the sum of the digits of a two-digit number is 14. the first digit is 4 less than twice the second digit. what is the number

Answer:

[tex]a = 8[/tex]  or [tex]a = -\frac{14}{3}[/tex]

Step-by-step explanation:

[tex]13(a+4)+24(a+5) = 3(a^{2} +9a+20)[/tex]

[tex]3a^{2} -10a-112=0[/tex]

[tex](3a+14)(a-8)=0[/tex]

[tex]a = 8[/tex]  or [tex]a = -\frac{14}{3}[/tex]

[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{13}{a + 5} + \cfrac{24}{a + 4} = 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{13(a + 4) + 24(a + 5)}{(a + 5)(a + 4)} = 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{13a + 52+ 24a + 120}{a {}^{2} + 5a + 4a + 20} = 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ 37a + 172}{a {}^{2} +9a + 20} = 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3( {a }^{2} + 9a + 20) = 37a + 172[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3 {a}^{2} + 27a + 60 = 37a + 172[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3 {a}^{2} + 27a - 37a + 60 - 172 = 0[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3 {a}^{2} - 10a - 112= 0[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3 {a}^{2} - 24a + 14a - 112 = 0[/tex]

[tex]\qquad \sf  \dashrightarrow \: 3a(a - 8) + 14(a - 8) = 0[/tex]

[tex]\qquad \sf  \dashrightarrow \: (a - 8) + (3a + 14) = 0[/tex]

So, required values of " a " are :

[tex]\qquad \sf  \dashrightarrow \: a = 8 \: \: \: or \: \: \: a = - \cfrac{ 14}{3} [/tex]

An example of the real world example of a rational function would be when a person would decide to share a cookie with their friend. Each of them would get half of the cookie each.

This is the term that is used to refer to the fraction that is represented by the use of a rational fraction. That is there is denominator and a numerator.

The denominator must have a variable and this variable has to be of degree 1. Hence the real world example of a rational function would be when a person would decide to share a cookie with their friend. Each of them would get half of the cookie each.

First you have to know if it is an inequality or not. Then you have to assign x and y variables to numbers. From here you form an equation. This equation is plotted on a graph.

Read more on the rational function here: https://brainly.com/question/12434659

#SPJ1

Profit function in terms of x = 33x + 390.

What is a profit in business?

Given that revenue, in dollars, of a company that makes skateboards can be modeled by the polynomial

[tex]2x^{3} + 30x - 130[/tex]

The cost, in dollars, of producing the skateboards can be modeled by

[tex]2x^{3} -3x - 520[/tex]

Profit function = Revenue - cost

                         = [tex]2x^{3} + 30x - 130 - ( 2x^{3} - 3x - 520)[/tex]

                          = 33x + 390

Therefore, profit function in terms of x = 33x + 390

Learn more about profit

brainly.com/question/13538701

#SPJ4

Answer: x = 2

Step-by-Step Solution:

Given:

OH = WO = LW

OH = 2x + 1

WO = 3x - 1

LW = 5

To Find:

The value of 'x'

Solution:

We know, OH = WO = LW

So, equate any two of the values given :-

2x + 1 = 5

2x = 5 - 1

2x = 4

x = 4/2

x = 2

Hence, x = 2.

Answer:

Zero

Step-by-step explanation:

The coefficient of t²s⁵ in the expansion of (2t + s)⁷ is d. 84

Binomial expansion is the expansion of two term expressions such as (a + b)ⁿ where n is a rational number.

Pascal's triangle is atriangle used in determining the coefficients of a binomial expansion.

The general term for the expansion (a + b)ⁿ = ∑ⁿCₓaˣbⁿ⁻ˣ.

So, each term is ⁿCₓaˣbⁿ⁻ˣ.

Comparing (a + b)ⁿ with (2t + s)⁷,

So, its general term is ⁿCₓ(2t)ˣsⁿ⁻ˣ = ⁷Cₓ(2t)ˣsⁿ⁻ˣ

= ⁷Cₓ(2)ˣtˣsⁿ⁻ˣ

So, the coefficient term is ⁷Cₓ(2)ˣ

Now for the term t²s⁵, x = 2.

So, the coeficient term is  ⁷Cₓ(2)ˣ =  ⁷C₂(2)²

= 7!/2!(7 - 2)! × 4

= 7!/2!5! × 4

= 7 × 6 × 5!/(2! × 5!) × 4

= 7 × 6/2 × 4

= 7 × 3 × 4

= 84

So, the coefficient of t²s⁵ in the expansion of (2t + s)⁷ is d. 84

Learn more about binomial expansion here:

https://brainly.com/question/25086608

#SPJ1

Applying the knowledge of fractions, the number of cups of water that will fill a bottle of 3 liters is calculated as: 13.5 cups.

Fractions are numbers that indicate a part of whole numbers. For example, 2/9 of a liter means 2/9 of a part of 1 liter.

In math, a proportion is simply and equation that sets to ratios equal to each other.

To solve the problem given, create a proportion then use cross product to find the wanted value, which is the number of cups of water that can be filled into a 3-liter bottle.

Let x represent the cups of water we are looking for.

1 cup = 2/9 liter

x = 3 liters

We would have:

(1)(3) = (2/9)(x)

3 = 2x/9

(3)(9) = 2x

27/2 = x

x = 13.5

The number of cups of water to fill a 3-liter bottle is: 13.5.

Learn more about fractions on:

https://brainly.com/question/78672

#SPJ1

If Mr. McConnell brings in 6 2/3 pizzas for his students' pizza party and he divided the pizza evenly among his 4 students, each student will get 5/3 pizzas.

Given the data in the question;

First we convert the mixed fraction into an improper fraction.

6 2/3 = ( 6×3 + 2 )/3 = 20/3

Since McConnell divides the pizza evenly among his 4 students.

We will divide 20/3 by 4

20/3 ÷ 4

20/3 × 1/4

(20 × 1) / (3 × 4 )

20/12

5/3

Therefore, if Mr. McConnell brings in 6 2/3 pizzas for his students' pizza party and he divided the pizza evenly among his 4 students, each student will get 5/3 pizzas.

Learn more about fractions here: brainly.com/question/10354322

#SPJ1

Answer:

[tex]4x^{6} y^{10}[/tex]  A.

Since Point R divides PQ in the ratio 1:3. the x-coordinate of Q is 3

The question has to do with line division

This is the division of a line into a given ratio.

Since Point R divides PQ in the ratio 1:3. If the x-coordinate of R is -1 and the x-coordinate of P is -3.

Let

Since R divides PQ in the ratio, 1:3, we have that

PR/RQ = 1/3

(x₂ - x₁)/(x₃ - x₂) = 1/3

Substituting the values of the variables into the equation, we have

(x₂ - x₁)/(x₃ - x₂) = 1/3

(-1 - (-3))/(x₃ - (-3)) = 1/3

(-1 + 3)/(x₃ + 3) = 1/3

2/(x₃ + 3) = 1/3

x₃ + 3 = 2 × 3

x₃ + 3 = 6

x₃ = 6 - 3

x₃ = 3

So, the x-coordinate of Q is 3

Learn more about line division here:

https://brainly.com/question/13018136

#SPJ1

Answer:

17.2 units

Step-by-step explanation:

You would use the Pythagorean Theorem to solve. The height (a) is 4 units. The base (b) is 6 units.

4^2 + 6^2 = sqrt(52)

This would mean that the hypotenuse is 7.211102550927979 units or 7.2 units (the square root of 4^2 + 6^2).

Now you would need to just add all three sides.
6 + 4 + 7.2 = 17.2 units

Hope that helps! Please mark me brainliest!

Answer:

The length of the diagonal of a rectangle is 10m

Step-by-step explanation:

There is a visual below,

The diagonal of a rectangle has formed 2 right triangles.

This diagonal is called hypotenuse side length of a triangle

To find the hypotenuse, we can use the Pythagorean Theorem

a² + b² = c²

we are given the length of a and b, where a = 6 and b = 8

(6)² + (8)² = c²

36 + 64 = c²

100 = c²

√100 = c

10 = c

Learn more about the Pythagorean Theorem here: https://brainly.com/question/343682

Answer:

Assunming that the equation is y = -16x² +85x, I suggest Dario find a new profession.  He kicks the ball 133 feet high, but it only travels 2.7 feet downfield at that point.  It will have travelled 2*2.7 feet or 5.4 feet horizontal when it returns to ground.

Step-by-step explanation:

We can answer the question of how hiogh and far the ball travels (in feet) by either of two methods.  The first is to differentiate the equation and set it equal to 0, and the second is the graph it and look for the vertex and x intercept.

Differentiate

y = -16x² +85x

The first derivative produces an equation that gives us the slope of the line at any point x.  At the ball's top height, it stops and reverses direction, a point at which the slope is zero.

y = -16x² +85x

d(y)/d(x) = -32x + 85

0 = -32x + 85

32x = 85

x = 2.66 feet horizontal distance when the ball reaches its peak.

At x = 2.66, y is 112.9 feet high

When the ball returns, it adds amnother 2.66 feet to the horizontal distance, for a total of 5.4 feet.

Graph:

See attached plot of the function.

Horizontal distance is 2.66 feet and height is 133 feet. The x-intercept is at 5.4 feet.

After all that, the ball only travelled downfield (we hope downfield) a total of 5.4 feet.

If a, b and c vectors are in cyclic order , the scalar product is equals to  zero which makes Option C to be correct.

A vector  can be regarded as quantities that posses both magnitude and a direction this is the opposite to scalar quantity which posses just the magnitude without any direction.

Geometrically,  the  vector can be described as the  line segment,  having its  length as  the magnitude  with the arrow over it  showing direction.

The scalar product  can be gotten with the  multiplication  of the magnitudes of vectors .

Hence, If a, b and c vectors are in cyclic order , the scalar product is equals to  zero .

Learn more about vectors at:

https://brainly.com/question/15519257

#SPJ1

CHECK THE COMPLETE QUESTION IN THE ATTACHMENT

Answer:

8.5 g/cm^3

Step-by-step explanation:

Density = mass/volume

          = 34 g / 4 cm^3 =  8.5 g/cm^3

Answer:

8.5 grams per cubic centimeter.

Step-by-step explanation:

Density = mass / volume

             =  34 / 4

             = 8.5 grams / cubic centimeter.

Answer:

C

Step-by-step explanation:

The inequalities y < 4 (x-1)² - 16 written in vertex form represents the graph region.

Option C is the correct answer.

It is a statement of an ordered relationship

- greater than,

- greater than or equal to,

- less than,

- less than or equal to between two numbers or algebraic expressions.

We have,

Inequalities where we need to find the coordinates and see which inequalities represent the graph.

y < 4 (x- 1)² - 11 ____(1)

y < -4 (x-1)² - 11 _____(2)

y < 4 (x-1)² - 16 ______(3)

From (1),

y =  4 (x- 1)² - 11

x = 0, y = 4 x 1 - 11 = 4 - 11 = -7

From (2),

y = -4 (x-1)² - 11

y = -4 x 1 - 11 = -4 - 11 = -15

From (3),

y = 4 (x-1)² - 16

x = 0, y = 4 x 1 - 16 = 4 - 16 = -12

We see that,

y < 4 (x-1)² - 16  gives us the coordinates (0, -12) on the graph.

Similarly,

x = 1, y = 4 x 0  - 16 = 0 -16 = -16

x = 1, y = 4 x 1 - 16 = 4 - 16 = -12

(1, -16) and (2, -12) coordinates are on the graph.

Thus,

The inequalities y < 4 (x-1)² - 16 written in vertex form represents the graph region.

Option C is the correct answer.

Learn more about inequality here:

https://brainly.com/question/22010462

#SPJ2

The given condition on the basis of truth table is true.

According to the statement

we have given that the some condition p and q. and we have to find that  the given condition is true or false on the basis of the truth table.

So, The given conditions are:

If 3 + 2 = 5, Then 5 + 5 = 10

Now,

The Truth table is a tabular representation of all the combinations of values for inputs and their corresponding outputs. It is a mathematical table that shows all possible outcomes that would occur from all possible scenarios that are considered.

So, the given conditions are properly verified with the truth table. So, it is true.

So, The given condition on the basis of truth table is true.

Learn more about Truth table here

https://brainly.com/question/18575348

#SPJ1

Shaded area = area of the hexagon – area of the pentagon + area of the square – area of the equilateral triangle. This can be obtained by finding each shaded area and then adding them.

From the question we can say that the Hexagon has three shapes inside it,

Also it is given that,

An equilateral triangle is shown inside a square inside a regular pentagon inside a regular hexagon.

From this we know that equilateral triangle is the smallest, then square, then regular pentagon and then a regular hexagon.

A pentagon is shown inside a regular hexagon.

An equilateral triangle is shown inside a square.

The expression for total shaded region would be written as,

Shaded area = Area of first shaded region + Area of second shaded region

Hence,        

⇒ Shaded area  = area of the hexagon – area of the pentagon + area of the  square – area of the equilateral triangle.

Learn more about area of a shape here:

brainly.com/question/16501078

#SPJ1

Answer:

Regular Hexagon

Regular Pentagon

Square

Equilateral Triangle

Step-by-step explanation:

E2020 Geometry B!! :3

The measures of variation that can be used by Paul are interquartile range and mean absolute deviation.

The interquartile range is the difference between the third quartile and the first quartile.

Third quartile = 3/4(n + 1)

First quartile = 1/4 x (n + 1)

The mean absolute deviation is the average of the distance of a data set from its mean.

Mean absolute deviation =  ∑ l x - m(x) l

Mean, median and mode are measures of tendency. They are used to measure the number at the center of a dataset.

Here are the options:

Mean absolute deviation

mean

interquartile range

median

mode

To learn more about interquartile range, please check:  https://brainly.com/question/3966385

#SPJ1

mean absolute deviation
mean

median

interquartile range

mode

Answer is Healthclub A costs $33 less in monthly fees than Healthclub B

The number next to the “x” is your rate.

So Healthclub A charges $33 less than B

A charges 27 monthly fee

B charges 60 monthly fee

60-27=33

Answer:

Step-by-step explanation:

4x - 10 + x + 7 = 4x

5x - 3 = 4x

-3 = -x

x = 3

4(3) - 10 = 12 - 10 = 2

3 + 7 = 10

10+2 = 12

4(3)= 12

The area of the given semicircle is 56.574 sq. units. Half of the area of the circle is the area of the semicircle.

The area of the semicircle is

A = [tex]\frac{1}{2}[/tex] πr² sq. units

Where r - radius

A semicircle is nothing but half part of the circle. So, its area is also half of the circle's area.

It is given that,

The radius of the semicircle is

r = 6

So, its area in terms of π is

A = [tex]\frac{1}{2}[/tex] πr²

   =  [tex]\frac{1}{2}[/tex] × π × 6²

   = 18π sq. units

On substituting π = 3.143, we get

A = 18 × 3.143 = 56.574 sq. units

Thus, the area of the given semicircle in terms of π is 18π or in terms of π =3.143 is 56.574 sq. units.

Learn more about semicircle here:

https://brainly.com/question/15656472

#SPJ1

Recall the Pythagorean identity,

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

Use it to rewrite the equation in terms of cos only.

[tex]12 (1 - \cos^2(x)) + \cos(x) - 6 = 0[/tex]

[tex]-12 \cos^2(x) + \cos(x) + 6 = 0[/tex]

Factorize the left side.

[tex]-(3\cos(x) + 2) (4\cos(x) - 3) = 0[/tex]

Solve the two cases for [tex]\cos(x)[/tex].

[tex]3 \cos(x) + 2 = 0 \text{ or } 4\cos(x) - 3 = 0[/tex]

[tex]\cos(x) = -\dfrac23 \text{ or } \cos(x) = \dfrac34[/tex]

From the first equation, we get one family of solutions:

[tex]\cos(x) = -\dfrac23[/tex]

[tex]x = \cos^{-1}\left(-\dfrac23\right) + 2n\pi \text{ or } x = -\cos^{-1}\left(-\dfrac23\right) + 2n\pi[/tex]

From the second equation, we get another family:

[tex]\cos(x) = \dfrac34[/tex]

[tex]x = \cos^{-1}\left(\dfrac34\right) + 2n\pi \text{ or } x = -\cos^{-1}\left(\dfrac34\right) + 2n\pi[/tex]

where [tex]n[/tex] is an integer.

We get solutions in the given domain for

[tex]n = 0 \implies \boxed{x = \cos^{-1}\left(-\dfrac23\right)} \text{ or } \boxed{x = \cos^{-1}\left(\dfrac34\right)}[/tex]

[tex]n=1 \implies \boxed{x = 2\pi - \cos^{-1}\left(-\dfrac23\right)} \text{ or } \boxed{x = 2\pi - \cos^{-1}\left(\dfrac34\right)}[/tex]

Equations can be used to find the number of pencils (x) and pens (y) she bought are X+1.5Y = 21, and X+Y = 18

The correct option is C.

One or two variables make up a linear equation. Neither the numerator nor the denominator of a fraction can be a variable in a linear equation raised to a power higher than 1. The lines that connect all the points on a coordinate grid when you identify the values that make a linear equation true are congruent.

Let x stand for the number of pencils Stella purchased, and y for the number of pens Stella purchased.

Since Stella purchased a total of 18 things, the quantity of pencils and pens purchased must equal 18 or one of the following:

x + y = 18

We also know she spent $21, that a pencil costs $1, and that a pen costs $1.50. Therefore,

1x + 1.5y = 21

Thus, the set of equations that may be utilized to determine the quantity of pencils (x) and pens (y) that she purchased is as follows:

X+1.5Y = 21, and X+Y = 18

equations can be used to find the number of pencils (x) and pens (y) she bought are X+1.5Y = 21, and X+Y = 18

To know more about liner equation visit:

https://brainly.com/question/1347950

#SPJ4

Answer:

20

Step-by-step explanation:

To solve this you have to make a system of equations.

Since the father and son's age sum up to 60, the first equation will be:

f + s = 60

Secondly, since the father's age is 5 times the age of the son 6 years ago the equation will be:

6 - (5s) = f

Now, you have to solve the first equation to let it equal to s

f + s = 60

f = 60 - s

Plug in

6 - 5s = 60 - s

     +s          +s

6 - 4s = 60

-6         -6

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

-4s = 54

-----   -----

 -4     -4

   s ≅ 14

14 + 6 = 20

Answer:

20.

Step-by-step explanation:

x = father's age and y = son's age,

x + y = 60

x - 6 = 5(y - 6)

x - 5y = -24

Subtract this from first equation:

6y = 84

y = 14

So the son's age is 14.

Six years time son will be 20.

The area of the resulting surface of this infinite curve is π units.

In this question,

The infinite curve is y = e^−2x, x ≥ 0.

The curve is rotated about x-axis.

Since x ≥ 0, the limits will be 0 to ∞.

Then the area of the resulting surface is,

[tex]A= 2\pi \lim_{b\to \infty} (\int\limits^\infty_0{e^{-2x} } \, dx )[/tex]

Now substitute,

u = -2x

⇒ du = -2dx

⇒ dx = [tex]-\frac{1}{2} du[/tex]

Then,

[tex]\int\limits{-\frac{1}{2}e^{u} } \, du =-\frac{1}{2}\int\limits{e^{u} } \, du[/tex]

Now substitute u and du, we get

⇒ [tex]-\frac{1}{2} \int\limits {e^{-2x} }(-2) \, dx[/tex]

⇒ [tex]-\frac{-2}{2} \int\limits {e^{-2x} } \, dx[/tex]

⇒ [tex](1) \int\limits {e^{-2x} } \, dx[/tex]

⇒ [tex]\int\limits {e^{-2x} } \, dx[/tex]

Thus the area of the resulting surface is

[tex]A= 2\pi \int\limits^\infty_0{e^{-2x} } \, dx[/tex]

⇒ [tex]A= 2\pi [{e^{-2x}(\frac{1}{-2} ) } \,]\limits^\infty_0[/tex]

⇒ [tex]A= \frac{2\pi}{-2} [{e^{-2(\infty)}-e^{-2(0)} } \,]\\[/tex]

⇒ [tex]A= -\pi [0-1} \,]\\[/tex]

⇒ [tex]A= \pi[/tex]

Hence we can conclude that the area of the resulting surface is π units.

Learn more about area of infinite curve here

https://brainly.com/question/3139455

#SPJ4

The steps to use to construct a frequency distribution table using sturge’s approximation is as below.

The steps to construct a frequency distribution table using Sturge's approximation are as follows;

Step 1: Find the range of the data: This is simply finding the difference between the largest and the smallest values.

Step 2; Take a decision on the approximate number of classes in which the given data are to be grouped. The formula for this is;

K = 1 + 3.322logN

where;

K= Number of classes

logN = Logarithm of the total number of observations.

Step 3; Determine the approximate class interval size: This is obtained by dividing the range of data by the number of classes and is denoted by h class interval size

Step 4; Locate the starting point: The lower class limit should take care of the smallest value in the raw data.

Step 5; Identify the remaining class boundaries: When you have gotten the lowest class boundary, then you can add the class interval size to the lower class boundary to get the upper class boundary.

Step 6; Distribute the data into respective classes:

Read more about  frequency distribution table at; https://brainly.com/question/27820465

#SPJ1

The expected error between the sample mean and the population mean is 3.

In this problem, we have been given :

population mean (μ) = 80,

standard deviation (σ) = 24,  

sample size (n) = 64

We know that,

The Central Limit Theorem states that, for a normally distributed random variable X, with mean μ and standard deviation σ, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean μ and standard deviation, which is also called standard error [tex]s=\frac{\sigma}{\sqrt{n} }[/tex]

We need to find the error between the sample mean and the population mean.

standard error

= σ /√n

= 24 /√64

= 24 / 8

= 3

Therefore, the expected error between the sample mean and the population mean = 3

Learn more about the Central Limit Theorem here:

https://brainly.com/question/15404124

#SPJ4

Hi there,

please see below for solution steps :

‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗

⨠ use the distributive property

[tex]\sf{7x-3(x-6)=30}[/tex]

[tex]\sf{7x-3x+18=30}[/tex]

⨠ combine like terms

[tex]\sf{4x+18=30}[/tex]

⨠ subtract both sides by 18

[tex]\sf{4x=30-18}[/tex]

[tex]\sf{4x=12}[/tex]

⨠ divide both sides by 4

[tex]\boxed{\sf x=3}[/tex]

‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗