What is the solution to this system of equations? 3⁢x+y+2⁢z=6 6⁢x−2⁢y=2 3⁢x+y−2⁢z=0

The measurement of the length of two sides and the included angle are given. Hence, using law of cosines, the approximate length of the lake is found to be 296.1 yards.

Given : b = 245 yards, c= 270 yards,

To find: The length of the lake which is the value of x

To find [tex]\angle A[/tex], we know that [tex]110^o\, and\, \angle A[/tex] lie on straight line. So

[tex]110^o + \angle A = 180^o\\\\\angle A = 180^o - 110^o = 70^o[/tex]

Using law of cosines

[tex]a^2 = b^2 +c^2 - 2bc\cdot cosA[/tex]

Plugging the values of b , c and A into the equation, we get

 [tex]a^2 =245 ^2 +270^2 - 2\times 245 \times 270 \times \cos 70^o \\[/tex]

The value of [tex]\cos 70^o[/tex] is 0.342 approximately

On substituting this in the equation, we get

a = 296.1 yards

Every triangle has three vertices and three sides. If in a triangle, the three vertices are A, B, and C, then, the sides opposite these vertices are a, b, and c, respectively. The angle at each vertex is given the same name as the vertex. Therefore, the three angles are also named A, B, and C. To solve such a  triangle is to find the lengths of each of its sides and all its angles.

The sine rule  or law of sines is used when we are given either

a) two angles and one side

or

b) two sides and the angle that is not included between these two sides.

[tex]\frac{\sin A}{a} =\frac{\sin B}{b} =\frac{\sin C}{c}[/tex]

The cosine rule or law of cosines is used when we are given either

a) three sides (SSS)

or

b) two sides and the included angle (SAS)

[tex]c^2=a^2+b^2-2ab\cdot\ cosC \\ b^2=a^2+c^2-2ac\cdot\ cosB\\ a^2=b^2+c^2-2bc\cdot\ cosA\\[/tex]

The measurement of the length of two sides and the included angle are given. Hence, using law of cosines, the approximate length of the lake is found to be 296.1 yards.

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Answer: $350 = t

Step-by-step explanation:

might help);   600-250 = t

t + 250 ≤ 600  INEQUALITY

Answer: $350.

Explanation: if you do  [tex]600 -250[/tex] you will end up with $350

Answer: 11 / 20

Step-by-step explanation:

Answer:

2 minutes

Step-by-step explanation:

8 : 4 = x : 1

x = 8/4

x = 2

Step-by-step explanation: To find the time it takes Elijah to run 1 lap, we need to divide the time it takes him to run 4 laps by the number of laps he runs in that time. Since it takes Elijah 8 minutes to run 4 laps, the time it takes him to run 1 lap is 8 minutes / 4 laps = ${2}$ minutes per lap.

This calculation illustrates the concept of unit rate, which is the rate at which a quantity changes over a given unit of time or distance. In this case, the unit rate is the time it takes Elijah to run 1 lap, which is 2 minutes per lap. This unit rate allows us to compare the time it takes Elijah to run different distances, and to determine how long it would take him to run a given distance at the same pace.

When f(x)=x4 and g(x)=2x+4 at x=2 for f(x) and x=-2 for g, the value of g(f(2)) is 8 (x).

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Four broad categories can be used to classify different types of functions. dependent upon element Function is a one-to-one relationship, a many-to-one relationship, onto function, one-to-one and into function.

Here,

f(x)=x-4

g(x)=-2x+4

f(2)=2-4

=-2

g(-2)=-2*-2+4

g(-2)=8

The value of g(f(2)) is 8 for f(x)=x−4 and g(x)=−2x+4 at x=2 for f(x) and x=-2 for g(x).

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Answer:

Step-by-step explanation:

17-8=9 but if 18-7=11 then 19-8=7

Each pound of fish and chicken will cost $3.75 and $2.25 respectively

A word problem is a mathematical exercise where significant background information on the problem is presented in ordinary language rather than in mathematical notation

Given that:

1. A customer pays a total of $11.10 for 1.6 pounds of chicken

and 2 pounds of fish.

2. Another customer pays a total of $12.15 for 2.4 pounds of

chicken and 1.8 pounds of fish

Let C and F represent the cost of one pound of chicken and fish respectively

Statement 1:

1.6C + 2F = 11.10  ---- (1)

Statement 2:

2.4C + 1.8F = 12.15  ---- (2)

we have simultaneous equations now, let's solve using the elimination method:

1.6C + 2F = 11.10      ---- (1)

2.4C + 1.8F = 12.15  ---- (2)

2.4 × (1.6C + 2F = 11.10) =     3.84C + 4.8F = 26.64

1.6 × (2.4C + 1.8F = 12.15) =  3.84C + 2.88F = 19.44

                                            ..........................................

                                                           1.92F  = 7.2

Thus, 1.92F  = 7.2

F = 7.2/1.92 = $3.75

Put F = $3.75 into (1):

1.6C + 2F = 11.10

1.6C + 2(3.75) = 11.10

1.6C + 7.5 = 11.10

1.6C = 11.10 - 7.5

1.6C = 3.6

C = 3.6/1.6 = $2.25

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Answer:   [tex]\frac{\sqrt{55}}{8}[/tex]

This is the same as writing sqrt(55)/8 where the 8 is not inside the square root.

=====================================================

Explanation:

Draw a right triangle in the quadrant Q1, which is in the northeast corner.

We're given [tex]\cos(\theta) = 3/8[/tex]. Recall that cosine is the ratio of adjacent over hypotenuse.

[tex]\cos(\text{angle}) = \text{adjacent/hypotenuse}[/tex]

This right triangle will have an adjacent side of 3 and hypotenuse of 8.

Refer to the diagram below.

Use the pythagorean theorem [tex]a^2+b^2 = c^2[/tex] to find the missing side is exactly [tex]\sqrt{55}[/tex], which is the opposite side of angle theta.

Then as one last step, we would say:

[tex]\sin(\text{angle}) = \text{opposite/hypotenuse}\\\\\\\sin(\theta) = \frac{\sqrt{55}}{8}[/tex]

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

A very similar approach is to use the pythagorean trig identity

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

Solving for sine gets us

[tex]\sin(\theta)=\sqrt{1-\cos^2(\theta) }[/tex]

Sine is positive in Q1.

Then plug in [tex]\cos(\theta) = 3/8[/tex] and simplify. You should get the answer mentioned above.

50 tickets were thus sold at the door. As a result, there were 175 advance sales of tickets and 50 at the door.

A linear equation has the algebraic form y=mx+b. where m is the slope and b is the y-intercept, consisting of a first-order (linear) term and a constant. If x and y are variables, the above is sometimes called a "two-variable linear equation".

Tickets for a high school dance were provided.

Purchases made before the dance cost $4.00 apiece.

If purchased at the door, they would cost $5 apiece.

225 tickets were sold, bringing in $950.

It's time to locate the advance-purchase and door-purchase tickets.

Let the quantity of advance tickets sold equal x.

And the quantity of tickets sold at the door will be equal to y.

hence, the overall quantity of tickets sold

[tex]x+y = 225\\y = 225 - x[/tex]......(1)

Now,

4x + 5y = 950

[tex]4x + 5y = 950\\4x + 5(225-x) = 950\\4x + 1125 - 5x = 950\\-x = -175\\x= 175\\y=50[/tex]

50 tickets were thus sold at the door.

As a result, there were 175 advance sales of tickets and 50 at the door.

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Answer:

f(-11) = -16

Step-by-step explanation:

We are given a function

f(x) = x-5

Substitute the value of x = -11 into the function

f(-11) = -11-5

        = -16

Answer:

-16

Step-by-step explanation:

Given that,

f ( x ) = x - 5

To find the value of f ( -11 ), you have to replace x with -11 & solve it.

Let us solve it now.

f ( -11 ) = x - 5

f ( -11 ) = -11 - 5

f ( -11 ) = -16

Answer: It is correct.

Step-by-step explanation:

For the given number line the inequality equation would be -10 < x ≤ 20.

What is inequality?

In mathematics, a linear inequality is an inequality that involves a linear function. A linear inequality contains one of the symbols of inequality. It shows the data which is not equal in graph form.

In the given number line, a hollow circle represents that the number is excluded and solid circle represents that the number is included in the range

and the range lies between -10 to 2.

Let's take a variable to describe the relationship,

By using inequality we can prepare the inequality equation as

                -10 < x ≤ 20

Hence, for the given number line the inequality equation would be -10 < x ≤ 20.

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The linear equation that is modeled by the table is:

y = 2x + 1

A general linear equation is written as:

y = a*x + b

Where a is the slope and b is the y-intercept.

If the line passes through two points (x1, y1) and (x2, y2), then the slope is:

a = (y2 - y1)/(x2 - x1)

Here we can use the first two points on the table (-3, -5) and (-1, -1), we will get the slope:

a = (-1 + 5)/(-1 + 3)

a = 4/2 = 2

Then the line is something like:

y = 2x + b

To find the value of b we can use one of the points (-3, -5), replacing these values we will get:

-5 = 2*-3 + b

-5 = -6 + b

-5 + 6 = b

1 = b

Then the linear equation is:

y = 2x + 1

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Answer: 166

Step-by-step explanation:

The sum of two odd numbers is an even number.

85 is odd, and so is the sum of an odd number and an even number.

2×83=166

Answer:

Below

Step-by-step explanation:

Value = 33000 ( .88)^t        each year, t , the value is only 88% ( or in decimal:  .88) of the prior year  ( 12 % decrease leaves 88% )

Answer:

Step-by-step explanation:

28

Answer: 64

Step-by-step explanation:

There are 16 ounces in a pound and there are 4 pounds and 4x 16 is 64

Answer:

64 oz.

Step-by-step explanation:

16 oz. = 1 lb.

16 oz. x 4 lbs. = 64 oz.

The system of equations is  y = 50 + 3x; y = 34 + 4x. The number of cups that Aunt Emily orders is 16. The cost of either shop for 16 cupcakes is $98.

What is substitution method?

The algebraic technique for resolving multiple linear equations at once is called the substitution method. As the name implies, this method involves substituting a variable's value from one equation into another. In this manner, a pair of linear equations are combined into a single, simple linear equation with just one variable.

Assume that x number of cups that ordered by Aunt Emily.

Given that, Clarksville Bakery charges $3 for each cupcake, plus $50 for the large cake.

Therefore, the cost of x cupcake is 50 + 3x.

y = 50 + 3x

Also, Sebastian's Sweet Shoppe charges $34 for the large cake and $4 for each cupcake.

Therefore, the cost of x cupcake is 34 + 4x.

y = 34 + 4x

The system of equations is

y = 50 + 3x  ....(i)

y = 34 + 4x  ...(ii)

To find the number of cupcakes that Aunt Emily orders, use substitution method to solve the system of equation.

Substitute y = 50 + 3x in equation (ii)

50 + 3x = 34 + 4x

Subtract 3x from both sides:

50 = 34 + x

x = 16

Putting x = 16 in equation (i)

y = 50 + (3 × 16) = 98

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Answer: True

Step-by-step explanation: Take a number on the list minus the previous one, and it shows a constant, then it is an arithmetic sequence.

The 25 teams can there be at most.

What is the LCM?

Least Common Multiple(LCM) is a method to find the smallest common multiple between any two or more numbers. A common multiple is a number which is a multiple of two or more numbers.

We have,

75 fifth graders

125 fourth graders.

They are going to be grouped in teams for a field day with the same number of students on each team.

the same number of teams if there must also be the same number of fifth graders on each team.

take the LCM of the 75  & 125 Which is the 25.

3 fifth graders each

5 fourth graders each

Hence, the 25 teams can there be at most.

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Answer:

8x + 9

Step-by-step explanation:

5x + 3x + 6 + 3

8x + 9

Answer:

√53 or 7.28

Step-by-step explanation:

To find the length of the line segment from the point (4, 2) to the point (-3, 4), you can use the distance formula, which is:

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

Plugging in the coordinates of the two points, you get:

distance = √((-3 - 4)^2 + (4 - 2)^2)

Simplifying this expression gives:

distance = √((-7)^2 + (2)^2)

Which simplifies to:

distance = √(49 + 4)

Which simplifies to:

distance = √53

So the length of the line segment from the point (4, 2) to the point (-3, 4) is approximately 7.28.

Answer: 1452 units^3

The required volume of the cylinder with a height of 12 units and radius of 11 units is 1452π cubic units.

Volume is defined as the mass of the object per unit density while for geometry it is calculated as profile area multiplied by the length at which that profile is extruded.

The formula for the volume of a cylinder is given as:

V = πr²h

Where V is the volume of the cylinder, r is the radius of the cylinder, and h is the height of the cylinder.

Substituting the given values, we get:

V = π(11)²(12)

= π(121)(12)

= 1452π cubic units

Therefore, the volume of the cylinder with a height of 12 units and radius of 11 units is 1452π cubic units.

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In a residual plot if the linearity assumption is the points are scattered and there is no obvious pattern.

Residual plot:

in statistics, residual value is a measure of how much a regression line vertically misses a data point.

Given,

Here we need to find what would you expect to see in a residual plot if the linearity assumption is correct.

As per the definition of residual plot, there is no obvious pattern for this plot and the positive and negative values are evenly spread across the whole range.

Therefore, in the regression line there are some points are misses the line on the vertical base.

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Answer:

30.9 cubic inches

Step-by-step explanation:

This is a volume question. You multiply 2 3/4, 2 1/4, and 5 inches all together.

First you turn it into an improper fraction and multiply. 11/4*9/4 = 99/16

Then multiply this by 5: 99/16 * 5 = 30.9375

Round to nearest tenth: 30.9

The question is an illustration of equation of circles.

The equation of the dartboard circle is:  [tex](x-12)^{2}+(y-\frac{17}{3}) ^{2} =\frac{9}{16}[/tex]

Brian's dart lands on the dartboard because D = 11.5 feet , 5 feet is within the circumference of the dartboard.

Given that,

h = 5 feet 8 inches ---- the height at which the dartboard was hung

d = 18 inches in diameter ---- the diameter of the dartboard

B = 12 feet --- the bull's eye

D = 11.5 feet , 5 feet --- Brian's dart

Equation of the circle

First, we convert all units to feet

This is done by dividing inches units by 12

h = 5 feet 8 inches

h = 5 ft + 8/12 ft

h = 5 ft + 2/3 ft

Take LCM

h = [tex]\frac{15+2}{3}[/tex] ft

h = [tex]\frac{17}{3}[/tex] ft

d = 18 inches

d = [tex]\frac{18}{12}[/tex] ft

d = [tex]\frac{3}{2}[/tex] ft

Divide by 2 to calculate radius

r = [tex]\frac{3}{2*2}[/tex] ft

r = [tex]\frac{3}{4}[/tex] ft

To find the equation of a circle when you know the radius and centre, use the formula as [tex](x-a)^{2} +(y-b)^{2} =r^{2}[/tex] , where represents the centre of the circle, and is the radius.

The equation of the circle is represented as:

[tex](x-a)^{2} +(y-b)^{2} =r^{2}[/tex]

In this case:

a = B = 12 feet -- the distance between the bull's eye and the wall

b = h = [tex]\frac{17}{3}[/tex] ft  ---- the height at which the dartboard was hung

So, we have:

[tex](x-a)^{2} +(y-b)^{2} =r^{2}[/tex]

[tex](x-12)^{2} +(y-\frac{17}{3}) ^{2} = (\frac{3}{4}) ^{2}[/tex]

Evaluate the exponents

[tex](x-12)^{2}+(y-\frac{17}{3}) ^{2} =\frac{9}{16}[/tex]

Hence, the equation of the circle is:

[tex](x-12)^{2}+(y-\frac{17}{3}) ^{2} =\frac{9}{16}[/tex]

Does Brian’s dart land on the dartboard?

Yes her dart lands on the dartboard because

D = 11.5 feet , 5 feet is within the circumference of the dartboard

Therefore,

The question is an illustration of equation of circles.

The equation of the dartboard circle is:  [tex](x-12)^{2}+(y-\frac{17}{3}) ^{2} =\frac{9}{16}[/tex]

Brian's dart lands on the dartboard because D = 11.5 feet , 5 feet is within the circumference of the dartboard.

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By using the concept of function, it can be seen that the given figure does not represents a function.

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here the values on the x axis represents the domain and the values on the y axis represents the Range.

From the figure it can be seen that

For x = 2, there are three points on the y axis.

So the point x = 2 maps to different points on the y axis

So this is not a function

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Sine and cosine are the fundamental trigonometric functions associated to the sides of a right triangle and tangent is a derivate trigonometric function.

Let be a right triangle, whose features are introduced in the figure attached below. The side with measure y is the side opposite to the angle θ, whereas the side with measure x is the side adjacent to the angle θ. The side with measure r represents the hypotenuse of the right triangle, which can be found by means of Pythagorean theorem:

r = √(x² + y²)

Trigonometric functions are trascendent functions that relates two sides of the right triangle. There are two fundamental trigonometric functions and four derivate trigonometric functions:

Fundamental

sin θ = y / r = (Opposite side) / (Hypotenuse)

cos θ = x / r = (Adjacent side) / (Hypotenuse)

Derivate

tan θ = sin θ / cos θ = y / x = (Opposite side) / (Adjacent side)

cot θ = 1 / tan θ = cos θ / sin θ = x / y = (Adjacent side) / (Opposite side)

sec θ = 1 / cos θ = r / x = (Hypotenuse) / (Adjacent side)

csc θ = 1 / sin θ = r / y = (Hypotenuse) / (Opposite side)

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