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1. Is it possible for the sequence t(n) = 5·2ⁿ to have a term with the value of 200? If so, which term is it? If not, justify why not.

2. Is it possible for the function f(x) = 5·2ˣ to have an output of 200? If so, what input gives this output? If not, justify why not.

The completed table with the values for composite functions in column 3, f(g(x)) and column 4, g(f(x))  included is presented as follows;

[tex]\begin{array}{|c|c|c|c|}f(x) & g(x) & f(g(x)) &g(f(x)) \\&&&\\x^2+1 &-2\cdot x + 5 &4\cdot x^2 -20\cdot x +26 & -2\cdot x^2+3 \\&&&\\2\cdot x^2 - 2\cdot x + 4 & x+3 & 2\cdot x^2 + 10\cdot x + 16& 2\cdot x^2 - 2\cdot x + 7 \\&&&\\ \sqrt{x-4} &2\cdot x^2+ 4 &x\cdot \sqrt{2} & 2\cdot x - 4 \\\end{matrix}[/tex]

f(g(x)) ≠ g(f(x)), because the operations and the order of operations in the functions are different

Composite functions are functions in which the input or argument are also functions.

The values of the composite functions based on the defined functions are found as follows;

f(x) = x² + 1, g(x) = -2·x + 5

Therefore; f(g(x)) is obtained by plugging in x = g(x) in f(x) as follows;

f(x) = x² + 1

f(g(x)) = (-2·x + 5)² + 1 = -2·x × (-2·x + 5) + 5 × (-2·x + 5) + 1

-2·x × (-2·x + 5) + 5 × (-2·x + 5) + 1 = 4·x² - 10·x - 10·x + 25 + 1

4·x² - 10·x - 10·x + 25 + 1  = 4·x² - 20·x + 26

When f(x) = x² + 1, and g(x) = -2·x + 5, f(g(x)) = 4·x² - 20·x + 26

g(f(x)) = is obtained by plugging in x = f(x) in g(x) as follows;

g(x) = -2·x + 5

f(x) = x² + 1

g(f(x)) = -2 × (x² + 1) + 5 = -2·x² - 2 + 5

g(f(x)) = -2·x² + 3

When f(x) = 2·x² - 2·x + 4, and g(x) = x + 3, we get;

f(g(x)) = 2×(x + 3)² - 2×(x + 3) + 4 = 2×(x² + 6·x + 9) - 2·x - 6 + 4

2×(x² + 6·x + 9) + 2·x + 6 + 4 = 2·x² + 10·x + 16

f(g(x)) = 2·x² + 10·x + 16

When f(x) = 2·x² - 2·x + 4, and g(x) = x + 3, f(g(x)) = 2·x² + 10·x + 16

g(f(x)) = is obtained by plugging in x = f(x) in g(x) as follows;

g(x) = x + 3

f(x) = 2·x² - 2·x + 4

g(f(x)) = 2·x² - 2·x + 4 + 3 = 2·x² - 2·x + 7

g(f(x)) = 2·x² - 2·x + 7

When f(x) = [tex]\sqrt{x - 4}[/tex], and g(x) = 2·x² + 4, we get;

f(g(x)) = [tex]\sqrt{2\cdot x^2 + 4 - 4} = x\cdot \sqrt{2}[/tex]

g(f(x)) = 2 × ([tex]\sqrt{x - 4}[/tex])² + 4 = 2 × (x - 4) + 4 = 2·x - 4

g(f(x)) = 2·x - 4

The values of the composite functions in column 3 and column 4 are included in the table in the first section of the response.

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The tip was $7.77 in total, and it is a 18.5% of the bill cost.

Here we know that the bill was $42, while the family spend a total of $49.77

Then the amount for the tip is equal to the difference between the bill and the amount that the family spends.

Tip = $49.77 - $42  = $7.77

Now we want to see which percentage of the bill this is, to check that, we need to take the quotient between the tip and the bill, and multiply that by 100%, we will get:

percentage = ($7.77/$42)*100%

percentage = (0.185)*100% = 18.5%

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On solving the provided question, we can say that  vertex form of the equation is in the form of y = a(x-h)^2 + k.

A vertex, or particular point, is a place where two or more lines or edges meet in a mathematical object. Angles, polygons, polyhedral, and graphs are where vertices are most frequently seen. Nodes and vertices in a graph are the same thing.

Recall that a parabola's General Form is y = ax2 + bx + c. The x-coordinate of the vertices, which is x = - b/2a, must first be discovered in order to find the vertex from this form. You will use this number to replace x in the parabola equation once you have determined the x-coordinate of the vertex.

a = -1/4 * (4 - 12) = -1

h = -b/(2a) = 12/(2(-1)) = -6

k = f(h) = -(-6)^2 + 12(-6) - 4 = 36

Therefore, the vertex form of the equation is y = -(x+6)^2 + 36.

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Answer:

Answer is in attached photo.

Step-by-step explanation:

The solution is in the attached photo. Do note that since this is a right-angled triangle, Pythagoras' Theorem can be used to find the length of one side：

[tex]c^{2} = a^{2} + b^{2}[/tex]

The manager of the clothing store always orders 2 small T-shirts and 3 small T-shirts for every 4 medium T-shirts.

What in mathematics is a linear equation?

According to Wolfram MathWorld A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

The manager plans to order 24 medium T-shirts and we want to find out how many small T-shirts the manager should order.

We can use the information given to set up an equation to represent the relationship between the number of small T-shirts and medium T-shirts. Let S be the number of small T-shirts and M be the number of medium T-shirts.

We know that:

S = 2 + (3/4)M

We are given that the manager plans to order 24 medium T-shirts, so we can substitute this value into the equation:

S = 2 + (3/4) * 24

We can simplify and solve for S:

S = 2 + (3/4) * 24

S = 2 + 18

S = 20

Therefore, the manager should order 20 small T-shirts.

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The differential equation is [tex]y=\frac{-7t^2+22t-7}{7t-22}[/tex]

We are given the Initial value problem:

y'=(t=y)²-1, y(3)=4

Substitute the value z=t+y

When t=3 and y=4 then z=3+4=7

y'=z²+1

Differentiate z w.r.t t

[tex]\frac{dz}{dt} =1+y'[/tex]

Then, we get [tex]z'=1+z'-1=z^2[/tex]

z⁻²dz=dt

Integrate on both sides:

-1/zdz=t+c

z=-1/t=c

Substitute t=3 and z=7

Then, we get

7=-1/3+c

21+7c=-1

7c=-1-21=-22

c=-22/7

Substitute the value of C then we get:

z=-1/t-22/7

z=-7/7t-22

y=z-t

y=-7/7t-22-t

y=-7-7t²+22t/7t-22

y=-7t²+22t-7/7t-22.

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The difference in the monthly payments for a 20-year mortgage and a 30-year mortgage on a $135,000 loan at 3.32% APR is $430.

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Option A, The degrees of freedom for a small-sample test of hypothesis for [tex]H_1[/tex] and [tex]H_2[/tex], assuming [tex]o_{12}[/tex], [tex]o_{22}[/tex], and [tex]n_1[/tex], [tex]n_2[/tex] is 15.

To calculate the degrees of freedom for a small-sample test of hypothesis for [tex]H_1[/tex] and [tex]H_2[/tex], assuming [tex]o_{12}[/tex], [tex]o_{22}[/tex] and [tex]n_1[/tex], [tex]n_2[/tex], you would use the following formula:

df = ([tex]o_{12}^2[/tex]/n_1) + ([tex]o_{22}^2[/tex]/n_2)

In this case, the degrees of freedom would be:

df = ([tex]o_{12}^2[/tex]/16) + ([tex]o_{22}^2[/tex]/16) = 15 and represents the number of values that are free to vary in the sample.

So, the answer would be A. 15

It's important to note that this formula is only used for small sample sizes. For large sample sizes, the degrees of freedom are approximated using the Welch-Satterthwaite approximation.

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The complement of A∪B is A′∩B ′.

A set of components in the universal set that are not a part of the initial set is known as the complement of a set in mathematics. Discover what a subset and its complement are, how to calculate a subset's complement, and the proper notation to use when writing a subset and its complement.

Given, universal set, U={1,2,3,4,5,6,7}

A={1,2,5,7}

B={3,4,5,6}

(A∪B) ′=U−(A∪B)

={1,2,3,4,5,6,7}−{1,2,3,4,5,6,7}=ϕ

A′∩B′=(U−A)n(U−B)

={3,4,6}n{1,2,7}=ϕ

Hence ( A∪B)′=A′∩B ′.

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The <HDI of the circle is 70 degree. A circle is divides into 360 equal degrees.

A circle is divides into 360 equal degrees.

In relation to a circle, angles are measured in degrees or radians, with one full rotation being equal to 360 degrees or 2 Pi radians.

so, <EDI = 140 degree.

so

A circle is  360 equal degrees.

360 - 140 =  220

< IDH = <EDF = 2x

<FDG = <GDH = 2y

so

2x + 2y = 220

2 * 70  + 2 * 40 = 220 degree

so,

<HDI = 70 degree.

so

The <HDI of the circle is 70 degree.A circle is divided into 360 equal degrees.

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Therefore , the circumference of the circle is 4 centimeters, while the length of the subtended arc is 16 centimeters.

Every point in the plane of a circle is equally separated from the center and is a closed, two-dimensional object. Each line tracing the circle contributes to the formation of the line of reflection symmetry. Additionally, each angle has rotational symmetry around the center.

Here,

calculation

The formula for arc length is (/2) 2r.

because of length of an arc equals r.

16 = 4θ

=> θ = 4

4r is the length of an arc.

Four times the radius, the subtended arc is longer.

Angle A is subtended by an arc if its length and radius are equal.

=> θ = 16 /4

=>4 rad

Therefore , the circumference of the circle is 4 centimeters, while the length of the subtended arc is 16 centimeters.

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a) The polygon has 11 sides

b) The Sum of the Interior Angles of the 11 sided Polygon = 1620 degrees

What is a polygon ?

The definition of a polygon is a closed, two-dimensional, plane shape created by connecting three or more line segments. During our study of geometry, polygons are a common sight.

A polygon's sides are constructed from segments of straight lines joined end to end. The sides or edges of a polygon are the line segments that make up its shape. The vertex or corners formed by two line segments are where an angle is created. A triangle with three sides is an illustration of a polygon.

There are 11  sides, So, n=11

Sum of the Interior Angles of a Polygon = 180 (n-2) degrees

So,

The Sum of the Interior Angles of the 11 sided Polygon
= 180 (11-2) degrees

= 180 ( 9)

= 1620 degrees

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The Fencing Mrs. Jones will need in centimeters is 1220 cm

The fencing is a measure of the perimeter of the region to be covered

With the dimensions in meters deduced form the problem as 3.4 m by 2.7 m. the perimeter is calculated using the formula

= 2( length * width)

= 2(3.4 + 2.7)

= 2(6.1)

= 12.2 m

To convert 12.2 m to cm we use the factor, 1 m i= 100 cm, therefore

= 12.2 * 100

= 1220 cm

the perimeter is 1220 cm

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Answer:

28.71

I looked it up.

Answer:

28.71

Step-by-step explanation:

To be honest with you, I looked it up

On solving the provided question we can say that the expression of the emaily number will be as =[tex]\frac{1}{4}n - 3[/tex]

In mathematics, it is possible to multiply, divide, add, or remove. The construction of an expression is as follows: Expression, number, and mathematical operator Numbers, variables, and functions are the building blocks of a mathematical expression (such as addition, subtraction, multiplication or division etc.) Expressions and phrases can be contrasted. Any mathematical statement with variables, numbers, and an arithmetic operation between them is called an expression or an algebraic expression. For instance, the expression 4m + 5 has the terms 4m and 5 as well as the variable m of the supplied expression, all of which are separated by the arithmetic sign +.

here,

the expression of the emaily number will be as =[tex]\frac{1}{4}n - 3[/tex]

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CuI's molar solubility is determined to be 1.27 x 10⁻¹².

The quantity of ions dissolved per litre of solution is measured by molar solubility. The quantity of ions that are dissolved in this situation's amount of solvent is represented as solubility.

Think about the equation.

Cu+(aq) CuI.(s) + I- (aq)

Let's assume that CuI (s) has a molar solubility of "S" mol/L.

Thus,

Product of solubility = [Cu+(aq)] + [ I-(aq)] → [Cu+(aq)] Ksp

[ I-(aq)] ———(1)

We are aware of

For CuI, the solubility product Ksp is 1.27 x 10⁻¹².

Consequently, from equation (1)

1.27 × 10⁻¹² = S.S

S² =1.27 × 10⁻¹²

= (1.27 × 10⁻¹² )½ = 1.127 x 10⁻⁶ M

Therefore, CuI has a molar solubility of 1.127 x 10-6 M.

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Answer:

9 and 4

Step-by-step explanation:

xy = 36

x+y = 13       or    13 - x = y    <====sub this into the first equation

x ( 13-x) = 36

-x^2 + 13x - 36 = 0  multiply the entire equation by -1  to make it easier solve

x^2 -13x+36 = 0     this factors to

(x -9)(x-4) = 0          showing x = 9 or 4     with y being 4 or 9

Answer:

[tex]\boxed{x = 8 \;m}[/tex]

Step-by-step explanation:

Nice drawing! :)

From the figure we see that the rectangle has a length of 30 m and a width of 20 m

The total area of the rectangle PQRS = 20 x 30 = 600 m²

The square footage of the planted area = area of figure MNSR = 388 m²

Therefore the rest of the area (the unshaded portion) is:
600 - 388 = 212 m²

This is the combined area of the two triangles ΔPNM and ΔMRG

Let's find the area of each of these triangles. Each of them is a right triangle which makes calculations easier

Area of a right triangle = (1/2) x base x height

ΔPNM has base = 30 - x and height = 20 -x

Area of ΔPNM

= (1/2) (30-x)(2-x)

We can use the FOIL method to evaluate (30-x)(2-x)

(30-x)(20-x)  

= 30·20 + (30)(-x) + x(20) + (-x)(-x)

= 600 - 30x + 20x + x²

= 600 -50x + x³

We usually rewrite with coefficients in decreasing magnitude of x degree

Area of ΔPNM

[tex]=\dfrac{x^2 - 50x + 600}{2}[/tex]

Let's now find the area of ΔMRQ with a base of x and a height of 20

Area of ΔMRQ
[tex]=\dfrac{1}{2}\cdot 20 = \dfrac{20x}{2}[/tex]

Adding both terms together we get
[tex]\dfrac{x^2 - 50x + 600}{2} + \dfrac{20x}{2} \\[/tex]

We have computed the area of the unshaded region as 212

So the above sum must be equal to 212

[tex]\dfrac{x^2 - 50x + 600}{2} + \dfrac{20x}{2} = 212[/tex]

Multiply throughout by 2 to get rid of the denominator:

[tex]\rightarrow \;\;x^2 - 50x + 600 + 20x = 212\times 2 = 424\\\\\rightarrow \;\;x^2 -30x + 600 =424\\[/tex]

Move 424 to the left:

[tex]x^2-30x+600-424=424-424\\\\x^2-30x+176=0[/tex][tex]\textrm{Factoring } x^2-30x+176=0\\\\\\\textrm{We get}\\\\x^2-30x+176=\left(x-8\right)\left(x-22\right)\\\\[/tex]

This is a quadratic equation which can be solved using the quadratic formula or by factoring

[tex]\textrm{Factoring } x^2-30x+176=0\\\\\\\textrm{We get}\\\\x^2-30x+176=\left(x-8\right)\left(x-22\right)\\\\[/tex]

So
[tex]x^2-30x+176=0 \rightarrow (x -8)(x-22) = 0\\\\[/tex]

So x = 8 or x = 22 are two possible solutions to this quadratic

If x = 22, it will be greater than the width of 20 and also 20-x = -2 so it is not a valid solution for this situation

Therefore we get the final answer as [tex]\boxed{x = 8 \;m}[/tex]

Answer:

the answer is 338.13

Step-by-step explanation:

289,00×.17=49.23

49.13+289.00

hope this helps:)

Answer: To determine how many hours John needs to work to earn R750, you would need to know his hourly wage. If John earns R50 per hour, he would need to work 15 hours to earn R750 (R750 / R50/hour = 15 hours). If his hourly wage is different, the number of hours he needs to work will be different.

Step-by-step explanation:

Answer:

Step-by-step explanation: The formula for the line that intersects at (8,-5) and is parallel to line x+y = 8 is given by the algebraic expression y = -x +3.

What is Algebraic expression?

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values.

Variables and constants can both be used in an algebraic expression.

A coefficient is any quantity that is added before a variable and then multiplied by it.

The Algebraic expression in this case is:

x + y = 8

which traverses points (8,-5)

Let's start by utilizing the point-intercept equation of line, which is provided by: to determine the slope of line m, that is parallel towards the line x + y = 8.

y = mx + c →(1)

x + y = 8

y = -x + 8

Comparing the aforementioned Algebraic expression to equation (1), we obtain

m = -1

The slope of the line parallel to the line x + y = 8 will now be the same, and it will be m = -1.

Let's use the point-slope equations of line to determine the linear equation now:

(y-y₁) = m(x-x₁)

Changing every value in the equation above to obtain the Algebraic expression for a line

(y-(-5)) = -1(x-8) (x-8)

(y+5) = -x+8

y + 5 = -x +8

y = -x +3

The formula for the line that intersects at (8,-5) and is parallel to line x+y = 8 is given by the algebraic expression y = -x +3

Answer:

116 months

Step-by-step explanation:

A $659,000 property is depreciated for tax purposes by its owner with the straight-line depreciation method. The value of the building, y, after x months of use is given by:

y = 659,000 − 1800x

After how many months will the value of the building be $450,200?

y = 659,000 − 1800x

450,200 = 659,000 − 1800x

subtract 659,000 from both sides:

450,200 - 659,000 = 659,000 − 1800x - 659,000

-208,800 =  − 1800x

divide both sides by -1800:

-208,800/1800 =  − 1800x/1800

116= x

so:

x  = 116

The terms in given expression like variables, coefficient, constant are respectively, y, 1, -9

An expression is a sentence with at least two numbers or variables having mathematical operation. Math operations can be addition, subtraction, multiplication, division.

For example, 2x+3

Given that,

An expression,

⇒ y-9

In given expression, y is term which is variable

And coefficient of y is 1

constant term is -9

Therefore, we can write our terms are,  y, 1, -9

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Therefore , the solution to the given problem of angles comes out to be

∠RQS is 59.5 degrees as ∠RQS = ∠PQR.

Define angles.

An angled structure in geometry is composed of two rays that converge at the vertex, or core, of the angle. These rays are referred to as the angle's faces. Depending upon where they are situated, two beams may be able to form any angle within a plane. The intersection of two planes also produces an angle. Diahedral angles are the name for them. Light beams or lines that share the same endpoint in plane geometry can have a wide variety of shapes or angles. The English term "angle" derives from the Latin term "angulus," which meaning "horn." The intersection of the two rays is known as the vertex, or vertex of the angle.

Here,

QS cuts an angle. PQR, followed by ∠SQR = ∠ PQS and

∠PQS + ∠RQS = ∠PQR.

The equation then changes to

∠PQR = ∠PQS + ∠PQS

∠PQR = 2∠PQS

∠PQS equals ∠PQR/2

assuming∠ PQR = 119°

Replace in the resulting expression from the above;

∠PQS equals ∠PQR/2

∠PQS = 119/2

∠PQS = 59.5°

∠RQS is 59.5 degrees as ∠RQS = ∠PQR.

Therefore , the solution to the given problem of angle comes out to be

∠RQS is 59.5 degrees as ∠RQS = ∠PQR.

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We can use the formula:

[tex]m = (y2 - y1) / (x2 - x1) = (5 - 3) / (-4 - 1) = 2/5[/tex]

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

20) The point-slope form of a line is[tex]y - y1 = m(x - x1)[/tex], where [tex](x1, y1)[/tex] is a point on the line and m is the slope. The slope of the line can be found by using the coordinates of two points on the line. To find the slope between the points [tex](1, 3)[/tex] and [tex](-4, 5)[/tex], we can use the formula:

[tex]m = (y2 - y1) / (x2 - x1) = (5 - 3) / (-4 - 1) = 2/5[/tex]

The point-slope form of the line is then:

[tex]y - 3 = (2/5)(x - 1)[/tex]

To convert this to slope-intercept form (y = mx + b), we can solve for y:

[tex]y = (2/5)x + (3 - (2/5)) = (2/5)x + (12/5)[/tex]

21) The line is parallel to [tex]y = -2x - 5[/tex], so we know that the slope of the line is -2. We can use the point [tex](4, -7)[/tex]and the slope -2 to write the equation in point-slope form:

[tex]y - (-7) = -2(x - 4)[/tex]

or

[tex]y + 7 = -2x + 8[/tex]

To convert this to slope-intercept form [tex](y = mx + b)[/tex], we can solve for y:

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

22) The line is perpendicular to [tex]y = -3/2x + 1[/tex], so we know that the slope of the line is the negative reciprocal of [tex]-3/2[/tex]. The slope of the line is [tex]2/3[/tex]. We can use the point [tex](3, 5)[/tex] and the slope [tex]2/3[/tex] to write the equation in point-slope form:

[tex]y - 5 = (2/3)(x - 3)[/tex]

or

[tex]y = (2/3)x + (5 - (2/3)3) = (2/3)x + (5 - 2) = (2/3)x + 3[/tex]

To convert this to slope-intercept form [tex](y = mx + b)[/tex], we can solve for y:

[tex]y = (2/3)x + 3[/tex]

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The first digit to the right of the decimal point indicates the number of tenths. For example, the decimal 0.3 is the same as the fraction 3/10. 

Answer:

(0,5) and (3,320)

Step-by-step explanation:

Plug in the ordered pair into the function and see if it makes sense.

Plug in 5 for y and 0 for x:

5= 5(4)^0
4^0 = 1 (anything to the power of 0 except 0 is one)
5x1 = 5
5=5
(0,5) works

320 = 5(4)^3

4^3 = 64

64x5 = 320

320=320

320=320 (3,320) works

(0,5) and (3,320) both work.

The quotient of the division (x^2-3x+5) divided (x-1) is x - 2 with a remainder of 3

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

(x^2-3x+5) divided (x-1)

Using the long division method of quotient, we have

x - 1 | x^2 - 3x + 5

The division steps are as follows

           x - 2

x - 1 | x^2 - 3x + 5

        x^2 - x

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

             -2x + 5

             -2x + 2

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

                         3

Hence, the quotient is x - 2

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