Prove algebraically that
(4n+ 1)²- (2n-1) is an even number
for all positive integer values of n.

Answer:Taylor is 28 and Julian is 26.

Step-by-step explanation: Because their ages are even integers, we know that Taylor is 2 years older than Julian. We can write this as an equation 132 = 4x + x + 2  where x is Julian's age.

Subtract 2 from both sides and you have 130 = 4x + x.

4x + x is the same as 5x, 130 = 5x

Divide both sides by 5 and you get 26, Julian's age.

Taylor is 28 years old and Julian is 26 years old.

An equation is made up of two algebraic expressions with the same value and the sign "=" in between.

Given, Taylor is older than Julian.

Their ages are consecutive even integers.

Let x be the age of Julian and x + 2 be the age of Taylor.

According to the question;

We have the equation,

x + 2 + 4x = 132

5x = 132 - 2

5x = 130

x = 130 / 5

x = 26

Julian's age is 26 and Taylor's age is 26+2 = 28.

Therefore, the Taylor is 28 years old and Julian is 26 years old.

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The Solution for the given quadratic equation is Cosx = 1 and Cosx = -3 .

In the question ,

it is given that ,

the quadratic equation is cos²x + 2 cosx - 3 = 0 ,

let cosx = a ,

So , the equation becomes , a² + 2a - 3 = 0

writing the factors using the splitting the middle term method ,

we get ,

a² [tex]+[/tex] 3a -a - 3 = 0

a(a + 3) -1(a [tex]+[/tex] 3) = 0

(a - 1)(a [tex]+[/tex] 3) = 0

which means a - 1 = 0      or    a + 3 = 0

a [tex]=[/tex] 1    or     a [tex]=[/tex] -3 ,

Substituting a = cosx , we get

cosx [tex]=[/tex] 1   or cosx [tex]=[/tex] -3 ,

Therefore , The Solution for the given quadratic equation is Cosx = 1 and Cosx = -3 .

The given question is incomplete , the complete question is

Solve the given quadratic equation for Cosx , cos²x + 2 cosx - 3 = 0 .

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Answer:

[tex]\textsf{1.} \quad (x-1)^2=12(y+2)[/tex]

     See attachment 1.

2.  See attachment 2.

Step-by-step explanation:

Given values:

As the x-value of the vertex and focus is the same, the parabola has a vertical axis of symmetry.

The focus is always on the inside of the parabola.

Since p represents the distance from the vertex to the focus, and the distance from the vertex to the focus is 1 - (-2) = 3, then p = 3.

If p > 0, the parabola opens upwards, and if p < 0, the parabola opens downwards.  Therefore, as p > 0, the parabola opens upwards.

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Standard form of a parabola}\\(with a vertical axis of symmetry)\\\\$(x-h)^2=4p(y-k)$\\\\where:\\ \phantom{ww}$\bullet$ $p\neq 0$\\ \phantom{ww}$\bullet$ Vertex: \;$(h,k)$\\ \phantom{ww}$\bullet$ Focus:\; $(h,k+p)$\\ \phantom{ww}$\bullet$ Directrix: \; $y=(k-p)$\\ \phantom{ww}$\bullet$ Axis of symmetry: \; $x=h$\\\end{minipage}}[/tex]

Therefore:

Substitute the values into the formula:

[tex]\implies (x-h)^2=4p(y-k)[/tex]

[tex]\implies (x-1)^2=4\cdot 3(y-(-2))[/tex]

[tex]\implies (x-1)^2=12(y+2)[/tex]

See attachment 1 for the graph of the parabola.

Given equation of a parabola:

[tex](y-4)^2=8(x+2)[/tex]

As the y-variable is contained within the squared part of the equation, the parabola has a horizontal axis of symmetry.

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Standard form of a parabola}\\(with a horizontal axis of symmetry)\\\\$(y-k)^2=4p(x-h)$\\\\where:\\ \phantom{ww}$\bullet$ $p\neq 0$\\ \phantom{ww}$\bullet$ Vertex: \;$(h,k)$\\ \phantom{ww}$\bullet$ Focus:\; $(h+p,k)$\\ \phantom{ww}$\bullet$ Directrix: \; $x=(h-p)$\\ \phantom{ww}$\bullet$ Axis of symmetry: \; $y=k$\\\end{minipage}}[/tex]

Therefore:

If p > 0, the parabola opens to the right, and if p < 0, the parabola opens to the left.   Therefore, as p > 0, the parabola opens to the right.

Find the y-intercepts of the graph by substituting x = 0 into the equation:

[tex]\implies (y-4)^2=8(0+2)[/tex]

[tex]\implies (y-4)^2=16[/tex]

[tex]\implies y-4=\pm4[/tex]

[tex]\implies y=4\pm4[/tex]

[tex]\implies y=0, y=8[/tex]

To sketch the graph of the parabola, plot:

Draw a curve through the vertex and y-intercepts, opening to the right.

Use the axis of symmetry to ensure the curve is symmetric.

See attachment 2 for the graph of the parabola.

Answer:

64

Step-by-step explanation:

the line BCD is a straight line, so angle BCA plus the angle ACD will add to 180. therefore 180 - 114 = 66.

all three angles in a triangle add to 180, so when angle A is 50 and angle C is 66, then 180 - 50 - 66 = 64

Answer:

m∠B = 64°

Step-by-step explanation:

We are told that m∠A = 50° and m∠ACD = 114°. We are then asked to find m∠B.

From the diagram, we can see the angles ACD and ACB lie on a straight line. Therefore, their measures add up to 180°. Hence:

∠ACD + ∠ACB = 180°

⇒ 114° + ∠ACB = 180°

⇒ ∠ACB = 180° - 114°

⇒ ∠ACB = 66°

∴ m∠C = 66°

We know that the angles inside a triangle add up to 180°.

Now that we know the measures of ∠A and ∠C, we can calculate m∠B:

∠A + ∠B + ∠C = 180°

⇒ 50° + ∠B + 66° = 180°

⇒ ∠B = 180° - 66° - 50°

⇒ ∠B = 64°

Answer:

a.)

[tex]\left \{ {{y=4} \atop {y=x}} \right.[/tex]

See Image 1 for graph

b.)

[tex]\left \{ {{y=2x+2} \atop {y=2x-3}} \right.[/tex]

See Image 2 for graph

Step-by-step explanation:

A system that has exactly one solution is called a consistent independent system. These systems consist of 2 lines that intersect once. Here's an easy one to graph:

[tex]\left \{ {{y=4} \atop {y=x}} \right.[/tex]

See Image 1 for the graph of this system!

A system with no solutions is one where the lines of equalities never cross. They are parallel! Here's one that will impress your professor!

[tex]\left \{ {{y=2x+2} \atop {y=2x-3}} \right.[/tex]

See Image 2 for the graph of this system! Good luck!

After the two reflections, the coordinates are:

k'' = (2, 5)

We start with the point K located at (-2, -5), and first, we apply a reflection over the y-axis.

It will only change the sign of the x-value, then we will get the new point:

k' = (- (-2), -5) = (2, -5)

And now we apply a reflection over the x-axis, this will only change the sign of the y-value,then we will get:

k'' = (2, -(-5)) = (2, 5)

These are the coordinates after the two reflections.

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The statistics z is -1.878 and the p value is 0.0302

Given,

Number of random sample selected, n = 6967

Number of surveys returned, x = 1331

Estimated proportion of return rate;

p = 1331/6967 = 0.192

Significance level, ∝ = 0.01

z would represent the statistic

We investigate the assertion that the return rate is less than 20%, and the following hypotheses are tested:

Null hypothesis, p₀ ≥ 0.2

Alternative hypothesis, p₀ < 0.2

The statistics, z = (p - p₀) / √(p₀(1 - p₀)/n)

That is,

z = (0.191 - 0.2) / √(0.2(1 - 0.2)/6967) = -1.878

Using the alternative hypothesis and the following probability, we can now get the p value:

p value  = p(z < - 1.878) = 0.0302

We fail to reject the null hypothesis when the p value exceeds the significance level of 0.01 and there is insufficient evidence to support the conclusion that the return rate is less than 20% at 1% of significance.

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The set of ordered pairs that represent dilation factor of -3 is

c. A' (-3, -18) B' (-18, -18) C' (-3, -6)

Dilation is a method of transformation that magnify or shrink the preimage depending on the scale factor

The transformation rule for dilation is as follows

(x, y) for a scale factor of r → (rx, ry)

From the graph the coordinates are read and following the given transformation rule we have

preimage                           image

A (1, 3)   → (2 * -3, 4 * -3) →   A' (-3, -8)

B (6, 6)  → (4 * -3, 4 * -3) →    B' (-18, -18)  

C (1, 6)  → (3 * -3, 2 * -3) →    C' (-3, -6)

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Answer:

A.  Circle.

     [tex](x+15)^2+(y-0)^2=13^2[/tex]

B.  Parabola.

     [tex](x-15)^2=-16(y-20)[/tex]

C.  Maximum height of Tunnel A = 13 ft.

     Maximum height of Tunnel B = 20 ft.

     The truck can only pass through Tunnel B without damage.

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6.3cm}\underline{General equation for any conic section}\\\\$Ax^2+Bxy+Cy^2+Dx+Ey+F = 0$\\\\where $A, B, C, D, E, F$ are constants.\\\end{minipage}}[/tex]

Circle:  A and C are non-zero and equal, and have the same sign.

Ellipse:  A and C are non-zero and unequal, and have the same sign.

Parabola:  A or C is zero.

Hyperbola:  A and C are non-zero and have different signs.

Tunnel A

[tex]x^2+y^2+30x+56=0[/tex]

As the coefficients of x² and y² are non-zero, equal and have the same sign, the conic section is a circle.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

Rewrite the given equation for Tunnel A in the standard form of the equation of a circle:

[tex]\implies x^2+y^2+30x+56=0[/tex]

[tex]\implies x^2+30x+y^2-56[/tex]

[tex]\implies x^2+30x+\left(\dfrac{30}{2}\right)^2+y^2=-56+\left(\dfrac{30}{2}\right)^2[/tex]

[tex]\implies x^2+30x+225+y^2=-56+225[/tex]

[tex]\implies (x+15)^2+(y-0)^2=169[/tex]

[tex]\implies (x+15)^2+(y-0)^2=13^2[/tex]

Therefore, the center of the circle is (-15, 0) and the radius is 13.

Tunnel B

[tex]x^2-30x+16y-95=0[/tex]

There is no term in y² so the coefficient of y² is zero.  Therefore, the conic section is a parabola.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Standard form of a parabola}\\(with a vertical axis of symmetry)\\\\$(x-h)^2=4p(y-k)$\\\\where:\\ \phantom{ww}$\bullet$ $p\neq 0$. \\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex.\\\end{minipage}}[/tex]

Rewrite the given equation for Tunnel B in the standard form a parabola:

[tex]\implies x^2-30x+16y-95=0[/tex]

[tex]\implies x^2-30x=-16y+95[/tex]

[tex]\implies x^2-30x+\left(\dfrac{30}{2}\right)^2=-16y+95+\left(\dfrac{30}{2}\right)^2[/tex]

[tex]\implies x^2-30x+225=-16y+95+225[/tex]

[tex]\implies (x-15)^2=-16(y-20)[/tex]

Therefore, the vertex is (15, 20).

Maximum height of Tunnel A

The maximum point of a circle is the sum of the y-value of its center and its radius:

Maximum height of Tunnel B

The maximum point of a downwards opening parabola is the y-value of its vertex:

As the truck is 13.5 feet high, it cannot pass through Tunnel A since the maximum height of Tunnel A is 13 feet.

The maximum height of Tunnel B is certainly adequate for the truck to pass through.  However, to determine if the truck can pass through Tunnel B safely, we also need to find the width of the tunnel when its height is 13.5 feet.  To do this, find the x-values of the parabola when y = 13.5.  If the difference in x-values is 8 or more, then the truck can pass through safely.

Substitute y = 13.5 into the equation for Tunnel B and solve for x:

[tex]\implies (x-15)^2=-16(13.5-20)[/tex]

[tex]\implies (x-15)^2=-16(-6.5)[/tex]

[tex]\implies (x-15)^2=104[/tex]

[tex]\implies \sqrt{(x-15)^2}=\sqrt{104}[/tex]

[tex]\implies x-15=\pm\sqrt{104}[/tex]

[tex]\implies x=15\pm\sqrt{104}[/tex]

Now find the difference between the two found values of x:

[tex]\implies (15+\sqrt{104})-(15-\sqrt{104})[/tex]

[tex]\implies 15+\sqrt{104}-15+\sqrt{104}[/tex]

[tex]\implies 2\sqrt{104}[/tex]

[tex]\implies 20.39607...[/tex]

Therefore, as the width of Tunnel B is 20.4 ft when its height is 13.5 ft, the 8 ft wide truck can easily pass through without damage since 20.4 ft is greater than the width of the truck.

Answer:

if you don't mind is there a starting number or a chart so i can calculate and solve this for you!

The value is 10.81

From the question, we have

[tex]\sqrt{3^{4} +6^{2} } \\=\sqrt{81+36 }\\=\sqrt{117 }\\ = 10.81[/tex]

The value is 10.81

Addition:

When we say "the addition," we mean adding two or more numbers together. The plus sign (+) denotes addition of two numbers, therefore three is written as three plus three. You also have control over how often the plus sign (+) is used. such as 3 + 3 + 3 + 3. Mathematical addition is an essential part of children's education. Students learn basic addition sums in primary school, including one-digit facts, math addition sums, and double-digit math addition sums. Sums in addition are one of the most fundamental and fascinating Math concepts for elementary school students. Children first learn fundamental mathematical ideas like adding numbers in their early years because math is an enthralling subject.

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Answer:

54°

Step-by-step explanation:

We know that the triangles are congruent by HL, meaning that [tex]m\angle QSR=m\angle QST[/tex].

[tex]9a+48=6a+50 \\ \\ 3a=2 \\ \\ \implies m\angle QST=2(2)+50=54^{\circ}[/tex]

The expression can be represented in terms of exponent as 4³ b² c³ if the expression is 4 · 4 · 4 · b · b · c · c · c.

In mathematics, integer exponents are exponents that should be integers. It may be a positive or negative number. In this situation, the positive integer exponents determine the number of times the base number should be multiplied by itself.

It is given that:

The expression is:

=  4 · 4 · 4 · b · b · c · c · c

By using the properties of exponent:

= 4³ b² c³

4³ = 64

=  64b² c³

Thus, the expression can be represented in terms of exponent as 4³ b² c³ if the expression is 4 · 4 · 4 · b · b · c · c · c.

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Miguel's sister's red gumdrops weigh 0.5 pounds.

Formulate based on the conditions stated: [tex]\frac{3}{8} + x = \frac{7}{8}[/tex]

Unknown terms should be moved to the left side of the equation: [tex]x = \frac{7}{8} - \frac{3}{8}[/tex]

Numerators written over common denominator: [tex]x = \frac{7-3}{8}[/tex]

Determine the total or difference: [tex]x = \frac{4}{8}[/tex]

Remove the connecting element: [tex]x = \frac{1}{2}[/tex]

get the outcome: or x = 0.5 or 50%

A fraction is a mathematical symbol representing any quantity of equal parts, and it also represents a portion of a whole. In common English, the word "fraction" is used to describe the quantity of parts of a particular size. Fractions speed up and simplify calculations by making it easier to divide and evaluate numbers.

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The value of x that makes m║n is 40°

In geometry, the line that intersects two straight lines at distinct points is known as a transversal.

The angles that are formed when two parallel lines are intersected by a third line i.e a transversal are corresponding angles.

Note:

Two corresponding angles are formed by a transversal with two parallel lines that are equal

Here we have

The angle made by the intersecting line with line m is 120°

And the angle made by the same line with line n is 3x°

Let us assume that m║n  

From above observations

The given angle are corresponding angles

=> 3x° = 120°

=> x° = 40°  

Therefore,

The value of x that makes m║n is 40°

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The possible outcomes in the sample space of this experiment are

{(h,1),,(h,2),(h,3),(h,4),(h,5),(h,6),(t,1),(t,2),(t,3),(t,4),(t,5),(t,6)}

Jo flips a coin sample space for a coin = {h, t}

and rolls a 6-sided die sample space for a die = {1,2,3,4,5,6}

at a time coin is tossed and the die is rolled

the possible outcomes in the sample space of this experiment are

{(h,1),,(h,2),(h,3),(h,4),(h,5),(h,6),(t,1),(t,2),(t,3),(t,4),(t,5),(t,6)}

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The most likely to generalize to it's population of interest is a stratified random sample of 120

a stratified random sample 120

Stratified random sampling is a method of sampling that involves the division of a population into smaller subgroups known as strata. In stratified random sampling, or stratification, the strata are formed based on members’ shared attributes or characteristics, such as income or educational attainment. Stratified random sampling has numerous applications and benefits, such as studying population demographics and life expectancy.

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The restriction of the given equation is [tex]cos^{2}x =cos^{2} x[/tex],  other than 0≤x≤2π.

What is restriction of equation?

Restriction of the equation is the value of domain where the value is limited.

Given,

[tex]sin^{2} xcos^{2}x +cos^4x =(1-sinx)(1+sinx)[/tex]

taking [tex]cos^2x[/tex] common, we get

[tex]cos^2x(sin^2x+cos^2x)=1+sinx-sinx-sin^2x[/tex]

we know

[tex]sin^2x+cos^2x=1\\cos^2x=1-sin^2x[/tex]

we get,

[tex]cos^2x=cos^2x[/tex]

and the restriction is 0≤x≤2π.

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The pair of triangles can be proved to be congruent using the ASA postulate of congruency.

In the two triangles , let the first right angled triangle be ABC and the second right angled triangle be DEF.

In ΔABC and ΔDEF

∠ABC = 90° and ∠DEF = 90°

∠ACB = ∠DFE

therefore by using the angle sum property of the triangle we can say that:

∠CAB = ∠FDE

Therefore in the pair of triangles: ΔABC and ΔDEF we have :

∠ACB = ∠DFE

∠CAB = ∠FDE

AC = DF is given which is also the common side to both the angles.

Hence we can say

ΔABC ≅ ΔDEF  ( By angle-side -angle postulate of congruency)

Therefore the two triangles are congruent by the ASA postulate of congruency.

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Answer:

Step-by-step explanation:

15.

it was a sad day in the stock market for Rudo ;(  

All plus went down 2.5 the previous day and the next day it went down 0.25 of that or

2.5*.25=0.625

2.5+.625=3.125    or -$3.13

16.

-13.4 * .5 = -6.7

-13.4-6.7 = -20.1

at 5 am  it's -20.1° F

The value of the same input x is  -7

What is linear equation?

An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation. A linear equation with one variable has the conventional form Ax + B = 0. In this case, the variables x and A are variables, while B is a constant. A linear equation with two variables has the conventional form Ax + By = C. Here, the variables x and y, the coefficients A and B, and the constant C are all present.

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1.A linear equation's graph will always be a straight line.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

Let output A = A

And output B = B

According to the question,

5x - 4 = A

3x + 8 = B

And A = 3B

from the above equations,

5x - 4 = 3(3x + 8)

5x - 4 = 9x + 24

4x = -28

x = -7

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The proportion of the pan of leftover brownies that now has icing is of 0.21 = 21%.

A proportion represents a fraction relative to a total amount, and this fraction is combined with the basic arithmetic operations to find the desired amounts in whichever context of the problem.

In the context of this problem, these amounts are given as follows:

Hence the part of the pan of leftover brownies that now has icing is obtained by the multiplication presented as follows:

0.7 x 0.3 = 0.21.

It can be interpreted by the multiplication of 7 and 3 = 21, and then the result moves two decimal places to the left as each factor of the multiplication has one decimal digit.

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Check the picture below on the left side.

now, let's notice on that table when y = 0, namely the points in red in the table, those are "zeros", well obviously, or solutions of the quadratic, and those occur when x = -9 and when x = -3, now, let's use another point hmmmm we also know that the quadratic goes through (-7 , 8) so let's use those fellows

[tex]\begin{cases} x=-9\implies &x+9=0\\\\ x=-3\implies &x+3=0 \end{cases}\hspace{5em}a(x+9)(x+3)=\stackrel{0}{y} \\\\\\ \textit{we know that} \begin{cases} x=-7\\ y=8 \end{cases}\implies a(-7+9)(-7+3)=8\implies a(2)(-4)=8 \\\\\\ -8a=8\implies a=\cfrac{8}{-8}\implies \boxed{a=-1} \\\\\\ -(x+9)(x+3)=y\implies -(x^2+12x+27)=y\implies \boxed{-x^2-12x-27=y}[/tex]

Check the picture below on the right side.

Answer:

y = 2m -1

Step-by-step explanation:

y = mx + b is the slope intercept form of a line.

Perpendicular slopes are opposite reciprocals of each other.  That means that you flip the fraction and take the opposite sign.  The slope will be 2.

Now us the slope, the x value and the y value from the give point to find the y-intecept (b)

Slope = 2 (m)

y = 7 from the point (4,7)

x = 4 from th point (4,7)

y = mx + b

7 = 2(4) + b

7 = 8 + b  Subtract 8 from both sides

-1 = b

y = 2x - 1

Answer:

Step-by-step explanation:

y = 2x - 1

The dynamic power consumption of your chip if it has an area of 70 mm² and runs at 450 MHz at vdd = 0.9V is 1.08 W.

It is given that,

The average activity factor = 0.1

The Average switching capacity = 450 pF/mm²

We know that,

              p = aCV²f

putting the values in the above equation:

   = 0.1× (450e⁻¹² × 70) × (0.9)² × 450e⁶

   = 1.08 W

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$ 1.31 should pay for the chance to win if it is a fair game.

Probability is simply how likely something is to happen.

Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

It is a fair game if what we gain equals what we stake.

There are 13 clubs in a standard deck of 52 cards.

Probability of 3 clubs = 13C3 / 52C3 = 286/22100 = 0.01294

For a fair game,

$100(0.01294) = x(1-0.01294)

Solving for x,

1.294 = x(0.98706)

x = 1.31

$ 1.31 should pay for the chance to win if it is a fair game.

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Answer: __________________________________________

3x + 9

Answer: 3x + 9

Step-by-step explanation:

The function to show the amount of tiles laid as a function of time is

The reasonable domain is 0 ≤ t ≤ 9.6

Information gotten from the question include

Jeremy and Travis are placing 1200 tiles in a building

hey have already place 200 tiles, and they are able to place 125 tiles every hour.

The relationship is expressed in the form.

y = mt + c

Definition of variables to suit the problem to be solved

y = Total number of tiles laid

m = tiles laid per hour = 125

t = number of time

c = tiles already placed

200 = 1200 - 125t

The reasonable domain is the range between zero tiles to laying up the 1200 tiles

0 = 1200 - 125t

1200 = 125t

t = 9.6

1200 = 1200 - 125t

0 = -125t

t = 0

the domain is 0 ≤ t ≤ 9.6

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The probability that tails appears on the next toss is 0.5

Given,

In the question:

If I toss a fair coin five times and the outcomes are TTTTT,

To find the probability that tails appears on the next toss is.

Now, According to the question:

The possible ordered outcomes are listed as elements in a sample space, which is commonly indicated using set notation.

A sequence of five fair coin flips has a sample space that contains all potential outcomes. [tex]2^3[/tex] {H, T}  is the sample of a fair coin toss. {HHHHH, HHHHT, HHHTH, HHTHH, HTHHH,...…TTTTT} .

The probability of a tails result on the next flip is always equal to 0.5 It makes no difference if previous outcomes were {TTTTT} , {HHHHH}  or {THTHT} In each of these and all other circumstances, the probability of the next being tails is still 0.5

Hence,  the probability that tails appears on the next toss is 0.5

Learn more about Probability at:

https://brainly.com/question/11234923

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