If p is a nonconstant polynomial, a is a number and p(a) = 0, then..
p(x) can be factored into (x-a)q(x), where is a polynomial of degree one less than the degree of p. p(x) must be equal to the zero polynomial, i.e. p(x) = 0. p(x) must be equal to (x-a)².
p(x) must be equal to (x-a)(x-b) where b is another number.

The number of minutes that Greg rode on the scooter is; 69 minutes

Algebraic word problems are defined as questions that require translating sentences to equations, then solving those equations. and a single variable.

The general form for the equation of a line in slope intercept form is;

y = mx + c

where;

m is slope

c is y-intercept

Initial fee paid is $2. Thus, y-intercept = $2.

Cost per minute = 12 cents of $0.12 and as such the slope is $0.12.

Thus, equation of line here is;

y = 0.12x + 2

where x is number of minutes

If total cost was $10.28, then;

10.28 = 0.12x + 2

0.12x = 8.28

x = 8.28/0.12

x = 69 minutes

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Answer:

Use the scale, 400:100

Step-by-step explanation:

400 on x axis, 100 on y axis
800 on x axis, 200 on y axis
Join the points with a line, extend it
That's it!

Answer:

[tex]\left(-\frac{3}{5},\frac{4}{5} \right)[/tex]

Step-by-step explanation:

We have to find [tex]n\in\mathbb{N}[/tex] such that [tex]\frac{1-n^2}{1+n^2} \neq 0[/tex] and [tex]\frac{2n}{1+n^2} \neq 0[/tex].

If we choose [tex]1\neq n\in\mathbb{N}[/tex]   [tex]\left(\frac{1-n^2}{1+n^2},\frac{2n}{1+n^2} \right)[/tex] is a rational point not on the x-axis or y-axis but on the unit circle.

Let [tex]n=2[/tex]

[tex]\left(\frac{1-2^2}{1+2^2},\frac{2*2}{1+2^2} \right)[/tex]

[tex]\left(\frac{1-4}{1+4},\frac{4}{1+4} \right)[/tex]

[tex]\left(-\frac{3}{5},\frac{4}{5} \right)[/tex]

Okun's law states that changes in unemployment and production are inversely related, with a coordinate of -2.

(a) When the unemployment rate stays the same in a given year, Okun's law's number 3.5 indicates the anticipated percent rise in yearly US national production.

(b) In accordance with Okun's law, a year in which unemployment increases from 5% to 8% results in a 1% decrease in national production.

(c) In accordance with Okun's law, a year in which production is equal to the previous year results in a decline of 1.75% in the unemployment rate.

(d) The inverse link between the unemployment rate and national output is represented by the coefficient -2 in Okun's law, which states that a rise in the unemployment rate would result in a fall in national production.

According to Okun's Law, the annual US national production's percent change (y) and the unemployment rate's percent change are connected (u). When the unemployment rate stays constant in a given year, the projected percent increase in national production is represented by the constant 3.5 in Okun's law. The inverse link between the unemployment rate and national output, represented by the coefficient -2 in Okun's law, states that a rise in the unemployment rate will result in a fall in national production. As a result, if unemployment increases from 5% to 8%, national production will decline by 1%. The change in the unemployment rate would need to be a decrease of 1.75 percent for national production to be the same as it was the previous year.

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The total number of fish caught during the festival can be calculated by adding the number of fish caught for each value of n.

For example, if 10 contestants caught 4 fish, 5 caught 8 fish, 7 caught 10 fish, and 8 caught 7 fish, the total number of fish caught would be 104 + 58 + 710 + 87 = 330.

The argument states that the total number of fish caught during the Frostbite Falls Fishing Festival can be determined by adding the number of fish caught by each contestant.

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Answer:

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

Step-by-step explanation:

for the equation to be true for all values of x , then both sides have to be the same.

expand both sides and equate like terms

left side

12(3n - [tex]\frac{1}{2}[/tex] x) ← distribute each term in the parenthesis by 12

= 36n - 6x

right side

- [tex]\frac{2}{3}[/tex] (9x - 18) ← distribute each term in the parenthesis by - [tex]\frac{2}{3}[/tex]

= - 6x + 12

we require

36n - 6x = - 6x + 12

that is

36n = 12 ( divide both sides by 36 )

n = [tex]\frac{12}{36}[/tex] = [tex]\frac{1}{3}[/tex]

for the equation to be true for all x then n = [tex]\frac{1}{3}[/tex]

One triangles can be made with these conditions.

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Answer: 17 units

Step-by-step explanation: A perimeter of a rectangle can be represented by the equation:

2(L)+2(W) = P

L represents length, W represents width and P represents perimeter

Lets say DC is width so AB has to be width aswell and then AD and BC are length.

We are given width and length as equations and P as 76 so we can substitute it into our equation to get:

2(3x-1) + 2(4x-3) = 76

We are trying to solve x so distributing would make it easy to obtain and we get:

6x - 2 + 8x - 6 = 76

Simplifying and solving for x gives us:

6x - 2 + 8x - 6 = 76

14x - 8 = 76

14x = 84

x = 6

Finally, we know AD represents our Width and AD = BC since we know BC is 3x-1, we can plug x in to get BC which is equal to AD to get:

3x - 1 = BC

3(6) - 1 = BC

17 = BC

since BC = AD

17 = AD

The system of equations that represents the given situation is:

q + d = 35

q*0.25 + d*0.10 = 4.90

Here we want to write a system of equations that represents the given situation.

First, let's define the two variables:

q = number of quarters.

d = number of dimes.

There are 35 coins in total, so the first equation is:

q + d = 35

And the value of these coins is $4.90, then the second equation is:

q*0.25 + d*0.10 = 4.90

Then the system of equations is:

q + d = 35

q*0.25 + d*0.10 = 4.90

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The sine function for this problem is defined as follows:

y = 2sin(2x).

The sine function for this problem is defined as follows:

y = Asin(B(x - C)) + D.

The parameters are described as follows:

The function has a minimum value of -2 and a maximum value of 2, for a difference of 4, hence the amplitude is obtained as follows:

2A = 4

A = 2.

The period is of π, hence the coefficient B is given as follows:

2π/B = π

B = 2.

The function does not have a phase shift (when x = 0, y = 0), neither a vertical shift (between 2 x -1 = -2 and 2 x 1 = 2), hence it is defined as follows:

y = 2sin(2x).

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Answer: 2:5:7

Step-by-step explanation:

You need to find the greatest common factor of all three of them which is 5. Then, you divide each number by 5

The motion of the train, obtained from the velocity time graph, can be described as follows;

a) In the first 15 minutes, the train is accelerating at 200 km/h²

b) The distance between the two cities is 125 kilometers

c) The average speed is 62.5 km/h

The velocity of the train is the rate at which the displacement of the train changes with time of motion.

The graph indicates that 8 units = 1 hour

1 hour = 60 minutes

Therefore; 8 units = 60 minutes

1 unit = 60/8 minute/unit = 7.5 minutes

a) Acceleration = Change in velocity/Time = Δv/Δt

Therefore, acceleration of the train in the first 15 minutes = The change in the velocity in the first 15 minutes, divided by the time

The velocity of the train after 15 minutes = 50 km/h

The initial velocity of the train = 0 km/h

The change in time, Δt = 15 minutes = 0.25 hours, therefore;

Acceleration = (50 - 0)/0.25 = 200

(b) The distance is the area under the velocity time graph, therefore, the distance between the two cities is found as follows;

d = 0.5 × 0.25 × 50 + 6 × 0.25 × 50 + (1 + 0.5) × 0.5 × 50 + 0.5 × 0.25 × 50 = 125

(c) The average speed = Total distance ÷ Time

The time it takes the train to complete the journey = 2 hours

The average speed = 125 km/(2 hrs) = 62.5 km/h

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The annual percentage rate, APR, for the loan is about 5.486%

The annual percentage rate is the cost of a loan or interest on an investment per year during the loan term or period.

Price of the car = $35,000.00

The down payment = 20% of $35,000 = $7,000

The installment loan details include;

The number of installment = 48 monthly installment

The amount paid each month = $651

The total payment, A= 48 × 651 = 31248

The equal monthly installment (EMI) formula is presented as follows;

[tex]EMI = P\cdot \frac{r\cdot (1+r)^n}{(1+r)^n -1}[/tex]

Where;

EMI = The equal monthly installment paid = $651

P = The principal amount of the loan

r = The interest rate

n = Number of payments to be made = 48

The principal is therefore;

P = $35,000.00 - $7,000.00 = $28,000.00

Therefore;

The

[tex]651 = 28000\times \frac{r\cdot (1+r)^{48}}{(1+r)^{48} -1}[/tex]

Using an online calculator and by trial and error, or by graphing we get;

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Answer:

[tex]\boxed{x^4+4x^3y+6x^2y^2+4xy^3+y^4\\}[/tex]

Step-by-step explanation:

Pascal’s triangle defines the coefficients which appear in binomial expansions. That means the nth row of Pascal’s triangle comprises the coefficients of the expanded expression of the polynomial (a + b)ⁿ

To create Pascal's triangle for any n in (a + b)ⁿ ,

Here is Pascal's triangle for the first 5 rows. The first row is row 0

Row #                                      

0                  1              

1               1    1            

2            1    2    1        

3         1    3    3    1      

4      1    4    6    4    1

[tex]\textrm{ Here row 0 represents $(a + b)^0$ }[/tex] =   [tex]\boxed{\textbf{1}}[/tex]

[tex]\textrm{Row 1 corresponds to $(a + b)^1$} =[/tex]  [tex]\boxed{1}\;a + \boxed{1}\;b[/tex]

[tex]\textrm{Row 2 corresponds to $(a + b)^2$} = \boxed{1}\;a^2 + \boxed{2} \;ab +\boxed{1}\;b^2[/tex]

[tex]\textrm{Row 3 corresponds to $(a + b)^3$} = \boxed{1}\;a^3 + \boxed{3}\;a^2b + \boxed{3}\;ab^2 + \boxed{1}\;b^3[/tex]

[tex]\textrm{Row 4 corresponds to $(a + b)^4$} = \boxed{1}\;b^4 + \boxed{4} \;a^3b +\boxed{6}\;a^2b^2 + \boxed{4} \;ab^3 + \boxed{1} \;b^4[/tex]

For the specific problem, (4x + 3y)³ use coefficients of row 3 and use 4x instead of a and 3y instead of b

[tex]\mbox{\normal(4x + 3y)^3= \boxed{1}\;(4x)^3 + \boxed{3}\;(4x)^2(3y) + \boxed{3}\;4x(3y)^2 + \boxed{1}\;(3y)^3}[/tex]

[tex]= \boxed{x^4+4x^3y+6x^2y^2+4xy^3+y^4\\} \;\;\; \textrm{ANSWER}[/tex]

Step-by-step explanation:

l don't know how to solve

it

Answer:

The dimensions of the strongest rectangular beam that can be cut from the log are:

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

From observation of the given diagram, the diameter of the circle is the hypotenuse of a right triangle (with legs w and d).  Given that the diameter is 48 cm, substitute the values into Pythagoras' Theorem and rearrange to create an expression for d²:

[tex]\begin{aligned}w^2+d^2&=48^2\\w^2+d^2&=2304\\d^2&=2304-w^2\end{aligned}[/tex]

Substitute the found expression for d² into the given strength equation:

[tex]\begin{aligned}s&=kwd^2\\s&=kw(2304-w^2)\end{aligned}[/tex]

Hence showing that s = kw (2304 - w²).

To find the value of w that will maximise the strength of the rectangular beam, first differentiate the equation for s in terms of w:

[tex]\begin{aligned}s&=kw(2304-w^2)\\s&=2304kw-kw^3\\\implies \dfrac{\text{d}s}{\text{d}w}&=2304k-3kw^2\\&=3k(768-w^2)\end{aligned}[/tex]

Set the differentiated equation to zero and solve for w:

[tex]\begin{aligned}3k(768-w^2)&=0\\ 768-w^2&=0\\w^2&=768\\w&=\sqrt{768}\\w&=\sqrt{256 \cdot 3}\\w&=\sqrt{256}\sqrt{3}\\w&=16\sqrt{3}\\\end{aligned}[/tex]

Substitute the value of w that maximises the strength of the beam, together with the given hypotenuse of 48 cm, into Pythagoras' Theorem and solve for d:

[tex]\begin{aligned}w^2+d^2&=48^2\\768+d^2&=2304\\d^2&=2304-768\\d^2&=1536\\d&=\sqrt{1536}\\d&=\sqrt{256 \cdot 6}\\d&=\sqrt{256}\sqrt{6}\\d&=16\sqrt{6}\end{aligned}[/tex]

Therefore, the dimensions of the strongest rectangular beam that can be cut from the log are:

Answer:

4/5

Step-by-step explanation:

The constant of proportionality is the number that relates the two quantities in a proportion. In the proportion (2/5) and (1/2), the constant of proportionality is the same for both fractions. To find the constant of proportionality, you can divide the two fractions and compare the results.

For example:

(2/5) / (1/2) = (2/5) * (2/1) = 4/5

So the constant of proportionality is 4/5.

The measure of angle POQ is 118°.

Having three edges and three vertices, a triangle is a three-sided polygon. The fact that a triangle's internal angles add up to 180 degrees is its most crucial characteristic. This characteristic is known as the triangle's angle sum property.

Given ΔPQR

∠R = 62°

QS and PT are altitudes,

Take ΔQRS

∠R = 62°

∠QSR = 90°

so, ∠SQR + ∠QSR + ∠R = 180°

∠SQR = 180 - 90 - 62

∠SQR = 28°

Taking ΔPRT

∠R = 62°

∠PTR = 90°

so, ∠RPT + ∠PTR + ∠R = 180°

∠RPT = 180 - 90 - 62

∠RPT = 28°

now let  ∠TPQ = x° and ∠SQP = y°

in ΔPQS,

∠SPQ = 28 + x

∠PQS = y°

∠QSP = 90°

∠QSP + ∠PQS + ∠SPQ = 180°

90 + y + 28 + x = 180

x + y = 180 - 90 - 28

x + y = 62°

now in  ΔPOQ

∠OPQ = x°, ∠OQP = y°

∠POQ + ∠OPQ + ∠OQP = 180°

∠POQ + x + y = 180

sunstitute value of x + y =  62°

∠POQ = 180 - 62

∠POQ = 118°

Hence the value of angle POQ is 118°.

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An integral for the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves y = tan² x, y = 3, and x = 0 about the line y = 3 is V = ∫[0, arctan(√3)] 2πx(3 - tan²(x)) dx.

The method of cylindrical shells can be used to set up the integral for the solid's volume after rotating the first quarter region defined by the curves y = tan² x, y = 3, and x = 0 about the line y = 3.

We must take into account an infinitesimally thin strip of width dx at a distance x from the y-axis in order to set up the integral.

This cylindrical shell's volume can be estimated by multiplying its height, radius, and thickness (dx) together. Consequently, each shell's volume is as follows:

dV = 2πx(3 - tan²(x)) dx

Setting 3 = tan²(x), we have:

tan²(x) = 3

tan(x) = √3

x = arctan(√3)

Thus, the integral for the volume of the solid is: V = ∫[0, arctan(√3)] 2πx(3 - tan²(x)) dx

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Answer:

Step-by-step explanation:

4. short leg= 4

      hyp= 8

5. leg 2= 5

    hypotenuse= 5√2

6. both legs = 2√2

The integer that represents the score in points is -42

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

Thumbs down = 42

A thumbs down indicates a negative rating

So, we have

Point = -42

Hence, the score is -42

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The calculation is performed in three stages, and the result represents the volume of the object in cubic units.

The expression ∭ e ⎛ ⎝y ln x z ⎞ ⎠dv represents a triple integral, which is a type of mathematical operation that calculates the volume of a 3-dimensional object bounded by certain conditions.

In this particular case, the volume is defined by the limits of integration (1 ≤ x ≤ e, 0 ≤ y ≤ ln x, 0 ≤ z ≤ 1), meaning that it considers only the points in the space that satisfy these conditions.

To understand the triple integral, it is important to understand the concept of single and double integrals.

The calculation of a triple integral is performed in three stages.

First, the integral with respect to x is calculated, then the integral with respect to y, and finally the integral with respect to z.

The expression e ⎛ ⎝y ln x z ⎞ ⎠ represents the integrand, or the function that we are integrating. In this case, it is a product of three functions of x, y, and z.

The ∭ symbol represents the triple integral and the dv represents the differential of the volume, which is used to calculate the volume in small increments.

In conclusion, the triple integral

=> ∭ e ⎛ ⎝y ln x z ⎞ ⎠dv

calculates the volume of a 3-dimensional object defined by the limits of integration.

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In Triangle ABC which is an Isosceles triangle. AB = BC where mLB = 34° so mLA = 73°

Triangles that have at least two sides of equal length are said to be isosceles.

Sum of all angle of a traingle is 180°

So mLA + mLB +mLC = 180

2(mLA) + mLB = 180

2(mLA ) = 180 - 34

2(mLA ) = 146

mLA = 73°

An isosceles triangle has three angles, which, like other triangles, sum up to 180 degrees. The angles other than the apex angle among the three inner angles are identical in size. According to the isosceles triangle theorem, an isosceles triangle has equal-sized angles on either side of its equal sides. As a result, we have B = C in the isosceles triangle ABC where AB = AC.

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7 + 6m is solution of parenthesizes equation .

What is parentheses in maths ?

These are also referred to as round brackets and are represented by the following symbols: ( ). The most typical brackets are those mentioned above.

                                 They are utilized to combine several numbers and formulae. In an equation, terms are grouped together or the order of operations is stated using parentheses or "round brackets."

Rewrite 12-(5+6m)

= 12 - 5 + 6m

=  7 + 6m

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The fish pulled the fishing line at a distance of 402.92 cm ≈ 402 cm.

We can use the formula for the circumference of a circle to find the distance the fishing line was pulled:

C = 2πr

Where C is the circumference, π (pi) is approximately equal to 3.14, and r is the radius of the spool.

The circumference of the spool is:

C = 2 × 3.14 × 4 = 25.12 cm

So, the distance the fishing line was pulled is:

25.12 × 16 = 402.92 cm

So, the fish pulled the fishing line at a distance of 402.92 cm ≈ 402 cm.

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--The given question is incorrect; the correct question is

"Bilal's fishing line was on a spool with a radius of 4cm. Suddenly, a fish pulled on the line, and the spool spun 16 times before Bilal began to reel in the fish. What is the distance the fish pulled the fishing line?"--

The imaginary part of the given expression 1/ ( 1+i ) is -i / 2.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the expression is 1 / ( 1 + i). The imaginary part will be calculated as:-

E = 1 / ( 1 + i)

Rationalize the numbers,

E = 1 / ( 1 + i)

E = [ 1 / ( 1 + i) ] [ ( 1 - i ) / ( 1 - i ) ]

E = ( 1 - i ) / ( 1² +1 )

E = ( 1 - i ) / 2

E = ( 1 / 2 ) - ( i / 2 )

The imaginary part will be -( i / 2 ).

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Answer:

519.2 in²

Step-by-step explanation:

First find the slant height  (s) of the triangle that makes up 1 side of the pyramid:

s² = 18² + (24/2)² = 468

s= √468 = 21.63

Now find the area of one side of the pyramid:

A = 1/2bh = 1/2(24)(21.63) = 519.2 / 2 = 259.56

Total area (all 4 sides) of the pyramid = 4(259.56) = 1038.2

1/2 the surface area is yellow:

1038.2 / 2 = 519.2 in²

The correct answer is option 3. [tex]4x^2 + 17[/tex] expressions represent the total perimeter of her sandwich, and if x = 1.2, the approximate length of the crust is 21.8 centimeters.

The first expression, [tex]4x^2 + 17[/tex] , represents the total perimeter of the sandwich. When x = 1.2, the expression evaluates to[tex]4 * 1.2^2 + 17 = 21.8[/tex].

So the total perimeter of the sandwich is 21.8 cm. However, this length is not the length of the crust, since the length of the crust would only be a portion of the total perimeter.

The expression [tex]4x^2 + 17[/tex] represents a mathematical formula for the total perimeter of the sandwich. The variable x represents the thickness of each slice of bread. The expression [tex]4x^2[/tex] represents the total length of the four edges of the bread slices, and the 17 represents the total length of the filling. When x = 1.2, the expression evaluates to 21.8, which represents the total perimeter of the sandwich.

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