A bus company uses a grid to locate bus stops. The bus station is at (3, 4). The point (7, 6) represents
the first bus stop. The location of the second bus stop is determined using the same relationships as
those between the coordinates of the bus station and the first bus stop.
City Bus Stops
North
Choose...
12
10-
City Blocks
6
8
4
2
N.
Bus Station
: increases by 2.
2
(7.6)
increases by 4, and the
(11,8)
4 6 8 10 12
City Blocks
Use the drop-down menus to describe the relationships between the coordinates of the first and second
bus stop.
The Choose...
East

The maximum area of the rectangle inside the ellipse is 2ab.

Given that,

Ellipse = x2 / a2 + y2 / b2 = 1

we know the general coordinates of any point in an ellipse of the equation

x2 / a2 + y2 / b2 = 1 is (acosθ,bsinθ)

So, we take general vertices of rectangles as

(acosθ,bsinθ), (acosθ,-bsinθ), (-acosθ,bsinθ), and (-acosθ,-bsinθ)

So, we can see that length and breadth of the rectangle is 2acosθ and 2bsinθ

So, its area will be ab4sinθcosθ (Length x Breadth)

Applying trigonometric formula 2sinθcosθ=sin2θ, let area equals to A

Now A=2absin2θ

Now as the area is the maximum given in the question, we have to differentiate it with respect to θ

dA/dθ=2abcos2θ×2 (using dsin2θ / dθ=2cos2θ)

Equating dA / dθ=2abcos2θ×2=0, it means cos2θ=0 , which means 2θ =  π / 2 and θ=π / 4

So now we to put θ=π / 4 in the equation of area which is A=2absin2θ

We get as A=2absin2π / 4=2absinπ / 2=2ab

Hence the max area of the rectangle inside the ellipse is 2ab.

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We can plot in the coordinate plane if an ordered pair is known to us.

The space or plane in where location of points is represented by a pair of numerical terms with respect to some reference lines or reference direction is called coordinate plane.

let we have some pair of numerical terms such as A (4, 9) and B (6, 10)

that is needed to plot in the coordinate plane.

At first, we will plot the point A (4, 9)

we will start from the origin and move horizontally along the direction of X axis up to 4 unit. In the next, start again and move up vertically in the direction of y axis up to 9 unit. And we will draw a point at the final location.

By the same way, we can plot the point B.

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Answer:

The zeros are -2.5 and 2.

Step-by-step explanation:

2x^2 +x-10 = 0

2x^2 - 4x + 5x - 10 = 0

2x(x - 2) + 5(x - 2) = 0

(2x + 5)(x - 2) = 0

2x + 5 = 0 gives x = -2.5.

x - 2 = 0 gives x = 2.

Answer:

The KOT club has 5 pledges and can send between 2 and 4 pledges to work at Goodwill on Helping-Out Day. This can be done in 10 different ways

Step-by-step explanation:

The evaluation of the expression a + 4 when a = 7 is 11

From the question, we have the following parameters that can be used in our computation:

Expression: a + 4

Value a = 7

Substitute the known values in the above equation, so, we have the following representation

a + 4 = 7 + 4

Evaluate the like terms in the equation

So, we have the following representation

a + 4 = 11

Using the above as a guide, we have the following:

The solution is 11

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Answer:  The resulting expression is 8+9y+4, which simplifies to 12+9y.

Step-by-step explanation: To combine like terms, we add the coefficients of the terms that are the same. In this expression, the like terms are 2y and 7y. Their coefficients are 2 and 7, respectively, so when we add them we get 2+7=9. The resulting expression is 8+9y+4, which simplifies to 12+9y.

Answer:

B) 2 + 3x = 8

Step-by-step explanation:

Option B is not an expression as expressions do not have equal signs.

The Pythagorean Theorem, often known as Pythagoras Theorem, is a crucial concept in mathematics that describes how the sides of a right-angled triangle relate to one another. Pythagorean triples are another name for the sides of the right triangle. Here, examples help to demonstrate the formula and proof of this theorem.

In essence, the Pythagorean theorem is used to determine a triangle's angle and length of an unknown side. This theorem allows us to obtain the hypotenuse, perpendicular, and base formulae.

In order to determine sin a without using the Pythagorean theorem, we will need to have the value of the opposite side and angle. Sin (a) = opposite / hypotenuseHere, the hypotenuse is not given, so we will need to have the value of the opposite side and angle (a) in order to determine sin a without using the Pythagorean theorem.

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Answer:

Step-by-step explanation:

the formula for the slope is y2 - y1 / x2 - x1

so 1 - 0 / -3 --3/7

on the calculator

- 7 / 18

The total pay when he sells 24 copies of Math exists Fun should be 3760.

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. A mathematical equation is a statement with two equal sides and an equal sign in between.

Numerical constants, symbolic names, mathematical operators, functions, and conditional expressions can all be found in expressions in some combination.

Calculation of the equation and the total pay:

His base salary is $2400, and he makes an additional $110 for every copy of Math is Fun he sells.

Here P means the total pay

So, here the equation be p = 2400 + 110 n

substitute the values in the above equation, we get

Now the total pay be

p = 2400 + 110(24)

simplifying the above equation, we get

= 2,400 + 2640

= 5040

The equation that should be related P to N should be considered as the 2400 + 110n.

Therefore, the total pay when he sells 24 copies of Math is Fun should be 5040.

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The total pay when he sells 24 copies of Math exists Fun should be 3760.

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. A mathematical equation is a statement with two equal sides and an equal sign in between.

Numerical constants, symbolic names, mathematical operators, functions, and conditional expressions can all be found in expressions in some combination.

Calculation of the equation and the total pay:

His base salary is $2400, and he makes an additional $110 for every copy of Math is Fun he sells.

Here P means the total pay

So, here the equation be p = 2400 + 110 n

substitute the values in the above equation, we get

Now the total pay be

p = 2400 + 110(24)

simplifying the above equation, we get

= 2,400 + 2640

= 5040

The equation that should be related P to N should be considered as the 2400 + 110n.

Therefore, the total pay when he sells 24 copies of Math is Fun should be 5040.

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On solving the provided question, we can say that since it it is right angled triangle so the relation that be formed is  [tex]m^2 = p^2 + z^2[/tex]

A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.

since it it is right angled triangle

so the relation that be formed is

[tex]m^2 = p^2 + z^2[/tex]

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15 degrees by eye i think                                                        

The height of the tree to the nearest hundredth of a meter is 10.5 meters.

The length of Jai's shadow = length of tree's shadow - distance of standing of Jai

Length of Jai's shadow = 39.55 - 34.1

Performing subtraction on Right Hand Side of the equation

Length of Jai's shadow = 5.45 meters

Tan theta = height of tree/length of shadow = height of Jai/length of his shadow

height of tree/39.55 = 1.45/5.45

Keep the values in formula

Height of tree = 39.55×1.45/5.45

Performing multiplication

Height of tree = 57.35/5.45

Performing division

Height of tree = 10.5 meters

Thus, the height of tree is 10.5 meters.

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The domain and range of the function on the graph are:

Remember that for any function y = f(x), we define the domain as the set of possible inputs (possible values of x) and the range is the set of the possible outputs of the function (possible values of y).

To find the domain we need to look at the horizontal axis. The minimum is at x = -2 with a colored circle, so x = -2 belongs to the domain.

The largest value is at x = 1, now with an open circle, then this value does not belong to the domain. Thus the domain is:

D: [-2, 1)

For the range we need to look at the vertical axis, the minimum is at zero and the maximum at 4, then the range is:

R: [0, 4]

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

Here's the solution ~

The two triangles in the shown figure are similar, therefore conclusion can be made that :

Ratio of its it's corresponding sides is equal.

[tex]\qquad \sf  \dashrightarrow \: \dfrac{2x - 4 + 6}{6} = \dfrac{24}{4} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{2x + 2}{6} = 6[/tex]

[tex]\qquad \sf  \dashrightarrow \: 2x + 2 = 6 \times 6[/tex]

[tex]\qquad \sf  \dashrightarrow \: 2x + 2 = 36[/tex]

[tex]\qquad \sf  \dashrightarrow \: 2x = 36 - 2[/tex]

[tex]\qquad \sf  \dashrightarrow \: 2x = 34[/tex]

[tex]\qquad \sf  \dashrightarrow \: x = 17[/tex]

Therefore, The value of x is " 17 "

Now, the measure of side BC is :

[tex]\qquad \sf  \dashrightarrow \: 2x - 4[/tex]

[tex]\qquad \sf  \dashrightarrow \:2 (17) - 4[/tex]

[tex]\qquad \sf  \dashrightarrow \: 34 - 4[/tex]

[tex]\qquad \sf  \dashrightarrow \: 30 \: \: units[/tex]

So, its perimeter will be :

[tex]\qquad \sf  \dashrightarrow \: p = AB + AB + BC [/tex]

[tex]\qquad \sf  \dashrightarrow \: p = 24 + 26 + 30[/tex]

[tex]\qquad \sf  \dashrightarrow \: p = 80 \: \: units[/tex]

Answer:

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

According to Triangle Proportionality Theorem we have the following equal proportions:

The missing side is:

The perimeter is:

Answer:

Step-by-step explanation:

We have,

x = 3y - 3 ---> x - 3y = -3 ------(1)

5x + 4y = 23 -----(2)

Multiplying eq. (1) by 5:

5x - 15y = -15 ------(3)

5x + 4y = 23

Subtracting (1) from (3):

-19y = - 38

y = 2

therefore, x - 6 = -3

x = 3

Example of perfect square trinomial is: x² + 6x + 9 it is expressed as

(x + 3)²

Trinomials are algebraic formulas that include three dissimilar terms, therefore the name "trinomial."

An algebraic expression called a trinomial contains three terms that are non-zero. Trinomial expression illustrations: The trinomial x + y + z has three variables: x, y, and z.

A trinomial is formed when three monomials are added together or subtracted from one another. A quadratic trinomial has the following form: ax² + bx + c, where a, b, and c are real, non-zero values.

Example:

  x² + 6x + 9

= x² + 3x + 3x + 9

= (x +3) (x + 3)

= (x + 3)²

so, this is a perfect square trinomial.

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A solid right prism ABCDEF  has a height of 16.  A part of the prism is sliced off with a straight cut through points  X, Y, and Z. CXYZ is a  triangular pyramid and its surface area is 48 + 9√3 + 3√91  or 92.2

The situation can be depicted in the attached picture. CXYZ is a triangular pyramid.

The surface area of CXYZ is equal to:

area CXYZ = area ΔCXY +  area ΔCYZ +  area ΔCXZ + area ΔXYZ

ΔABC and ΔCXY are congruent, hence ΔCXY is a equilateral triangles with its sides are 1/2 those of ΔABC.

CX = CY = XY = 1/2 x 12 = 6

To find the area of ΔCXY, find its height first.

√3/2 = h/6

h = 3√3

Area ΔCXY = 1/2 x 3√3 x 6 = 9√3

ΔCXZ is a right triangle at C. Hence,

Area ΔCXZ =  1/2 x CX x CZ

                   = 1/2 x 6 x 8 = 24

ΔCYZ = ΔCXZ

Hence,

Area ΔCYZ =  24

To find the area of ΔXYZ, find XZ first using the Pythagorean Theorem.

XZ² = CX² + XZ²   (XZ = 8, since Z is midpoint of CD)

XZ² =  6² + 8²

XZ² = 10²

XZ = 10

Find the height of ΔXYZ

h = √(10² - 3² )

h = √91

Therefore,

Area ΔXYZ = 1/2 x XY x h

                   = 1/2 x 6 x √91 =3√91

Finally,

area CXYZ = area ΔCXY +  area ΔCYZ +  area ΔCXZ + area ΔXYZ

                  = 9√3 + 24 + 24 + 3√91

                  = 48 + 9√3 + 3√91

                  = 92.2

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The similar photo should be 4.5 inches wide.

Direct variation of a quantity {y} with respect quantity {x} means that the value of {y} increases as the the value of {x} increases. Mathematically -

y = Kx

K = y/x

where -

{K} is scale factor. It also tells by how many times is {y} greater than {x}.

Given is that Gwen scans a photo that is 4 inches wide by 6 inches long into her computer. Now, she scales the length down to 3 inches.

We can write -

y₁/x₁ = y₂/x₂

6/4 = 3/x₂

x₂ = 6 x 3/4

x₂ = 6 x 0.75

x₂ = 4.5

Therefore, the similar photo should be 4.5 inches wide.

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Santiago can make 7 dumplings

To find out how many dumplings Santiago can make, we need to divide the total amount of dough (5 11/16 cups) by the amount of dough used per dumpling (13/16 cups).

To do this, we can first convert the fractions to have a common denominator. In this case, the denominator of 13/16 is 16, so we can convert 5 11/16 to have a denominator of 16 as well:

5 11/16 = (5 * 16) + 11/16 = 80 + 11/16

Now we can divide:

(80 + 11/16) ÷ (13/16) = (80 + 11) ÷ 13 = 91 ÷ 13

So Santiago can make 91/13 dumplings. This can be simplified further to 7 dumplings. So Santiago can make 7 dumplings with 5 11/16 cups of dough.

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The value of exponent x, if [tex]$\sqrt[\text X]{81} = 3[/tex], is 4, in the given equation.

Exponent is the process of expressing large numbers as powers, according to the definition. Accordingly, the exponent describes the number of times a number has been multiplied by itself. Using 6 as an example, multiply it by itself four times, or 6×6×6×6. 6⁴ can be used to represent this. In this case, the base is 6 and the exponent is 4. You can interpret this as 6 raised to the power of 4.

Exponents are represented by the symbol. The word "carrot" refers to this symbol (). 4 raised to a power of two, for instance, can be expressed as 4^2 or 4². Thus, 4^2 = 4 × 4 = 16. A few exponent-based numerical expressions are represented in the table below.

The value of x, if [tex]$\sqrt[\text X]{81} = 3[/tex]

3 × 3 = 9

9 × 3 = 27

27 × 3 = 81

Thus, The value of x, if [tex]$\sqrt[\text X]{81} = 3[/tex], is 4.

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The coefficient of XY in 7xy is 7

Any integer or symbol that multiplies the variable of a single term or the terms of a polynomial to represent a constant value is known as a coefficient in mathematics. In some phrases, a letter might be substituted for the usual number. For example, x is the variable and a and b are the coefficients in the formula: ax2 + bx + c.

What is a coefficient?

A coefficient is a quantity or number that is coupled with a variable. Frequently, an integer is multiplied by the variable and written next to it. The variables that don't have a corresponding number are presumed to have a coefficient of 1.

Seven is the coefficient of XY in 7xy.

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Therefore , As a result, y=Sin will be the equation of the sinusoid that answers the supplied equation problem (2x).

An equation is a representation of two equal variables in mathematics, one on each side of a "equals" sign. Common difficulties can be solved using equations. We commonly seek pre algebra help to resolve problems in real life. Pre-algebra lessons address the fundamental concepts of mathematics.

Here,

Given: Maximum: 210

Minimum: 10

Time : 90

In order to create a sinusoid equation:

When x=0, sin(2*0)=sin(0)=0.

If x = 4 then sin(4 / 2) = 1.

Sin(π) = 0 if x =π/2 and

Consequently, the sinusoid's equation will be y=Sin (2x)

Therefore , As a result, y=Sin will be the equation of the sinusoid that answers the supplied equation problem (2x).

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We get the number of bananas as 8 and the number of mangoes as 4.

The general equation of a straight line is -

y = mx + c

{m} - slope.

{c} - intercept along the y - axis.

Given is that Chee and her children went into a grocery store and she bought $12 worth of bananas and mangos. Each banana costs $0.50 and each mango costs $2. She bought twice as many bananas as mangos.

We can write the system of equations as -

x = 2y               .....  (1)

(1/2)x + 2y = 12   ..... (2)

We can write equation (2) as -

(1/2) x 2y + 2y = 12

3y = 12

y = 4

then -

x = 8

Therefore, we get the number of bananas as 8 and the number of mangoes as 4.

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The number of adults at the public pool that day is 284 and the number of children is 147.

Linear equations are equation involving one or more expressions including variables and constants and the variables are having no exponents or the exponent of the variable is 1.

Linear equations may include one or more variables.

Let x be the number of adults and y be the number of children who used the swimming pool.

Total number of people used the swimming pool = 431

x + y = 431

Daily price for adults = $2.00

Daily price for children = $1.50

Total amount for admission = $788.50

2x + 1.5y = 788.5

We got two linear equations here :

x + y = 431 and 2x + 1.5y = 788.5

x + y = 431 ⇒ x = 431 - y

Substituting this in second equation,

2(431 - y) + 1.5y = 788.5

862 - 2y + 1.5y = 788.5

-0.5y = -73.5

y = 147

x = 431 - 147 = 284

Hence the number of adults is 284 and the number of children is 147.

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Answer:x=5

Step-by-step explanation:

f(x) = g(x)

[tex]\frac{1}{2}(3x-1)=7\\\\ \frac{3}{2}x-\frac{1}{2} =7\\\frac{3}{2}x-\frac{1}{2}+\frac{1}{2} =7+\frac{1}{2} \\\\\frac{3}{2}x=\frac{15}{2} \\\frac{3}{2}x*\frac{2}{3} =\frac{15}{2}*\frac{2}{3} \\\\x=5[/tex]

Answer:

x = 4, y = 12

Step-by-step explanation:

xy = -16

y = x + 8

substitute y = x + 8 into xy = -16

x(x + 8) = -16

x^2 +8x + 16 = 0

x = 4

sub x = 4 into y = x + 8

y = 4 + 8

y = 12

If you double the amount of fluid ounces, the amount of cups also double

There is a proportional relationship between the number of fluid ounces and the number of cups. So, the number of cups would change by the same factor as the number of fluid ounces.

The donation is proportional to the dinner bill

The donation for a dinner bill of $50 is $9

If there are 8 fluid ounces in one cup

8 : 1 = 16 : x

8/1 = 16/x

8 × x = 16 × 1

8x = 16

x = 16/8

x = 2

The difference between weight = 1

The difference between cost = $0.68 - $0.47

= $0.21

$0.89 - $0.68

= $0.21

$1.10 - $0.89

= $0.21

Cost of mailing 6-ounce letter

1 : 0.47 = 6 : x

1/0.47 = 6/x

1 × x = 6 × 0.47

x = $2.82

The donation for a dinner bill $50 is

Dinner bill : donation

25 : 4.50 = 50 : x

25/4.50 = 50/x

25 × x = 50 × 4.50

25x = 225

x = 225/25

x =$ 9

Therefore, there is a proportional relationship between two quantities if there is an equal change in the values of both quantities.

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The length of each piece of wood needed to build each brace of the bookcase will be 4.24 feet.

Based on the dimensions, the shape of bookcase is square. So the diagonal of the bookcase will be calculated from the formula -

Diagonal = ✓side² + side²

Keep the values in formula to find the value of diagonal

Diagonal = ✓ 3² + 3²

Taking square on Right Hand Side of the equation

Diagonal = ✓9 + 9

Performing addition on Right Hand Side of the equation

Diagonal = ✓ 18

Finding square root on Right Hand Side of the equation

Diagonal = 4.24 feet

Thus, each diagonal will be 4.24 feet.

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