is approximately 440 square centimeters.

The subject in this case is Mathematics, and the question pertains to the use of geometry to determine the area of a label on a cylindrical coffee can. Specifically, the question asks to calculate the area of the cylindrical surface (lateral surface), which will be used as the label area. To calculate this, one needs to use the formula for the lateral surface area of a cylinder, which is 2πrh, where r is the radius and h is the height of the cylinder.

Given a cylinder with a height of 14 centimeters and a radius of 5 centimeters, the area of the label can be calculated as follows:

Area = 2π(5 cm)(14 cm) = 2π×5 cm×14 cm = 2×3.1416×5 cm×14 cm ≈ 440 cm².

Answer:

8x10^-7

Step-by-step explanation:

First one you have correct however the second one you were close lets figure it out:

0.0000008

In order to move the decimal in front of the eight we have to "jump" 7 times to the right.  Furthermore since we have to move the decimal to the right our exponent will be negative.  That means that we have:

[tex]0.0000008=8 \times 10^{-7}[/tex]

~~~Brainliest Appreciated~~~

To express in scientific notation, move the decimal so it is after the first non-zero digit, and count the number of steps as the power of 10. 42,670,000 becomes 4.267 * 10^7 and 0.0000008 becomes 8 * 10^-7.

To convert a value to scientific notation, you need to express it as a product of a number between 1 and 10 and a power of 10.

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Answer:

1/2 of his book

Step-by-step explanation:

1/3 of an hour = 1/6 of the book

3/3 of an hour = 3/6 or 1/2 of the book read

Answer:

option 4

Step-by-step explanation:

tan30° = 17/x

=>1/√3 = 17/x

=>x = 17√3

for y,

sin30° = 17/y

=>1/2 = 17/y

=>y = 34

For this case we must find the value of y (Hypotenuse) and the value of x (leg adjacent to the 30 degree angle) of the triangle shown.

By definition:

[tex]Tangent (30) = \frac {17} {x}\\x = \frac {17} {Tangent (30)}\\x = \frac {17} {\frac {\sqrt {3}} {3}}\\x = \frac {17 * 3} {\sqrt {3}}\\x = \frac {51 \sqrt {3}} {3}\\x = 17 \sqrt {3}[/tex]

Now, we look for the value of y:

[tex]Sine (30) = \frac {17} {y}\\y = \frac {17} {sine (30)}\\y = \frac {17} {\frac {1} {2}}\\y = 34[/tex]

Answer:

[tex]x = 17 \sqrt {3}\\y = 34[/tex]

Option D

Answer:

36 cm²

Step-by-step explanation:

1. ΔBEF & ΔADF are similar; 2. the height of ΔABE is 3height of ΔBEF; 3. area of ΔABE is 3 area of ΔBEF; 4. area of ΔABE= area of ΔACE; 5. area of ΔABC = 2 area of ΔACE; 6. Area of ABCD = 2 area of ΔABC = 12 area of BEF.

36 cm²

Step-by-step explanation:

1. ΔBEF & ΔADF are similar; 2. the height of ΔABE is 3height of ΔBEF; 3. area of ΔABE is 3 area of ΔBEF; 4. area of ΔABE= area of ΔACE; 5. area of ΔABC = 2 area of ΔACE; 6. Area of ABCD = 2 area of ΔABC = 12 area of BEF.

y = (1/2)x - 5

Try the answers:

... -4 ≠ (1/2)·2 + 5

... -4 ≠ (1/2)·2 - 3

... -4 = (1/2)·2 - 5 . . . . . the third choice works

... -4 ≠ (1/2)·2 + 3

___

You can write the point-slope form equation and simplify.

... y -k = m(x -h) . . . . . . equation for line of slope m through point (h, k)

... y -(-4) = (1/2)(x -2) . . . filled in with your values, m=1/2, (h, k) = (2, -4)

... y = (1/2)x -1 -4 . . . . subtract 4, eliminate parentheses

... y = (1/2)x - 5 . . . . . simplified. (Matches the 3rd selection.)

Answer: y = one half x − 5

Step-by-step explanation:

We know that , the equation of line that passes through point (a,b) and has slope m is given by :-

[tex](y-b)=m(x-a)[/tex]

Given : Point = (a,b)=(2, -4)

Slope = [tex]\dfrac{1}{2}[/tex]

Then, the equation of the line will be (substitute all the values in the above formula) :-

[tex](y-(-4))=\dfrac{1}{2}(x-2)[/tex]

[tex]\Rightarrow\ y+4=\dfrac{1}{2}(x-2)[/tex]

[tex]\Rightarrow\ y+4=\dfrac{1}{2}x-1[/tex]

Subtract 4 from both sides , we get

[tex]y=\dfrac{1}{2}x-1-4[/tex]

[tex]y=\dfrac{1}{2}x-5[/tex]

Hence, the correct answer is y = one half x − 5

Answer:

C - 1

Step-by-step explanation:

The mean is the sum total of all the given value within the data set divided by the number of values in the data set. By counting, you can find that we have 6 values in our data set.

First thing you need to do is add all the numbers given together in order to get the sum of the values in the given data set. Follow these steps presented below:

-2 + 9 - 10 + 3 + 5 + 1

7 - 10 + 3 + 5 + 1

- 3 + 3 + 5 + 1

0 + 5 + 1

5 + 1

6

Now, in order to find the mean, take the sum (6) and divide by the amount of numbers there are; 6.

6/6 = 1

Therefore, the mean is 1.

Hope this helps!

Answer:

t = 3

Step-by-step explanation:

The ball thrown vertically upward reaches its maximum height after 3 seconds, as calculated by setting the derivative of the position function, which represents the velocity, to zero.

The question is about determining how many seconds it will take for a ball thrown vertically upward to reach its maximum height. The ball's position as a function of time is given by 30t - 5t2 meters. To find when the ball reaches its maximum height, we need to find the time at which the velocity is zero. The velocity function is the derivative of the position function, which is 30 - 10t. Setting the velocity equal to zero gives us the time when the ball reaches its maximum height:

0 = 30 - 10t

t = 3 seconds

The ball will reach its maximum distance from the ground after 3 seconds.

(a) To make use of the forms of the equation for a line this problem requires, we must first calculate the slope (m) of the line between the two given points. That is computed by ...

... m = (change in y)/(change in x)

... m = (8 -3)/(2 -0) = 5/2

The second given point tells us the y-intercept (b) is 3, so the equation for the line is ...

... y = mx + b

... y = 5/2x + 3

(b) As in part (a), we must calculate the slope of the line. That is ...

... m = (change in y)/(change in x)

... m = (4 -3)/(1 -3) = 1/(-2) = -1/2

Using the first point (h, k) in the point-slope form of the equation of a line, we have ...

... y -k = m(x -h)

... y -4 = -1/2(x -1) . . . . . point-slope equation

Simplifying this, we get

... y = -1/2x +1/2 +4

... y = -1/2x +9/2 . . . . . slope-intercept equation

(c) The slopes of the two equations are different, so they must intersect at exactly one point. The equations are consistent.

(d) We can substitute the "y" from one equation into the other to give ...

... 5/2x +3 = -1/2x +9/2

... 3x = 3/2 . . . . . . . add 1/2x -6/2

... x = 1/2 . . . . . . . . divide by 3

Using the first equation to find y:

... y = (5/2)·(1/2) +3 = 5/4 +12/4

... y = 17/4

The point of intersection is (x, y) = (1/2, 17/4).

Answer:

The correct answer is 1 / 18.

Step-by-step explanation:

We are given the following equation for which we need to solve for the variable n:

[tex]-\frac{7}{3n} =-42[/tex]

So for solving this, we will rearrange the equation in a manner that isolates the variable n, making it the subject of the equation.

[tex]-\frac{7}{3n} =-42[/tex]

[tex]-\frac{7}{42} =3n[/tex]

[tex]\frac{1}{6} =3n[/tex]

[tex]n=\frac{1}{6*3} \\\\n=\frac{1}{18}[/tex]

Therefore, the value of n is equal to 1 / 18.

Answer:

option C

Step-by-step explanation:

to find the solution for the equation 4x + 2 = x + 3, graph the equations

y= 4x+2  and y=x+3

Lets make a table for each equation

x          y= 4x+2

-1          4(-1) + 2= -4+2 = -2

0           4(0) + 2= 2

1             4(1) + 2= 6

plot all the points on graph

Lets make table for second equation

x          y= x+3

-1          -1+3 = 2

0           0+3 = 3

1             1+3 = 4

Plot the point on the graph and make a line

option C is correct

Answer:

The required graph of the equation is shown below:

Step-by-step explanation:

Consider the provided equation.

[tex]4x + 2 = x + 3[/tex]

Solve the equation for x.

[tex]4x-x=3-2[/tex]

[tex]3x=1[/tex]

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

Now substitute x=1/3 in [tex]y=4x + 2[/tex]

[tex]y=\frac{4}{3} + 2=\frac{10}{3}[/tex]

So, the graph will be the lines which intersect at (1/3,10/3)

The correct graph is 3rd one.

Alternate method:

The equation [tex]4x + 2 = x + 3[/tex] can be written as:

[tex]y= 4x + 2 \\y= x + 3[/tex]

Draw the graph for both equation.

For [tex]y= 4x + 2[/tex]

Substitute x=0 in above equation,

[tex]y= 4(0) + 2[/tex]

[tex]y=2[/tex]

Substitute y=0 in above equation,

[tex]0= 4x + 2[/tex]

[tex]-2=4x[/tex]

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

Now use the coordinate (0,2) and (-1/2,0) to draw the graph of straight line.

For [tex]y=x + 3[/tex]

Substitute x=0 in above equation,

[tex]y=3[/tex]

Substitute y=0 in above equation,

[tex]0=x + 3[/tex]

[tex]x=-3[/tex]

Now use the coordinate (0,3) and (-3,0) to draw the graph of straight line.

Hence, the required graph of the equation is shown below:

"Easiest" depends on several factors, including your understanding of what you're trying to do and of how the mean is calculated.

The most straight-forward way is to

One of my favorite ways is this.

If the distribution is reasonably symmetrical, this second method gives you fewer (and smaller) products to compute, and you can probably do them in your head.