All 210 students in a school went roller skating, to a water park, or both. The number of students who went only to the water park is four times the number of students who went only roller skating. The number of students who went only to the water park is two times the number of students who did both. How many students did both?

On solving the provided question we can say that the rectangular form of the equation is y = -1

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.

Ф = [tex]\pi[/tex]/2

y = - 1

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The intervals that are equivalent to each inequality are given as follows:

The interval notation of an inequality of lower bound a and upper bound b is given as follows:

(/[a,b]/).

The closing notation can be in either parenthesis or brackets, as follows:

For the first interval, 2 < x ≤ 5, it is an open interval at the lower bound of 2 and closed at the upper bound of 5, hence the notation is given as follows:

(2,5].

For the second interval, x > 7, we have that it has only a lower bound of 7, which is open. Hence, the upper bound is of infinity, and an infinite bound is always open, hence the notation is given as follows:

(7, ∞).

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The point(s) on the surface at which the tangent plane is horizontal are (0, 0, 1).

To find the point(s) on the surface at which the tangent plane is horizontal, we need to find the points at which the partial derivatives of z with respect to x and y are both equal to 0. This means we need to solve the system of equations:

Solving this system of equations yields x = 0, y = 0, and z = 1. Therefore, the point(s) on the surface at which the tangent plane is horizontal is (0, 0, 1).

To understand why solving the system of equations yields the points for which the tangent plane is horizontal, it is important to understand what the partial derivatives represent. The partial derivative of z with respect to x is the rate of change of z with respect to x, and similarly for y.

When the partial derivatives are both equal to 0, it means that the rate of change of z with respect to x and y is 0, which implies that the tangent plane is horizontal.

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Answer: 12

Step-by-step explanation: 3/4 divid by 3

Alex is standing somewhere in between Gina and Dana. Thus, the least possible number of people between Dana and Alex are:

20.

In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one.

A number line is useful because it acts as a visual math aid. It can support teachers and parents as they teach children how to count and write numbers. It's also a tool that can help build understanding behind adding, subtracting, multiplying, and dividing numbers.

Let's assume that Gina is standing on first number while Dana on the last. Hence the total number of people in the line are: n= 23.Alex is standing somewhere in between Gina and Dana. Thus, the least possible number of people between Dana and Alex are: (n-2) = 20.

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50-50-80 is a Isosceles acute triangle.

An isosceles acute triangle is a triangle in which all three angles are less than 90°, and at least two of its angles are equal in measurement. One example of the angles of an isosceles acute triangle is 50°, 50°, and 80°.

Properties of Isosceles Acute Triangle

It is easy to point to an isosceles acute triangle if we know its properties. The properties of the isosceles acute triangle are noted below:

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The quadrilateral's image points will be,A(3, 3) → A'(-3, 3) R(5, 7) → R'(-7, 5) M(7, 5) → M'(-5, 7) Y(5, 1) → Y'(-1, 5).

If a point (x, y) is rotated 180 degrees anticlockwise with respect to the origin, the rotational rule is:

(x, y) → (-y, x) (-y, x)

Consequently, the image point's coordinates following rotation will be (-y, x).

Using this guideline,

The quadrilateral's image points will be,

A(3, 3) → A'(-3, 3)

R(5, 7) → R'(-7, 5)

M(7, 5) → M'(-5, 7)

Y(5, 1) → Y'(-1, 5)

By graphing the image points we can get the graph of the image quadrilateral A'R'M'Y'.

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The question is incomplete, complete question:

Quadrilateral ARMY is rotated 180 degrees about the origin. Draw the image of this rotation.

The vertex of V' had a preimage of V(0, 3) as V' has a dilation of 3.

Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.

Dilation: The preimage is scaled up or down to create the image.

Reflection: The picture is a preimage that has been reversed.

Rotation: Around a given point, the preimage is rotated to create the final image.

Translation: The image is translated and moved a fixed amount from the preimage.

Given, On a coordinate plane, polygon S'T'U'V' has points (- 2, 4), (- 4, 1),

(- 2, - 2), and (0, 1) with a dilation rule  DO,1/3 (x, y) to [(1/3)x, (1/3)y].

Therefore, The coordinates of the vertex V had a preimage of,

3(V'), which is 3(0, 1) = (0, 3).

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Answer:

Mason can make 20 sandwiches.

Step-by-step explanation:

Therefore , the solution of the given question of triangle comes out to be y=2.

You can link any three points in a plane to form triangles, which have three sides and three angles. They were among of the first geometric forms to be examined. Since triangles may be divided into any polygon, regardless of whether it has four, five, six, or n sides, they are very significant.

Here,

Since it is an isosceles trapezoid the length of the two legs (i.e.  AB =  4y - 6 and CD = 3y - 4) are equal.  In other words,

4y - 6  = 3y - 4.

Subtract 2y from each side:

4y - 3y - 6= -4

Now add 10 to each side

=> 4y - 3y - 6 + 4  

=> y - 2 =0

=> y = 2

Therefore , the solution of the given question of triangle comes out to be y=2.

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Projective tests are created so that the test taker cannot tell what they are testing for.

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For the described situation, y  = 6sin ( (π / 10) x )  -  10 is the sine function.

The trigonometric functions in mathematics are real functions that connect the right-angled triangle's angle to the ratios of its two side lengths.

They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.

Asin(Bx)+C is the usual sine function formula.

A buoy bobs 6 meters up and down.

That is, the amplitude of the buoy shifts by 6 meters both vertically and horizontally.

A=6 Amplitude

The x-axis distance that constitutes one complete oscillation is known as the period.

We are aware that it takes a buoy 10 seconds to go from its greatest position to its lowest point (half oscillation).

20 seconds make up the complete oscillation period.

⇒ 20 (B) = 2π

B=2π /20

= π/ 10

Given that the buoy is 10 meters under the ocean's surface, C equals –10.

Using the formula's whole range of values:

⇒y  = 6sin ( (π/ 10) x )  -  10

The midline is the initial point.

x=0 is the first point.

⇒y=6sin(0)-10

y= -10

The line is drawn from the origin at y = -10.

Either the highest or smallest value is at the second point.

Therefore, (5,-4) and (15, -16.)

Therefore, for the described situation, y  = 6sin ( (π / 10) x )  -  10 is the sine function.

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Complete question:
At an ocean depth of 10 meters, a buoy bobs up and then down 6 meters from the ocean's depth. Ten seconds pass from the time the buoy is at its highest point to when it is at its lowest point. Assume at x = 0 the buoy is at normal ocean depth. Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

Line segment theorem states that if a line segment is bisected by another line, then the two parts of the line segment are equal.

This theorem is also known as the Midpoint Theorem. The formula for the Line Segment Theorem is A/B = C/D, where A and C are the two halves of the line segment, and B and D are the distances from the midpoint to the two endpoints of the line segment.

For example, consider a line segment AB with endpoints at A(-3, 2) and B(3, 2). The midpoint of the line segment is at (0,2). To calculate the length of each half of the line segment, we would use the distance formula and plug in the coordinates of the two endpoints and the midpoint.

For half A, the distance is d(A, M) = sqrt[(-3 - 0)^2 + (2 -2)^2] = 3.

For half C, the distance is d(M, B) = sqrt[(3 - 0)^2 + (2 -2)^2] = 3.

Therefore, A/B = 3/3 = 1 and C/D = 3/3 = 1. This confirms that the two halves of the line segment are equal in length.

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B. The function shown on the graph has a smaller rate of change, but a higher starting point.

Rate of change = slope of a linear graph.

Given the linear function, [tex]y=4x+2[/tex] , the rate of change here is represented by the slope of the line, which is 4.

Let's find the rate change of the function represented on the graph using slope formula:

Rate of change = [tex]\frac{y2-y1}{x2-x1}[/tex]

Use coordinates of any two points on the line. Let's take,

[tex](0,4)= (x1,y1)\\(1,7)=(x2,y2)[/tex]

Rate of change = [tex]\frac{7-4}{1-0} = 3[/tex]

Rate of change of the linear function of the graph = 3.

Thus the function on the graph has a smaller rate of change, than the function, y = 4x + 2.

The starting value of a function is the y-intercept, that is, the value of y, when x = 0. It is the point at which the line intercepts the y-axis.

The starting value of the linear function, y = 4x + 2 is 2.

The starting value of the function on the graph is 4. At this point, x = 0.

Thus, the function on the graph has a higher starting point.

The statement that correctly compares the function shown on this graph with the function y= 4x+2 is:

B. The function shown on the graph has a smaller rate of change, but a higher starting point.

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The probabilities are given as follows:

a) No injuries: 0.4493 = 44.93%.

b) More than two injuries: 0.0474 = 4.74%.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following equation:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

The parameters are:

The mean for this problem is of:

[tex]\mu = 0.8[/tex]

The probability of no injuries is of P(X = 0), hence:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-0.8}(0.8)^{0}}{(0)!} = 0.4493[/tex]

The probability of more than two injuries is of:

P(X > 2) = 1 - P(X <= 2)

In which:

P(X <= 2) = P(X = 0) + P(X = 1) + P(X = 2)

Hence:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-0.8}(0.8)^{0}}{(0)!} = 0.4493[/tex]

[tex]P(X = 1) = \frac{e^{-0.8}(0.8)^{1}}{(1)!} = 0.3595[/tex]

[tex]P(X = 2) = \frac{e^{-0.8}(0.8)^{2}}{(2)!} = 0.1438[/tex]

Then:

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If a  box of 6 donuts has 3 plain donuts, 2 chocolate donuts, and 1 blueberry donut. The number of  visually distinct arrangements of the donuts that are there is: 60.

Using this formula to find the  number of  visually distinct arrangements of the donut that are there.

Number  visually distinct arrangements = Box of  donuts!/ Plain donuts! × Chocolate donuts! × Blueberry donut!

Let plug in the formula

Number  visually distinct arrangements = 6!/ 3! × 2! × 1!

Number  visually distinct arrangements = 720/ 6 ×2 × 1

Number  visually distinct arrangements = 720 /12

Number  visually distinct arrangements = 60

Therefore the number of Number  visually distinct arrangements  is 12.

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Scrooge sees out of his window when Marley's ghost leaves as all the spirits of people whom Scrooge has not met before in his life. Thus, option B is correct.

Although ghosts are typically thought of as lone, human-like essences, and there have been reports of ghostly battalions, ghosts that resemble animals instead than people.

Here, we have,

Scrooge discovers "the air filled with phantoms" outside s windowsill after Marley's Apparition hath left it. Some of these imprisoned spirits were people Scrooge had indeed known. It resembles an incredible image of the metropolis that Scrooge has already been familiar with. Scrooge is compelled to agree.

Scrooge responds with the ghost's request to peek out just the balcony. He observes a mass of ghosts, each one imprisoned in bonds. As both they and Marley vanish into darkness, they sob at their failure to live noble, compassionate lives and their incapacity to help other people in need.

Therefore, option B is the correct option.

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The real and complex solutions of the polynomial equation are 2,-3i and 3i

Polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables

The given polynomial is x³-2x²+9x-18

Factor out x² from  x³-2x² which becomes  x²(x-2)

Factor out 9 from 9x-18 which becomes 9(x-2)

x³-2x²+9x-18

x²(x-2)+9(x-2)

Factor out common term x-2

( x²+9)(x-2)=0

x²+9=0

x=-3i and 3i

x-2=0

x=2

Hence, the real and complex solutions of the polynomial equation are 2,-3i and 3i

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Answer:

Step-by-step explanation:

(100xInterest)(RatexTime)

hope this helps

Step-by-step explanation:

The formula for calculating Principal amount would be P = 100I / (RT) where Interest is Interest Amount, R is Rate of Interest and T is Time Period.

The steps for measuring the angle b is shown below, <b= 41.

If a triangle side is created, the outside angle that results is equal to the sum  of the two opposite internal angles.

If an equivalent angle is measured at each triangle vertex, the exterior angles always add up to 360 degrees.

Given:

exterior angle = 113

and, opposite interior angle =72

Using Exterior angle property we have

72 + b = 113

Subtract 72 from both side

72 + b -72 = 113 -72

b = 113- 72

b = 41

Hence, the measure of b is 41 degree.

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Answer:

6[tex]x^{2}[/tex] + 1

Step-by-step explanation:

[tex]\frac{6x^{3}+ 2x-x }{x}[/tex]   combine 2x -x = x

[tex]\frac{6x^{3} + x}{x}[/tex]

[tex]\frac{x(6x^{2} + 1) }{x}[/tex]  the x's cancel out

6[tex]x^{2}[/tex] + 1

34.7 is the monthly phone use will Plan A cost no more than Plan B and 2.1 is the value of m in the inequality.

a relationship between two expressions or values that are not equal to each other is called 'inequality.

In Plan A, the customer pays a monthly fee of $20 and then an additional 7 cents per minute of use

20+7m

In Plan B, the customer pays a monthly fee of $26.30 and then an additional 4 cents per minute of use

26.3+4m

We need to find the amounts of monthly phone use will Plan A cost no more than Plan B

20+7m<26.3+4m

7m-4m<26.3-20

3m<6.3

m<2.1

Hence, 34.7 is the monthly phone use will Plan A cost no more than Plan B and 2.1 is the value of m in the inequality.

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The graph of the inequality 2x+3y>6 is a half-plane the neither contains origin nor the points of the the line 2x+3y=6.

A linear inequality is a mathematical statement involving a linear function and one of the symbols <, >, ≤, or ≥. It represents a range of values for the variables that satisfy the inequality. The solution to a linear inequality is the set of all points that make the inequality true. These points form a region in the coordinate plane, which is known as the solution set. The solution set can be represented graphically as a half-plane, a line, or a combination of lines and half-planes, depending on the inequality.

The graph of the inequality 2x+3y>6 is the set of all points (x,y) that satisfies the inequality 2x+3y>6. It can be represented by a line 2x+3y=6 and all the points on one side of the line. This side is determined by the inequality sign ( > in this case).The side is away from the origin and the line 2x+3y=6 is the boundary line, and all the points on one side of the line satisfy the inequality while all the points on the other side do not.

The line 2x+3y=6's origin and its points are not present on the half-plane that represents the inequality 2x+3y>6.

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[tex]16r^2s^2[/tex] as the product of power is written as [tex](4rs)^2[/tex]

The power of a product rule states that [tex](a^m)(a^n) = a^(m+n)[/tex], where "a" is the base, "m" is the exponent of the first term, and "n" is the exponent of the second term. This rule allows us to simplify expressions where multiple variables are being raised to a power and multiplied together by adding the exponents.

For example, [tex](a^2)(a^3) = a^(2+3) = a^5[/tex].

It is also important to remember that when we are using the power of a product rule, the bases must be the same. If we have different bases, we cannot use this rule.

For example, [tex](a^2)(b^3) = a^2 * b^3, not a^(2+3) = a^5[/tex].

Rewritten as power of product:

[tex]16r^2 s^2= 4*4*r*r*s*s\\16r^2s^2= 4^2*r^2*s^2\\16r^2s^2=(4rs)^2[/tex]

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Answer:

Step-by-step explanation:

a.50%=[tex]\frac{1}{2}[/tex]

b.so 50% of 188 means 188·[tex]\frac{1}{2}[/tex]=94

answer is 94

The number of  pastries will be offered in one year could be 26 and the algebraic expression could be  x = 2 + 2y

What is Arithmetic progression ?

Arithmetic progression could be defined as the sequence of number where the difference between two consecutive number is same.

Given,

Mamas bakery just opened and is currently selling only two types of pastry.

Each month mamas bakery will add two more types of pastry to their menu.

let the number of pastries be x.

let y be the given number of pastries.

Hence the algebraic expression could be

x = 2 + 2y

Number of pastries will be offered in one year could be

x = 2 + 2 * 12

x = 26

Hence, the number of  pastries will be offered in one year could be 26 .

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Answer:

u just do a variable minus 7

s-7=

Answer: x=(y+9)5−32 simplified

Step-by-step explanation:

Answer:

a) 4.2 - 2.8 = 1.4

b) 72.1 - 45.6 = 26.5

c) 82.2 - 41.3 - 29.2 = 11.7

d) 17.1 - 11.7 - 0.8 = 4.6

e) 29.2 - 12.5 - 0.9 = 15.8

f) 21 - 14 = 7

g) 71 - 37 - 18 = 16

h) 50 - 19 - 8 = 23

i) 92 - 57 - 16 = 19

j) 44 - 18 - 9 = 17