Explain how to translate the phrase into an algebraic expression. The difference between four times a number and 10

The critical numbers of the function are; θ = π/6, 5π/6

We are given the trigonometric function as;

f(θ) =  6secθ + 3tanθ ; 0 < θ < 2π

The critical points of a function are defined as those points of the domain where the function is not differentiable or those points for which the derivative is equal to zero.

Let us find the first derivative;

f'(θ) = 6(-sin θ/cos² θ) + 3(1/cos² θ)

f'(θ) = (3 - 6sin θ)/cos² θ

At f'(θ) = 0, we have;

3 - 6sin θ = 0

θ = sin⁻¹(3/6)

Since sin is positive, then;

θ = π/6, 5π/6

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ax + by = 6000 and a + 5 = b are two equations that might be used to calculate how many soup cans and frozen dinners were purchased.

There are several methods to define a formula. A mathematical statement that establishes the equivalence of two mathematical expressions is the definition of an equation in algebra.

This is best shown by the formula 3x + 5 = 14. In this instance, the word equal comes between the phrases 3x + 5 and 14. Even the most basic algebraic equations include the usage of mathematical variables.

Let frozen supper be y and soup cans be x.

X now has 300 mg of sodium in it.

x = 300

Y also has 450 mg of sodium in it.

y = 450

Samuel reportedly buys five more frozen dinners than soup cans, and they all have a combined salt content of 6000 mg.

Let the number of frozen dinners be "b," and the number of soup cans be "a."

ax + by = 6000

⇒ a(300) + b(450) = 6000 ...1

Samuel bought 5 more frozen dinners than soup cans, according to the inquiry, which results in equation

a + 5 = b ...2

Now, by changing the value of b in equation 1 we obtain,

⇒ a(300) + (a + 5)(450) = 6000

⇒ 300a + 450a = 6000 - 2250

⇒ 750a = 3750

⇒ a = 3750/750

⇒ a = 5

By changing the value of an in equation 2, we obtain

⇒ 5 + 5 = b

⇒ b = 10

So, ax + by = 6000 and a + 5 = b are two equations that might be used to calculate how many soup cans and frozen dinners were purchased, respectively, where x and y stand for soup cans and frozen dinners, respectively, and a and b stand for the quantity of soup cans and frozen dinners.

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Answer:

C

Step-by-step explanation:

it is still C, as the angle-angle-angle information of D is not enough. the sides could all grow or shrink (in the same relation, but still) and the angles would still be the same.

so, D is not enough (although we have 3 pieces of information, but this is the special case, where even that is not enough - we need at least one side in the mix).

Answer:

$27.69

Step-by-step explanation:

The length of x in the similar quadrilateral is 1.5 units.

Quadrilaterals are polygons with 4 sides. The sum of the interior angles of a quadrilateral is 360 degrees.

Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same and two adjacent sides have equal ratios.

Therefore, quadrilaterals ABCD and JKLM are similar.

Hence, let's find the side x of the quadrilaterals.

Therefore,

CD / LM = BC / KL

CD = 5

LM = 2.5

BC = 3

KL = x

Hence,

5 / 2.5 = 3 / x

cross multiply

5x = 2.5 × 3

5x = 7.5

x = 7.5 / 5

x = 1.5

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Explanation:

Choices A and B are ruled out because they are cube root functions, which have a domain and range of "all real numbers".

Choice C is ruled out because the domain is [tex][a, \infty)[/tex]

To determine this domain, we set the radicand (aka stuff under the square root) to be greater than or equal to zero.

[tex]x-a \ge 0 \ \ \text{ solves to } \ \ x \ge a[/tex]

[tex]x \ge a \ \ \text{ turns into the interval notation } \ \ [a, \infty)[/tex]

The square bracket is used to include the value 'a' as part of the interval. This is the interval from x = a to positive infinity.

The domain of choice D is [tex][b, \infty)[/tex] found using similar steps as shown above.

The range is [tex](-\infty, a][/tex] since this square root function is decreasing, due to the negative out front, which means y = a the largest output possible. I recommend graphing it out using a tool like desmos or geogebra. These tools offer a way to use parameters.

Answer:

[tex]\huge\boxed{\sf v = 0.4\ mph}[/tex]

Step-by-step explanation:

Distance = S = 4/5 miles = 0.2 miles

Time = t = 1/2 hour = 0.5 hours

Speed = v = ?

v = S/t

v = 0.2/0.5

[tex]\rule[225]{225}{2}[/tex]

Answer: 8/5 mph

Step-by-step explanation:

According to definition, an object's speed is determined by:

s=d/t where,

d is the object's travel distance, and t is the amount of time it takes.

We can replace these values with:

Since, s= (4/5) / (1/2) mph

=> s= 8/5 mph

The object's speed to cover 4/5 miles in 30 minutes is: 8/5 mph

In differentiation , qA = 13/5  , qB = 11/5 are  values of qA and qB maximize the revenue .

What does mathematics differentiation serve?

In mathematics, differentiation is used to compute rates of change. For instance, in mechanics, velocity is the amount at which a displacement changes over time.

                   The acceleration is the speed at which velocity is changing with respect to time.

R = 40qA + 50qB − 6qA2 − 9qB2 − 4qAqB

Let aA = x , qB = y

so,

 R= 40x + 50y - 6x² - 9y² - xy

Partially differentiate above equation with respect to x and y .

Rx = 40 - 12x - 4y

Ry = 50 - 18y - 4y

Set Rx = 0 and Ry = 0 solve x and y .

40 - 12x - 4y = 0

50 - 18y - 4x = 0

Solving above questions

 x = 13/5 , y = 11/5

therefore,

      qA = 13/5  , qB = 11/5

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By AAS congruence theorem, ΔABD is congruent ΔCDB.

Quadrilateral ABCD is given in the figure.

Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. The word congruent means exactly equal in shape and size no matter if we turn it, flip it or rotate it.

Given, AB║DC

Consider ΔABD and ΔCDB

m∠A=m∠C=90° (Given)

m∠ABD=m∠BDC (Alternate interior angles)

BD = BD (Common)

Using AAS congruence theorem, ΔABD ≅ ΔCDB

Hence, by AAS congruence theorem, ΔABD is congruent ΔCDB.

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Answer:

x = 79

Step-by-step explanation:

To find the value of x we first need to find the missing angle of the triangle in the image.

101 + 29 + missing angle = 180

130 + missing side = 180

missing side = 50

Now, the sum of two interior angles in a triangle is equal to an exterior angle that is supplementary to the third interior angle.

So the value of x:

29 + 50 = 79

Answer:

When factoring, you want to get all of the terms on one side first and set all of it equal to 0. Many of the problems here are already in this format =0, but there are some where you'll have to move the terms around.

In these examples, it also looks like the highest power you have is x^2. This makes things easier to work with.

Let's just start with the first one, x^2 - 9x + 18 = 0.

We need to find two numbers that multiply to give us 18, and add together to give us -9. That would be -6 and -3.

-6 + -3 = -9

- 6 * -3 = 18

We can rewrite it now. This is the factored form.

(x - 6) * (x - 3) = 0

If you want to, you can go back to expanded form using the acronym 'FOIL'. First, Outside, Inside, Last. This refers to the order of multiplying the terms.

(x - 6) * (x - 3)

x * x = x^2

x * -3 = -3x

-6 * x = -6x

-6 * -3 = 18

Simplify

x^2 - 3x - 6x + 18 = 0

And we're back to where we started.

x^2 - 9x + 18 = 0

Hope this helps.

To find the two points having an x-coordinate of 3 whose distance from the point (-2, -1) is 13, you can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the coordinates of the first point, and (x2, y2) is the coordinates of the second point.

In this case, you can use the distance formula to find the y-coordinates of the two points having an x-coordinate of 3:

Distance = √((3 - (-2))^2 + (y - (-1))^2)

= √(5^2 + (y + 1)^2)

= √(25 + y^2 + 2y + 1)

= √(26 + y^2 + 2y)

= 13

If you set the distance equal to 13 and simplify the equation, you get:

y^2 + 2y + 12 = 0

This is a quadratic equation, which means it can be solved using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation.

Substituting the values into the quadratic formula, you get:

y = (-2 ± √(-2^2 - 4 * 1 * 12)) / (2 * 1)

= (-2 ± √(-4 - 48)) / 2

= (-2 ± √(-52)) / 2

Since the square root of a negative number is not a real number.

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By using fraction, it can be calculated that

30 boards are required to cover a floor of width 205 inches wide

What is fraction?

Suppose there is a collection and a part of collection has to be taken.

The part which is taken is called fraction. In other words part of a whole is called fraction.

The upper part of the fraction is the numerator and the lower part of the fraction is the denominator.

This is a word problem on fraction

Width of each board = [tex]6\frac{5}{6}[/tex] inches = [tex]\frac{41}{6}[/tex] inches

Total width of floor = 205 inches

Number of boards required = [tex]205 \div \frac{41}{6}[/tex] = [tex]205 \times \frac{6}{41}[/tex] = 30

30 boards are required

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The inequality is y > 2x-3

option C is correct

What is inequality in math?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa. Literal inequalities are relationships between two algebraic expressions that are expressed using the inequality symbols.

The line intersects the y- axis at point (0,-3)

i.e., The y-intercept =-3

We know that,

Equation of lime, y= mx + b

                                                        b: y-intercept

                                                         m: Slope of the line

Slope of the line,

m = 0-(-3)/3/2 = 2

m ⇒ 2

∴ Equation of line,  y = 2x - 3

The shaded are is above the line, and line is represented with dotted lines,

hence the inequality is,

y > 2x - 3

Hence, option C is correct

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$9200\\ r=rate\to 4.9\%\to \frac{4.9}{100}\dotfill &0.049\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &5 \end{cases} \\\\\\ A=9200\left(1+\frac{0.049}{4}\right)^{4\cdot 5}\implies A=9200(1.01225)^{20} \implies A \approx 11736.63[/tex]

The collection of all logical values for each independent variable under consideration.

The set of all independent variable values that are logical in the particular circumstance. the collection of every possible value for the dependent variable the collection of all feasible dependent variable values that are appropriate in the situation.

Let y = f(x) represent a function where x and y are independent variables. The chosen x-value is said to fall within the domain of f if a function f offers a means of successfully producing a single value y while using a value for x to that end.

The set of possible input values is referred to as the domain, and the domain of a graph is made up of all the input values displayed on the x-axis. The set of potential output values that make up the range is represented by the y-axis.

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slope formula  can be illustrated  as ,  (y₂ - y₁) / (x₂ - x₁)

It is given in question that-

x₁ = -1 and y₁ = 1

x₂ = -2and y₂ = -3

so by using given formula for slope

slope= (y₂ - y₁) / (x₂ - x₁)

slope = (-3- (1) / (-2 - (-1) = (-3 - 1) / (-2 + 1) = -4/-1 = 4

final answer , slope =4

What is slope ?

The ratio of how much y grows as x grows by a certain amount is known as a line's slope. The slope of a line indicates how steep it is, or how much y rises as x rises. Anywhere along the line, the slope remains constant (the same).

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Answer:

[tex] \sf \: slope(m) = 4[/tex]

Step-by-step explanation:

Now we have to,

→ find the required slope.

Formula we use,

→ Slope(m) = (y2 - y1)/(x2 - x1)

Then the slope will be,

→ m = (y2 - y1)/(x2 - x1)

→ m = (-3 - 1)/(-2 -(-1))

→ m = (-4)/(-2 + 1)

→ m = (-4)/(-1)

→ m = 4/1

→ [ m = 4 ]

Hence, the slope is 4.

Answer:

See below

Step-by-step explanation:

I would start off writing the equation in the slope intercept form of a line

y + 4 = 1/4(x + 10)  Distribute the 1/4

y + 4 = 1/4x + 10/4    Simplify 10/4

y + 4 = 1/4 x + 5/2  Subtract 4 from both sides

y = 1/4 x + 5/2 -4

y = 1/4 x + 5/2 - 8/2

y = 1/4 x - 3/2

Then I used the free graphing calculator Desmos to graph.

By solving a linear equation, we will see that they sold a total of 288 cookies.

An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario. It is vital to note that an equation is a mathematical statement which is made up of two expressions that are connected by an equal sign.

We know that Phil and Matt sell cookies for $0.25 each, so, if they sell x cookies, the total revenue is: R(x) = $0.25*x

Now, we know that they made a total of $72.00, so now we need to solve the linear equation:

$0.25*x = $72.00x

= $72.00/$0.25 = 288

They sold 288 cookies.

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Simplify the [tex]2\sqrt{10}[/tex] · [tex]5\sqrt{12}[/tex] we will get  20 · √30.

What do you mean by exponent and radicals?

Exponents are a convenient way of describing repeated multiplication of the same number. Radicals include the use of radical characters √

These are sometimes called surdos. The exponent is the power p of an expression of the form [tex]a^{p}[/tex]

The process of raising a base number to a given power is called exponentiation. (where a ≠ 0)

Given:

[tex]2\sqrt{10}[/tex] · [tex]5\sqrt{12}[/tex]

= (2 × 5)· [tex](\sqrt{10}\times \sqrt{12)}[/tex]

= 10 · [tex]\sqrt{120}[/tex]

= 10 · 2 · [tex]\sqrt{30}[/tex]

= 20 · √30

Therefore, [tex]2\sqrt{10}[/tex] · [tex]5\sqrt{12}[/tex] = 20 · √30

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I will have the fraction of 1/6 cups of one cup, if 3 cups of half 2/6 fill is what I have.

We can solve word problems in mathematics using the 4 basic operations (addition, subtraction, multiplication and division) on fractions.

From the question, we are have 3 cups of half 2/6 filled cups, this implies that;

2/6 filled cups = 1/3

half 2/6 filled cups = (1/3)/2

half 2/6 filled cups = 1/3 × 1/2 {changing the division to multiplication}

half 2/6 filled cups = 1/6

Therefore, I have 1/6 for one cup and for 3 cups of half 2/6 filled cups, I will have 1/2.

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So each friend will get by dividing  3*10 + 5 + 6/2 = 30 + 5 + 3 =38 blueberries.

One of the fundamental mathematical operations is division, which involves breaking a bigger number into smaller groups with the same number of components. How many groups will be created, for instance, if 30 students need to be separated into groups of five for a sporting event? The division operation may be used to quickly and simply fix such issues. In this case, we must divide 30 by 5. 30 x 5 = 6 will be the outcome. There will thus be 6 groups with 5 students each. The initial number, 30, which is obtained by multiplying 6 by 5 may be used to confirm this figure.

The names of the phrases connected with the division process are referred to as division parts. The division is made up of the following four components: dividend, divisor, quotient, and remainder.

Two friends can divide 76 blueberries by dividing 76 divided by 10 .

The value we get is Divisor is 10 , Dividend is 76

So, The Quotient is 7 and Remainder is 6

So there will be 7 box of 10 blueberries and 1 box with 6 berries.

Pictorial Representation is :

(10), (10), (10), (10), (10), (10), (10), (06).

Now since there are 7 bowls of 10 berries so 7 divide by 2 is 3 as quotient and 1 as remainder so 1 box will be split apart into 10 divided by 2 is 5 berries each.

So each friend will get 3*10 + 5 + 6/2 = 30 + 5 + 3 =38 blueberries.

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So each friend will get by dividing  3*10 + 5 + 6/2 = 30 + 5 + 3 =38 blueberries.

What is Division ?

One of the fundamental mathematical operations is division, which involves breaking a bigger number into smaller groups with the same number of components. How many groups will be created, for instance, if 30 students need to be separated into groups of five for a sporting event? The division operation may be used to quickly and simply fix such issues. In this case, we must divide 30 by 5. 30 x 5 = 6 will be the outcome. There will thus be 6 groups with 5 students each. The initial number, 30, which is obtained by multiplying 6 by 5 may be used to confirm this figure.

The names of the phrases connected with the division process are referred to as division parts. The division is made up of the following four components: dividend, divisor, quotient, and remainder.

Two friends can divide 76 blueberries by dividing 76 divided by 10 .
The value we get is Divisor is 10 , Dividend is 76
So, The Quotient is 7 and Remainder is 6
So there will be 7 box of 10 blueberries and 1 box with 6 berries.
Pictorial Representation is :
(10), (10), (10), (10), (10), (10), (10), (06).
Now since there are 7 bowls of 10 berries so 7 divide by 2 is 3 as quotient and 1 as remainder so 1 box will be split apart into 10 divided by 2 is 5 berries each.
So each friend will get 3*10 + 5 + 6/2 = 30 + 5 + 3 =38 blueberries.

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Answer:

x=9.8

Step-by-step explanation:

Answer:

Bachelor's Degree: $55,000
Master's Degree: $61,000

Step-by-step explanation:

First start defining variables for each of the salaries

Let M be the salary of the master's individual

Let B be the salary of the bachelor's individual

Given:
Average yearly salary of an individual whose final degree is a master's is $49 thousand less than twice that of an individual whose final degree is a bachelor's

can be translated into the equation

M = 2B - 49,000

Subtract M from both sides and add 49,000 to both sides:

49,000 = 2B - M

Switch sides:

2B - M = 49,000   [1]

Together they earned 116,000 becomes

B + M = 116,000   [2]

Add equations [1] and [2] to eliminate M
2B - M + B + M = 49,000 +116000
3B = 165000

B = 165000/3 = 55,000

Using equation 2, B + M = 116000 we get

55000 + M = 116000

M = 116000 - 55000 = 61,000

Solution
Bachelor's Degree: $55,000
Master's Degree: $61,000

Total percentage of students in each department, to the whole of Lee Business School is 85.5298308   11.48709 0 2.983081     100.

What is percentage ?

A % is a quantity or ratio that, in mathematics, represents a portion of one hundred. A dimensionless connection between two numbers can be represented in a variety of ways, such as through ratios, fractions, and decimals. The sign "%" is frequently put after the number to indicate percentages.

Department Undergrad Master Doctoral Total

 Total % Total % Total % Numbers %

Accounting 390 17.364203 83 3.695459 0 0 473 21.05966

Economics 467 20.79252 20 0.890472 0 0 487 21.68299

Finance 215 9.5725735 7 0.311665 0 0 222 9.884239

Management, E&T 590 26.268923 75 3.33927 1 0.044524 666 29.65272

Marketing and I.B. 259 11.531612 10 0.445236 1 0.044524 270 12.02137

MBA 0 0 63 2.804987 65 2.894034 128 5.69902

Total 1921   258   67   2246  

Percentage of total students 85.5298308   11.48709 0 2.983081     100

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Answer: 24

Step-by-step explanation:

A=5 so F(x)=5^2-1

F(x)=24

La franja amarilla del rectángulo tiene un área de 30 centímetros cuadrados.

En este problema tenemos un rectángulo formado por dos cuadrados que se traslapan uno al otro. La franja amarilla es el área en la que los cuadrados se traslapan. La anchura del rectángulo es descrita por la siguiente ecuación:

(10 - x) + 2 · x = 17

Donde x se mide en centímetros.

A continuación, despejamos x en la ecuación descrita:

10 + x = 17

x = 7

Ahora, el área de la franja amarilla se determina mediante la fórmula de area de un rectángulo:

A = b · h

Donde:

A = (10 - 7) · 10

A = 3 · 10

A = 30

El área de la franja amarilla del rectángulo es igual a 30 centímetros cuadrados.

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Answer:

24-3-32233'2]/2

Step-by-step explanation: