A man is 4 times as old as his son. In 5 years time he will be 3 times as old as his son. What is the present age of the son in years.​

Answer: The sum of the first 10 terms of the given geometric sequence is 3069.

Step-by-step explanation:

A geometric sequence is a sequence of numbers such that any two consecutive terms are in a constant ratio.

The first term of the given sequence is 3 and the common ratio is 2 (6/3 = 12/6 = 24/12 = ...).

To find the sum of the first 10 terms of a geometric sequence, we can use the formula:

S = a(1 - r^n)/(1 - r)

where a is the first term, r is the common ratio and n is the number of terms.

So for this geometric sequence:

S = 3(1 - 2^10)/(1 - 2) = 3(1 - 1024)/(-1) = 3(-1023)/(-1) = 3069

Explanation: By using the formula for the sum of a geometric sequence, the sum of the first 10 terms of the sequence was found by substituting the first term, common ratio and number of terms into the formula.

Answer:

C)  3069

Step-by-step explanation:

A geometric series is the sum of the terms of a geometric sequence.

[tex]\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}[/tex]

Given geometric sequence:

From inspection of the sequence, the first term is 3:

[tex]\implies a=3[/tex]

To find the common ratio, divide consecutive terms:

[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{6}{3}=2[/tex]

To find the sum of the first 10 terms, substitute the found values of a and r together with n=10 into the geometric series formula:

[tex]\implies S_{10}=\dfrac{3(1-2^{10})}{1-2}[/tex]

[tex]\implies S_{10}=\dfrac{3(1-1024)}{1-2}[/tex]

[tex]\implies S_{10}=\dfrac{3(-1023)}{-1}[/tex]

[tex]\implies S_{10}=\dfrac{-3069}{-1}[/tex]

[tex]\implies S_{10}=3069[/tex]

Therefore, the sum of the first 10 terms of the given geometric sequence is:

Answer:

Step-by-step explanation:

S= 2πrh

<=> 3520= 2x3.142xrx40

=> r ≈ 14

The radius of the cylindrical lampshade is approximately 13.98 cm.

The formula for the curved surface area of a cylinder is 2πrL, where r is the radius, L is the height and π is pi. Given that the curved surface area of a cylindrical lampshade is 3520 cm² and the height is 40cm.

We know that:

2πrL = 3520cm²

So we can solve for the radius by dividing both sides by 2πL:

r = 3520cm² / (2π * 40cm)

Substituting the value of pi we have

r = 3520cm² / (6.284 * 40cm) = 3520cm² / 251.36cm = 13.98cm

The radius of the cylindrical lampshade is approximately 13.98 cm.

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The missing angle x of the interior angles of the quadrilateral is; x = 40°

The sum of interior angles of a quadrilateral is 360°. Now, we are given the interior angles of this quadrilateral as;

(2x + 20)°

2x°

2x°

(2x + 20)°

Thus;

(2x + 20)° + (2x + 20)° + 2x° + 2x° = 360°

8x + 40 = 360

8x = 360 - 40

8x = 320

x = 320/8

x = 40°

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Answer:

[tex]g(x)=-4\sqrt{\dfrac{x}{5}}-6[/tex]

Step-by-step explanation:

Given function:

[tex]f(x)=4\sqrt{x}+1[/tex]

1.  Horizontal stretch

[tex]f\left(\dfrac{1}{a}x\right) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $a$}.[/tex]

Therefore, if f(x) is horizontally stretched by a factor of 5:

[tex]\implies f\left(\dfrac{1}{5}x\right)=4\sqrt{\dfrac{x}{5}}+1[/tex]

2.  Reflection across the x-axis

[tex]-f(x)\implies f(x) \: \textsf{reflected in the $x$-axis}.[/tex]

Therefore, if f(x/5) is reflected in the x-axis:

[tex]\begin{aligned}\implies -f\left(\dfrac{1}{5}x\right)&=-\left(4\sqrt{\dfrac{x}{5}}+1\right)\\\\&=-4\sqrt{\dfrac{x}{5}}-1 \end{aligned}[/tex]

3.  Translation

[tex]f(x)-a \implies f(x) \: \textsf{translated $a$ units down}[/tex]

Therefore, if -f(x/5) is translated 5 units down:

[tex]\begin{aligned}\implies -f\left(\dfrac{1}{5}x\right)-5&=-4\sqrt{\dfrac{x}{5}}-1 -5\\\\&=-4\sqrt{\dfrac{x}{5}}-6\end{aligned}[/tex]

Therefore:

[tex]g(x)=-4\sqrt{\dfrac{x}{5}}-6[/tex]

Answer: The value represented by the digit 6 is 100 times greater.

Step-by-step explanation: The digit 6 in the number 84,362 is in the tens place, whereas the digit 6 in the number 6,419 is in the thousandths place. One thousand is equal to 10 x 100, meaning that the value is 100 times greater. Hope this helps and have a great day!

The value of digit 6 in 6419 is 100 times greater than the value of the digit 6 in 84362.

Place value describes the position or place of a digit in a number. Each digit has a place in a number.

When we represent the number in general form, the position of each digit will be expanded.

Those positions start from a unit place, or we also call it one’s position. The order of place value of digits of a number of right to left is units, tens, hundreds, thousands, ten thousand, a hundred thousand, and so on.

Given that are two numbers 6419 and 84362, we need to determine that the value digit 6 in 6419 is how much times greater than the value of the digit 6 in 84362.

So, the expanded form of the numbers =

6419 = 6000 + 400 + 10 + 9

84362 = 80000 + 4000 + 300 + 60 + 2

Place values of 6 in each =

In 84362 = 60

In 6419 = 6000

∴ 6000 / 60 = 100

Hence, the value of digit 6 in 6419 is 100 times greater than the value of the digit 6 in 84362.

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Answer: A, B, and E can be used to satisfy the equation

Step-by-step explanation:

A applies because 45 is the same exact number as 45, so the equal sign accurately compares them.

B applies because that symbol means less than or equal to, and 45 is not less than but is equal to 45.

C does not apply because that symbol means is not equal to, but 45 is equal to 45.

D does not apply because that is the less than symbol, and 45 is not less than 45.

E applies because it is greater than or equal to symbol, and 45 is not greater than but is equal to 45.

F does not apply because that is the greater than symbol and 45 is not greater than 45.

Hope this helps!

An ordered pair (x, y) that is a solution to the equation include the following: (x, y) = (-3, 0).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

In order to determine the ordered pairs that represent points on the graph of the given function, we would plot its equation by using an online graphing calculator and then read all the ordered pairs that lie on the line.

By critically observing the graph of the given function (see attachment), the required solutions are include following;

Ordered pair = (0.75, 0).

Ordered pair = (-3, 0).

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The domain and range of the the function are given below -

Domain → [-5, 10)

Range → [-3, 3)

Given is a graph of the function.

The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that function takes. So, we can write the domain as -

Domain → [-5, 10)

Range → [-3, 3)

Therefore, the domain and range of the the function are given below -

Domain → [-5, 10)

Range → [-3, 3)

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Answer: Hello how are you doing today?

Step-by-step explanation: How may I help you?

Given polynomials are Binomial and Trinomial.

What are polynomials?

Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable. An example of a polynomial with one variable is x^2+x-12. In this example, there are three terms: x^2, x and -12.

Based on the number of terms present in the expression, it is classified as

Monomial:-A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term. A Examples of monomials are: 5x,3.

Binomial:-A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials.  Examples of binomials are:

– 5x+3, 6a^4 + 17x.

Trinomial:-A trinomial is an expression which is composed of exactly three terms. A Examples of trinomial expressions are:

– 8a^4+2x+7, 4x^2 + 9x + 7.

Now,

Given polynomials are 2x ^ 2 - 2, 3x - 9,  - 3i ^ 2 - 6i + 9.

As the number of terms in first two polynomials is 2, they are binomial.

and the number of terms in third polynomial is 3, it is trinomial.

hence,

          Given polynomials are Binomial and Trinomial.

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[tex]\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2 - b^2}=a \qquad \begin{cases} c=\stackrel{hypotenuse}{12}\\ a=adjacent\\ b=\stackrel{opposite}{9}\\ \end{cases} \\\\\\ \sqrt{12^2 - 9^2}=a\implies \sqrt{144 - 81}=a\implies \sqrt{63}=a[/tex]

Answer:To estimate the number of steps the average man takes in 1 week, we can multiply the number of steps per day by the number of days in a week.

A week has 7 days so the number of steps the average man takes in a week would be 7,912 steps/day * 7 days/week = 55,384 steps/week

So the correct answer for the number of steps the average man takes in one week is 55,384 steps/week or in scientific notation is 5.5 x 10^4. So the answer is D. 7.9 x 10^4

Step-by-step explanation:

Answer:

A: 5.5*10⁴

Step-by-step explanation:

1 week = 7 days

7 * 7912 = 55384 steps/week

Scientifin notation:

55384 =

              5538.4*10¹

               553.84*10²

               55.384*10³

                5.5384*10⁴

Then:

the answer is:

5.5*10⁴

The height of the radio tower observed by the person is approximately 554.75 feet.

The tangent function is a mathematical function that maps an angle to its corresponding tangent value. It is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

We can use trigonometry to solve this problem. The angle of elevation from the window to the top of the tower is 38°, and the angle of depression from the window to the bottom of the tower is 29°. We can use these angles to form two similar right triangles, one with the tower and the window and one with the height of the tower and the difference between the height of the window and the height of the tower.

Let H be the height of the tower. Then the difference between the height of the window and the height of the tower is H - 300.

Using the tangent function, we can write:

tan(38°) = H / 300

and

tan(29°) = (H - 300) / 300

Solving for H, we find:

H = 300 * tan (38°) = 554.75 feet

So, the height of the tower is approximately 554.75 feet.

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The required it would take 6 years for the amount to be 2000000.

Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan.

here,

Celestine invested 1500000 simple interest at a rate of 5% per annum which amounted to 2000000.

Amount = 20,000,00
Principal = 15,000,00
Rate = 5% = 0.05

Time =  n years

Amount  = Principal [1 + rate×time]
20,000,00 = 15,000,00 [1 + 0.05×n]
n = 6.6 years

Thus, the required, it would take 6 years for the amount to be 2000000.

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The indicated congruency of the angles ∠BEC and ∠AED (∠BEC ≅ ∠AED) and ∠BEA and ∠CED (∠BEA ≅ ∠CED), according to the alternate interior angles postulate, shows that [tex]\overline{AB}[/tex]║[tex]\overline{CD}[/tex] and  [tex]\overline{BC}[/tex]║[tex]\overline{AC}[/tex] are parallel.

Two segments on a plane are parallel if the distance between the segments remain the same, through the lengths of the segments.

The specified details of the dimensions of the segments in ABCD are;

Segments [tex]\overline{AC}[/tex] bisects segment [tex]\overline{BD}[/tex] and vice versa

Required to prove that [tex]\overline{AB}[/tex] ║ [tex]\overline{CD}[/tex] and [tex]\overline{BC}[/tex] ║ [tex]\overline{AD}[/tex]

The two column consisting of the statements and reasons to prove that the specified sides are parallel is presented as follows;

Step [tex]{}[/tex]        Statement                                    Reasons

1.    [tex]{}[/tex]   [tex]\overline{AC}[/tex] and [tex]\overline{BD}[/tex] bisect each other           Given

2. [tex]{}[/tex]    [tex]\overline{BE}[/tex] ≅ [tex]\overline{ED}[/tex] and  [tex]\overline{CE}[/tex] ≅ [tex]\overline{AE}[/tex]                   Definition of bisected segments

3.   ∠BEC ≅ ∠AED and  ∠BEA ≅ ∠CED     Vertical angles theorem

4. [tex]{}[/tex] ΔBCE ≅ ΔADE and ΔBEA ≅ ΔCED      SAS congruency rule

5.  [tex]{}[/tex] ∠EBC ≅ EDA and ∠EBA ≅ ∠EDC        CPCTC

6.  [tex]{}[/tex] [tex]\overline{AB}[/tex] ║ [tex]\overline{CD}[/tex] and [tex]\overline{BC}[/tex] ║ [tex]\overline{AD}[/tex]                     Converse of alt. int. ∠s theorem

The details of the reasons in the above table are;

Vertical angles theorem

The vertical angle theorem states that the vertical angles formed by two intersecting straight lines are congruent.

SAS congruency rule

The SAS (acronym for Side-Angle-Side) congruency rule states that if two sides and an included angle in one triangle are congruent to two sides and an included angle in a second triangle, the two triangles are congruent

CPCTC

CPCTC is an acronym for Corresponding Parts of Congruent Triangle are Congruent.

Converse Alt, Int, ∠s theorem

The converse of the alt. int. ∠s (abbreviation for alternate interior angles) theorem state that if the the alternate interior angles formed by two lines and a common transversal are congruent, then the two lines are parallel

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Answer:

Step-by-step explanation:

m∠5 + m∠3 = m∠4 is always true, because in a triangle, the sum of all the angles is always 180°.m∠3 + m∠4 + m∠5 = 180° is always true, because in a triangle, the sum of all the angles is always 180°.m∠5 + m∠6 =180° is not always true, since it depends on the specific diagram and the measure of m∠6.

let's take it from the basic one dolla!!  So how long for one dolla to turn into 3?

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 3\\ P=\textit{original amount deposited}\dotfill &\$1\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year} \end{array}\dotfill &4\\ t=years \end{cases}[/tex]

[tex]3 = 1\left(1+\frac{0.08}{4}\right)^{4\cdot t} \implies 3=1.02^{4t}\implies \log(3)=\log(1.02^{4t}) \\\\\\ \log(3)=t\log(1.02^{4})\implies \cfrac{\log(3)}{\log(1.02^{4})}=t\implies \stackrel{years}{13.870\approx t} \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 3\\ P=\textit{original amount deposited}\dotfill & \$1\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ t=years \end{cases} \\\\\\ 3 = 1e^{0.08\cdot t} \implies \log_e(3)=\log_e(e^{0.08t})\implies \log_e(3)=0.08t \\\\\\ \ln(3)=0.08t\implies \cfrac{\ln(3)}{0.08}=t\implies \stackrel{years}{13.733\approx t}[/tex]

Answer:

To calculate the interest earned on an investment, we can use the formula:

A = P(1 + r)^t

Where A is the amount in the account after time t, P is the initial principal or deposit, r is the interest rate, and t is the time in years.

In this case, the initial deposit is $8,241.78, the interest rate is 5.5% (or 0.055 as a decimal), and the time is 31 days.

Since the interest is compounded daily, we need to calculate the number of compounding periods for 31 days, which is 31 days.

Therefore, the formula becomes:

A = 8241.78(1+0.055)^31

To calculate the interest earned, we will subtract the initial deposit from the final amount in the account

Interest = A - P

So the interest earned in 31 days will be:

$8241.78 * (1+ 0.055)^31 - $8241.78 = $35094.06

The possible lengths for the third side 8² + 11² < x²,  8² + 11² = x²

and 8² + 11² ≥ x².

If a² + b² < c² then it is an acute angle triangle.

If a² + b² = c² then it is a right-angle triangle.

If a² + b² ≥ c² then it is an obtuse angle triangle.

Given, Two sides of a triangle have lengths 8 and 11.

Let, The third side be 'x'.

Therefore, The possible lengths for the third side are,

8² + 11² < x²...(i)

8² + 11² = x²...(ii)

8² + 11² ≥ x²...(iii)

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Answer:

To determine whether the three points P, Q, and R are colinear, we can calculate the distances between pairs of points using the distance formula:

d(P,Q) = √((-8 - (-7))^2 + (4 - 6)^2 + (-11 - (-8))^2) = √((-1)^2 + (-2)^2 + (-3)^2) = √14

d(Q,R) = √((-9 - (-8))^2 + (3 - 4)^2 + (-14 - (-11))^2) = √((-1)^2 + (-1)^2 + (-3)^2) = √9

d(P,R) = √((-9 - (-7))^2 + (3 - 6)^2 + (-14 - (-8))^2) = √((-2)^2 + (-3)^2 + (-6)^2) = √19

If the three points are collinear, the distance between any two points should be a multiple of the distance between the other two points. Since √14, √9, and √19 are not multiples of each other, it can be concluded that the three points P, Q, and R are not collinear.

Answer:

$5520×6÷100×7=2318.40

By logical equivalence theorem, we find that composite proposition p ∨ q → r is equivalent to composite proposition (p → r) → (q → r).

Propositions are structures that contains a truth value, and can be, but not exclusively, a sentence. Propositions can be simple or composite, that is, a combination of simple propostions and operators.

In this problem we must determine that a given composite proposition is equivalent to another composite proposition by means of logical equivalence theorem. First, write the complete proposition:

p ∨ q → r

Second, use a conditional formula:

¬ (p ∨ q) ∨ r

Third, apply the DeMorgan's theorem:

(¬ p ∧ ¬ q) ∨ r

Fourth, use distributive property:

(¬ p ∨ r) ∧ (¬ q ∨ r)

Fifth, use conditional formula once again:

(p → r) → (q → r)

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The number of zeroes that the product has at the end 50 x 49 x 48 x ... x 3 x 2 x 1 will be 12.

The factor of n is given as the production of the number n, (n - 1), (n - 2)... and 1. Then the factorial of n is given as

n! = n x (n - 1) x (n - 2) x ... 3 x 2 x 1

The expression is given below.

⇒ 50 x 49 x 48 x ... x 3 x 2 x 1

⇒ 50!

Then the number of zeroes in the product at the end will be given as,

⇒ 50 / 5 + 50 / 5²

⇒ 50 / 5 + 50 / 25

⇒ 10 + 2

⇒ 12

The number of zeroes that the product has at the end 50 x 49 x 48 x ... x 3 x 2 x 1 will be 12.

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Answer:

Rachel bought 6 songs on Friday and 8 more songs on Saturday. The total number of songs she bought is 6+8 = 14

Rachel paid $2 for each song, the amount of money Rachel payed for all the songs is 14*2 = $28

Therefore, the expression that represents the amount of money that Rachel payed is "14*2 = $28" or simply "28"

- C

Sara's average speed for the entire race is

3.6 meters per second

The first 50 meters of the race is ran at a constant speed of 4.6 m/s.

Let's compute the time taken for this portion

distance = rate*time

time = distance/rate

time = 50/(4.6)

time = 10.869

It takes approximately 10.869 seconds to run the first 50 meters (at a constant speed of 4.6 m/s).

Add on the other time values from the other sections (10 and 23) to get 10.869+10+23 = 43.869

The entire 200 m race takes about 43.869 seconds.

So Sara's average speed over the entire race is about 200/43.869 = 4.559 which rounds to 4.6 m/s when rounding to one decimal place.

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Answer: x is 17

Step-by-step explanation:

Distribute the -5: -5x + 50 = -35

Subtract 50 from both sides: -5x = -85

Divide both sides by -5: x = 17

Answer: x is representing here.

Step-by-step explanation: as the equation is -5(x-10) = -35

(-) sign in both sides will cancel out each other so the equation will become 5(x-10)= 35

now we will solve the equation

5x-50 = 35

5x = 35+50

5x = 85

x=85/5 = x =

The computation of sin^3(x) + cos^3(x) is mathematically given as (2/5)(1 - sin(x)cos(x))

Generally, The study of the connections between the lengths of the sides of triangles and the angles of those triangles is the subject of the mathematical discipline known as trigonometry.

Applications of geometry to astronomical research gave rise to the discipline in the Hellenistic civilization during the third century B.C.

To compute sin^3(x) + cos^3(x), we can use the identity:

sin^3(x) + cos^3(x) = (sin(x) + cos(x))(sin^2(x) - sin(x)cos(x) + cos^2(x))

We know that sin(x) + cos(x) = 2/5, so we can substitute this into the above equation: sin^3(x) + cos^3(x) = (2/5)(sin^2(x) - sin(x)cos(x) + cos^2(x))

We also know that sin^2(x) + cos^2(x) = 1, so we can substitute this into the above equation:

sin^3(x) + cos^3(x) = (2/5)(1 - sin(x)cos(x))

Then we can simplify the above equation:

sin^3(x) + cos^3(x) = (2/5)(1 - sin(x)cos(x))

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