The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government 101010 denarius per day to support 333 legionaries and 333 archers. It only costs 333 denarius per day to support one legionary and one archer. Use a system of linear equations in two variables.

Can we solve for a unique cost for each soldier?

Answer:

[tex]x = 2\sqrt{10}[/tex]

Step-by-step explanation:

We have to solve for x from the logarithmic equation as follows:

[tex]\log x^{2} + \log 25 = 3[/tex]

⇒ [tex]\log 25x^{2}  = 3[/tex]  

{Since, using logarithmic property [tex]\log A + \log B = \log AB[/tex]}

Now, converting this logarithmic equation above into exponential equation we get,

[tex]25x^{2}  = 10^{3}[/tex]  

{Since we know that if [tex]\log_{10}a = b[/tex] then, we can write [tex]a = 10^{b}[/tex]}

⇒ [tex]25x^{2}  = 1000[/tex]

⇒ [tex]x^{2}  = 40[/tex]

⇒ [tex]x = 2\sqrt{10}[/tex] (Answer)

The factorization of [tex]\(3x^3+7x^2-18x+8\)[/tex] is [tex]\((x-1)(3x^2 + 10x - 8)\)[/tex]. So, the factors are [tex](x-1), (3x^2 + 10x - 8)[/tex], and any additional factors that [tex]\(3x^2 + 10x - 8\)[/tex] may have, which can be further factored if possible.

Given that  x-1 is a factor of the expression [tex]\( 3x^3+7x^2-18x+8 \)[/tex], we can use the long division method to find the other factors.

Set up the division as follows:

            [tex]3x^2 + 10x - 8[/tex]

       ______________________

[tex]x - 1 | 3x^3 + 7x^2 - 18x + 8 \\ - (3x^3 - 3x^2)[/tex]

        ___________________

                [tex]10x^2 - 18x + 8 \\ - (10x^2 - 10x)\\[/tex]

                 ______________

                          -8x + 8

                          - (-8x + 8)

                          __________

                                 0

The quotient is [tex]\(3x^2 + 10x - 8\)[/tex], and since there is no remainder, \(x-1\) is indeed a factor.

Therefore, The factorization of [tex]\(3x^3+7x^2-18x+8\)[/tex] is [tex]\((x-1)(3x^2 + 10x - 8)\)[/tex]

So, the factors are [tex](x-1), (3x^2 + 10x - 8)[/tex], and any additional factors that [tex]\(3x^2 + 10x - 8\)[/tex] may have, which can be further factored if possible.

The complete factorization of the original expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] is:[tex]\[ (x - 1)(x + 4)(3x - 2) \][/tex]

To factorize the expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] given that one of the factors is [tex]\(x - 1\)[/tex], we can use polynomial division to divide the expression by [tex]\(x - 1\)[/tex] and find the quotient. The factors of the expression will then be [tex]\(x - 1\)[/tex]  and the factors of the quotient.

The expression to be factorized is: [tex]\(3x^3 + 7x^2 - 18x + 8\).[/tex]

Let's perform the division step-by-step.

1.Divide the first term of the dividend (3x³) by the first term of the divisor (x):

  [tex]\(3x^3 ÷ x = 3x^2\)[/tex].

Write this as the first term of the quotient.

2. Multiply the divisor by this term and subtract the result from the dividend:

Multiply [tex]\(x - 1\)[/tex] by [tex]\(3x^2\)[/tex] to get [tex]\(3x^3 - 3x^2\)[/tex].

Subtract this from the original polynomial: [tex]\(3x^3 + 7x^2\)[/tex]becomes [tex]\(10x^2\)[/tex].

3.Bring down the next term of the original polynomial to form a new dividend:

The new dividend is [tex]\(10x^2 - 18x\)[/tex].

4.Repeat this process for the new dividend**:

  Divide [tex]\(10x^2\) by \(x\) to get \(10x\)[/tex].

  Multiply [tex]\(x - 1\) by \(10x\)[/tex] to get [tex]\(10x^2 - 10x\)[/tex].

  Subtract this from the new dividend:[tex]\(10x^2 - 18x\)[/tex]  becomes [tex]\(-8x\)[/tex].

5.Bring down the next term of the original polynomial to form a new dividend:

  The new dividend is [tex]\(-8x + 8\)[/tex].

6.Repeat this process for the new dividend**:

  Divide [tex]\(-8x\) by \(x\)[/tex] to get [tex]\(-8\)[/tex].

  Multiply [tex]\(x - 1\) by \(-8\)[/tex] to get [tex]\(-8x + 8\)[/tex].

  Subtract this from the new dividend: [tex]\(-8x + 8\)[/tex] becomes 0.

The quotient we obtain from this division is [tex]\(3x^2 + 10x - 8\)[/tex]. Now, we need to factorize this quadratic expression. Let's proceed with the factorization.

The roots of the quadratic expression [tex]\(3x^2 + 10x - 8\)[/tex] are [tex]\(-4\)[/tex] and[tex]\(\frac{2}{3}\)[/tex]. This means that the quadratic expression can be factored as [tex]\((x + 4)(x - \frac{2}{3})\)[/tex].

However, to express the factors in a more standard form, we'll rewrite the factor [tex]\(x - \frac{2}{3}\) as \(3x - 2\)[/tex], which is obtained by multiplying the numerator and denominator of [tex]\(\frac{2}{3}\)[/tex] by 3.

Therefore, the factorized form of [tex]\(3x^2 + 10x - 8\)[/tex] is [tex]\((x + 4)(3x - 2)\)[/tex].

Combining this with the given factor [tex]\(x - 1\)[/tex], the complete factorization of the original expression [tex]\(3x^3 + 7x^2 - 18x + 8\)[/tex] is:[tex]\[ (x - 1)(x + 4)(3x - 2) \][/tex]

There is no way to determine the rate of change [slope], so it is impossible to answer this question. I apologise.

Answer:

Apple Produce pays employees  38.5 for per bushels of apples picked.

Step-by-step explanation:

Given:

Apple produce pays it's employees by the formula;

[tex]P(b) = \frac{7}{2}b+35[/tex]

[tex]P(b)[/tex] ⇒ Employee's Total daily pay

[tex]b[/tex] ⇒ Number of bushels of apples

We need to find the Rate employees are paid per bushels of apples picked.

Rate employees are paid per bushels of apples picked can be calculated by substituting the value of "b = 1" in above formula.

Substituting the value of b = 1 in above formula we get;

[tex]P(1) = \frac{7}{2} \times 1+35\\[/tex]

Now We will take LCM to make the Denominator common.

[tex]P(1) = \frac{7}{2} +\frac{35\times 2}{2} = \frac{7}{2} +\frac{70}{2} = \frac{77}{2}= 38.5[/tex]

Hence Apple Produce pays employees  38.5 for per bushels of apples picked.

Answer:

34.4

Step-by-step explanation:

Answer:

34.4

Step-by-step explanation:

USE A CALCULATOR!!!!

You will place the sticker [tex]\frac{5}{8}[/tex] inches from the edge of the switch board

Step-by-step explanation:

Given data:

Sticker’s length =  [tex]2 \frac{1}{2}[/tex]

Breadth of switch box = [tex]3 \frac{3}{4}[/tex]

Subtracting the above value, we get

    [tex]\text { Switch Box's total area }=3 \frac{3}{4}-2 \frac{1}{2}=\frac{15}{4}-\frac{5}{2}=\frac{15-10}{4}=\frac{5}{4}[/tex]

Divided [tex]\frac{5}{4}[/tex] by 2 gives you the distance from the edge if you put the sticker in the centre). Therefore,

    [tex]\frac{\left(\frac{5}{4}\right)}{\left(\frac{2}{2}\right)}=\frac{5}{8}[/tex]

So, will place the sticker [tex]\frac{5}{8}[/tex] inches far from the edge of the switch board.

Answer:

The Cardboard dimensions are 13cm in length and 12cm in breadth

Step-by-step explanation:

We can use Pythagoras Theorem to find the length the Cardboard (a²+b²=c²). Let A be 5 and B be 12, we find that C is equivalent to 13. The height is 12, thus the breadth of the Cardboard is also 12. Hope this helps :)

Final answer:

The dimensions of the cardboard rectangle are 5 cm by 12√2 cm.

Explanation:

To determine the dimensions of the cardboard rectangle inserted diagonally into a box with given dimensions, we first need to calculate the diagonal of the box that lies in the length-height plane. The box is 12 cm long and 12 cm high.

By using the Pythagorean theorem, we can find the length of the diagonal 'd' using the formula √(l² + h²), where 'l' is the length of the box and 'h' is the height.

In this case, 'd' = √(12² + 12²) = √(144 + 144) = √(288) = 12√2 cm.

Thus, the dimensions of the cardboard rectangle are its width, which is 5 cm (the width of the box), and its diagonal, which is 12√2 cm.

Answer:

There is not solution to that it is just 12 because there is no variable

Step-by-step explanation:

Answer:

I would say one because the solution is 12.

12 = 12.

Answer:

- [tex]\frac{24}{25}[/tex]

Step-by-step explanation:

Given 630 < A > 720 then A is in the fourth quadrant where

cosA > 0 and sinA < 0

Given

cosA = [tex]\frac{3}{5}[/tex] = [tex]\frac{adjacent}{hypotenuse}[/tex]

Then the triangle is a 3- 4 - 5 with opposite side 4, thus

sinA = - [tex]\frac{opposite}{hypotenuse}[/tex] = - [tex]\frac{4}{5}[/tex]

Using the trigonometric identity

sin2A = 2sinAcosA

         = 2 × - [tex]\frac{4}{5}[/tex] × [tex]\frac{3}{5}[/tex]

         = [tex]\frac{2(-4)(3)}{5(5)}[/tex] = - [tex]\frac{24}{25}[/tex]

Answer:

5x(5x-7)+y(7-y)

Step-by-step explanation:

25x^2-35x+7y-y^2

5x(5x-7)+y(7-y)

Answer:

(5x-7y) (5x-y)

Step-by-step explanation:

Answer:

[tex]\frac{5}{7}-(-\frac{1}{7})=\frac{6}{7}[/tex]

Step-by-step explanation:

To evaluate :

[tex]\frac{5}{7}-(-\frac{1}{7})[/tex]

Solution:

Two negatives multiply to become a positive.

Thus, we can remove parenthesis by reversing the signs of the fraction by multiplying the negative outside.

⇒ [tex]\frac{5}{7}+\frac{1}{7}[/tex]

Since the denominators are same for both fractions, so we simply add the numerators.

⇒ [tex]\frac{5+1}{7}[/tex]

⇒ [tex]\frac{6}{7}[/tex]  (Answer)

The determinant of given matrix is zero

Step-by-step explanation:

Given matrix is:

[tex]\left[\begin{array}{ccc}1&4&4\\5&2&2\\1&5&5\end{array}\right][/tex]

The determinant of a 3x3 matrix is calculated by selecting a single row.

We are choosing the first row.

So,

[tex]= 1\left|\begin{array}{ccc}2&2\\5&5\\\end{array}\right|-4\left|\begin{array}{ccc}5&2\\1&5\\\end{array}\right|+4\left|\begin{array}{ccc}5&2\\1&5\\\end{array}\right|\\=1(10-10) -4(25-2)+4(25-2)\\=0-4(23)+4(23)\\=-92+92\\=0[/tex]

The determinant of given matrix is zero

Keywords: Matrices, determinant

Learn more about matrices at:

#LearnwithBrainly

the value of x is x=1/12'

HOPE IT HELPS U.

Answer:

The water she need is [tex]7\frac{7}{8} \ cups.[/tex] and strawberry syrup [tex]11\frac{13}{16}\ cups[/tex].

Step-by-step explanation:

Given:

Dina wants to make 15 3/4 cups of strawberry drink by mixing water and strawberry syrup with a ratio of 2 1/4 cup of water for every 3/4 cup of syrup.

Now, to find the quantity of water and syrup she need to use.

As given in question ratio so:

Strawberry syrup = 2 1/4 = 9/4.

Water = 3/4.

Total cups of strawberry drink = 15 3/4 = 63/4.

Let the strawberry syrup be [tex]\frac{9}{4} x[/tex].

And let the water be [tex]\frac{3}{4} x[/tex].

According to question:

[tex]\frac{9x}{4} + \frac{3x}{4}=\frac{63}{4}[/tex].

On adding the fractions:

⇒[tex]\frac{9x+3x}{4} =\frac{63}{4}[/tex]

⇒[tex]\frac{12x}{4} =\frac{63}{4}[/tex]

Multiplying both sides by 4 we get:

⇒[tex]12x=63[/tex]

Dividing both sides by 12 we get:

⇒[tex]x=\frac{63}{12}[/tex]

Dividing numerator and denominator by 3 on R.H.S we get:

⇒[tex]x=\frac{21}{4}[/tex]

Now, putting the value of [tex]x[/tex] on ratios:

Strawberry syrup =  [tex]\frac{9}{4}\times x=\frac{9}{4}\times\frac{21}{4}[/tex]

                             =  [tex]\frac{189}{16}[/tex]

                             =  [tex]11\frac{13}{16}\ cups[/tex]  

Water = [tex]\frac{3}{4}\times x =\frac{3}{4} \times\frac{21}{4}[/tex]

          = [tex]\frac{63}{8}[/tex]

          = [tex]7\frac{7}{8} \ cups.[/tex]

Therefore, the water she need is [tex]7\frac{7}{8} \ cups.[/tex] and strawberry syrup [tex]11\frac{13}{16}\ cups[/tex].

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

4x-3y=-1 ...(1)  

2x+3y=13..(2)

add (1) and (2) : 4x-3y + 2x+3y = -1 +13

6x =12

x = 12/6 = 2

put x = 6 in (2) : 2(2)+3y = 13

4+3y =13

3y =9

y=3

Final answer:

The ratio of legs to ears for Santa's reindeer is 2 legs/ear.

Explanation:

The question is asking for the ratio of legs to ears as it relates to Santa's reindeer. We know that Santa has 9 reindeer and reindeer typically have 4 legs and 2 ears. So the total number of legs would be 9 reindeer * 4 legs/reindeer = 36 legs. And the total number of ears would be 9 reindeer * 2 ears/reindeer = 18 ears. To find the ratio of legs to ears, we divide the number of legs by the number of ears: 36 legs / 18 ears = 2 legs/ear.

Para levantar 500 toneladas de mineral a una altura de 90 pies en una hora, se requieren aproximadamente 90.91 caballos de fuerza (HP), calculados utilizando la fórmula de trabajo y potencia.

Para calcular la potencia requerida en caballos de fuerza (HP) para levantar 500 toneladas de mineral hasta una altura de 90 pies en una hora, necesitamos utilizar la fórmula de trabajo y la potencia. La potencia se mide en unidades de trabajo por unidad de tiempo.

Primero, convirtamos la altura a pies a la misma unidad que la tonelada, que es la tonelada-pie (ton-ft). 1 tonelada-pie es igual al trabajo necesario para levantar una tonelada a una altura de un pie.

500 toneladas * 90 ft = 45,000 ton-pie

El trabajo total necesario es de 45,000 toneladas-pie.

Dado que el trabajo se realiza en una hora (3600 segundos), podemos usar la siguiente fórmula para calcular la potencia en HP:

[tex]\[Potencia (HP) = \frac{Trabajo (ft-lbf)}{Tiempo (s)} \times \frac{1}{550}.\][/tex]

Donde 1 HP es igual a 550 ft-lbf/s.

Sustituyendo los valores conocidos:

[tex]\[Potencia (HP) = \frac{45,000 \text{ ton-pie} \times 2,000 \text{ lbf/ton} \times 1 \text{ ft}}{3600 \text{ s} \times 550 \text{ ft-lbf/s}} = \frac{180,000,000 \text{ lbf-ft}}{1,980,000 \text{ ft-lbf/s}} \approx 90.91 HP.\][/tex]

Por lo tanto, se requieren aproximadamente 90.91 caballos de fuerza (HP) para realizar este trabajo en una hora.

For more such information on: mineral

https://brainly.com/question/15844293

#SPJ2

Answer:

The number of footballs, basketballs and volleyballs were sold are 75, 36 and 15 respectively.

Step-by-step explanation:

Consider the provided information.

A football costs $35, a basketball costs $25  and a volleyball costs $15.

Let F represents the football, B represents the basketball and V represents the volleyball.

On a given day, the store sold 5 times as many footballs as volleyballs.

[tex]F=5V[/tex]......(1)

They  brought in a total of $3750 that day,

[tex]35F+25B+15V=3750[/tex]......(2)

The money made from basketballs alone was 4 times the money.

[tex]25B=4(15V)[/tex]......(3)

By equation 1, 2 and 3.

[tex]35(5V)+4(15V)+15V=3750[/tex]

[tex]250V=3750[/tex]

[tex]V=15[/tex]

Substitute the value of V in equation 1 and 3.

[tex]F=5(15)=75[/tex]

[tex]25B=4(15\times 15)\\B=36[/tex]

Hence, the number of footballs, basketballs and volleyballs were sold are 75, 36 and 15 respectively.

The system of equations is as follows:

35F + 25B + 15V = 3750

1 F = 5V

25B = 60V

Let's define the variables first:

We are given the following information:

We can write the system of equations as:

35F + 25B + 15V = 3750

F = 5V

25B = 60V

Complete question:

A sporting goods store sells footballs (F), basketballs (B), and volleyballs (V). A football sells for $35 a basketball sells for $25, and a volleyball sells for $15. On a given day, the store sold 5 times as many footballs as volleyballs. The sales brought in a total of $3750 that day, and the money made from basketballs alone was four times the money made from volleyballs. Write the system of equations to determine how many of each type of ball were sold by entering relevant numbers in the boxes provided below to complete each equation. Do not solve the system.

___ F + ___ B + ___ V = 3750

___ F = ___ V

____ B = ____ V

Answer:

y = 30 + 1.5x and y = 2x

Step-by-step explanation:

Website 1 has a plan for a yearly fee of $30 and $1.5 for each download.

Therefore, if x is the number of downloads for a year and y is the total cost for the year, then we can model the conditions as  

y = 30 + 1.5x ......... (1)  

Website 2 has a plan of $2 for each download.

Therefore, we can models the condition as  

y = 2x ........ (2)

Therefore, equations (1) and (2) represent the costs for one year. (Answer)

Answer:

77

Step-by-step explanation:

Simplify the following:

52 + 7×3 + 4

7×3 = 21:

52 + 21 + 4

| 5 | 2

| 2 | 1

+ | | 4

| 7 | 7:

Answer: 77

To evaluate the expression 52 + 7 ⋅ 3 + 4, follow the order of operations (PEMDAS). First, multiply 7 by 3 to get 21, then add 52 and 4 to get the final result of 77.

Evaluation of the Expression:

To evaluate the expression 52 + 7 ⋅ 3 + 4, it is important to follow the order of operations (PEMDAS/BODMAS):

Using these rules, solve the expression step-by-step:

Therefore, the correct answer is:

77

There are approximately 275.59 inches in 7 meters.

To find out how many inches are in 7 meters, we will first convert meters to centimeters, and then centimeters to inches.

Since there are 100 centimeters in one meter, we can calculate:

7 meters × 100 centimeters/meter = 700 centimeters

Now, we know there are 2.54 centimeters in one inch, we can convert the 700 centimeters to inches:

700 centimeters × 1 inch/2.54 centimeters ≈ 275.59 inches

To the nearest inch, there are approximately 275.59 inches in 7 meters.

Answer:

b.) 8 on ed2020

Answer:

Step-by-step explanation:

2 + (-3) + 7 = 2 - 3 + 7 = 9 - 3 = 6 <==

a positive multiplied by a negative will be negative

Answer:

x= 33.49

Step-by-step explanation:

Answer:

15 according to the test

Step-by-step explanation:

Answer:

Step-by-step explanation:

4 + x = 133......to solve this, u would subtract 4 from both sides

so that would be the subtraction property of equality

Answer:

Step-by-step explanation:

4 + x = 13              subtract 4 from both sides

4 - 4 + x = 13 - 4

The cost of the camera if Sharon paid $78 sales tax at a rate of 6.5% is $1200

What was the cost of the camera?

Amount of sales tax = $78

Sales tax rate = 6.5%

Cost of the camera = x

.

Amount of sales tax = Sales tax rate × Cost of the camera

78 = 6.5% × x

78 = 0.065 × x

78 = 0.065x

Divide both sides by 0.065

x = 78/0.065

x = 1200

Therefore, the camera cost $1200

Answer:

y = -3x + 14

Step-by-step explanation:

Answer:

For a )    y = 6x - 3

[tex]slope = m = 6\\y-intercept = c = -3\\[/tex]

For b )     y = -2x - 10

[tex]slope = m = -2\\y-intercept = c = -10\\[/tex]

For c )    y = -4x + 4

[tex]slope = m = -4\\y-intercept = c = 4\\[/tex]

Step-by-step explanation:

Given:

a. y = 6x - 3

b. y = -2 (x + 5)

   y = -2x - 10

c. y = 4 (-x + 1)

  y = -4x + 4

To Find:

y-intercept and the slope for each equation = ?

Solution:

Slope-intercept Formula is given by

[tex]y=mx+c[/tex]

Where,

m = slope

c = y-intercept

So on comparing the Given equations with the above Equation we get

For a )   y = 6x - 3

[tex]slope = m = 6\\y-intercept = c = -3\\[/tex]

For b )    y = -2x - 10

[tex]slope = m = -2\\y-intercept = c = -10\\[/tex]

For c )    y = -4x + 4

[tex]slope = m = -4\\y-intercept = c = 4\\[/tex]