In a triangle ABC with an orthocenter O, prove that angle BOC plus angle BAC equal to 180°​.
Prove: <BOC + <BAC = 180°​

We have the appropriate response to the inquiry b) The polygon's area is 99 inch2.

A polygon is a two-dimensional, closed shape that is flat or plane and is bounded by straight sides. Its sides are not curled. The edges of a polygon are another term for its sides. The vertices (or angles) of a polygon are the places where two sides converge.

According to the diagram we have the following:

DP = 10 ₋ 4 ₋ 6inch EP = 12 ₋ 5 = 7inch

Area of rectangle = l x b

Area of Triangle = [tex]\frac{1}{2}[/tex] x base x height

Area of polygon Area MBCD + Area AMPE + Area EPD = (5x10) + (4x27) + ([tex]\frac{1}{2}[/tex]x7x6) .

= 50 + 28 + 21 = 99 inch²

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The total cost of either plan is, 42

The number of DVD's rented is, 31

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

For example, 3x+2y=0.

Types of equation

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

Plan 1- membership fee of $11 and charges $1 for every DVD

Plan 2- $42 per month for unlimited DVD rentals

The two plans cost the same amount

The total cost of either plan = ?

The number of DVD's rented = ?

Suppose, the number of DVD's is n

So, cost in plan 1,

=  membership fee  +  n x (charges $1 for every DVD)

=   11 + n x 1

=   11 + n

And, cost in plan 2,

=  42

According to given condition,

cost is equal in both cases,

11 + n = 42

n = 31

Hence, the total cost and number of DVD's are respectively,  45 & 31

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The surface area of the cone is 113.09 square units.

A three-dimensional object's surface area is the sum of all of its faces. Real-world applications of the concept of surface areas include wrapping, painting, and eventually building things to achieve the best possible design.

Given that the height of the cone is 3 units and the radius of the base is 4 units.

The surface area of the cone is calculated by the formula:-

SA = πrl + πr²

Slant height will be calculated as:-

l² = h² + r²

l² = 3 ² + 4²

l² = 25

l = 5 units

The surface area will be:-

SA = πrl + πr²

SA = ( πx 4 x 5 ) + ( π x 4² )

SA = 113.09 square units

Therefore, the surface area will be 113.09 square units.

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Answer:

60 + 5x = 125 - 10x

Step-by-step explanation:

Jeter- 60 + 5x

Callie- 125 - 10x

Set them equal to each other to find out when they will equal the same amount

Answer:

The answer can be expressed in multiple forms

[tex]4.\overline3\overline3[/tex] weeks

[tex]4\frac{1}{3}[/tex] weeks

Step-by-step explanation:

Lets write down what we are given

Jeter Initial Amount = $60.00

Callie Initial Amount = $125.00

Deposit/Withdrawal Per Week

Jeter + $5.00

Callie - $10.00

We can set up an expression for Jeter's money

[tex]60+5w[/tex]

We can set up an expression for Callie's money

[tex]125-10w[/tex]

[tex]w[/tex] represents the number of weeks

We can set each expression equal to each other to find out when they will have the same amount of money.

[tex]60+6w=125-10w[/tex]

Lets solve for [tex]w[/tex].

Add [tex]10w[/tex] to both sides of the equation.

[tex]60+5w+10w=125-10w+10w[/tex]

Combine like terms and simplify.

[tex]60+15w=125[/tex]

Subtract 60 from both sides of the equation.

[tex]60+15w-60=125-60[/tex]

Evaluate the subtraction and simplify..

[tex]15w=65[/tex]

Divide both sides of the equation by 15.

[tex]\frac{15w}{15}=\frac{65}{15}[/tex]

Simplify the left side by cancelling the common factor of 15.

[tex]w=\frac{65}{15}[/tex]

[tex]w=\frac{13}{3}[/tex]

[tex]w=4.3333[/tex]

[tex]w=4\frac{1}{3}[/tex]

The sum of 15 and 60 using the greatest common factor and the distributive property is 15(1+4).

Greatest common factor, as the name implies, is greatest among all common factors among the quantities given.

We can factorize the quantities in smallest relative factors possible (one level of factors over which all quantities can form themselves), and collect as many common factors as we could. Those will together compose GCF(Greatest common factor).

Given;

(15+60)

The common factor of 15 and 60 = 15

Taking 15 common;

=15(1+4)

Therefore, the answer by taking factor will be 15(1+4).

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The probability of the event is calculated by taking the probability of the event occurring 0.53, or about 0.03125.

A coin being flipped five times is an example of a Bernoulli trial, which is a type of experiment where the outcome can be either success or failure. The probability of success (a heads) is equal to the probability of failure (a tails). So, for this experiment, the probability of a heads is 0.5 and the probability of a tails is 0.5.

The event of a coin being flipped five times can be expressed as a set in roster notation as {HHHHT, HHHTH, HHTHH, HTHHH, THHHH, HHHTT, HHTTH, HTHHT, THHHT, HTTHH, TTHHH, HHTTT, HTTHT, THHTT, TTHHT, TTTHH, HTTTT, THTTT, TTHTT, TTTHT, TTTTH, TTTTT}.

The probability of the event is calculated by taking the probability of the event occurring (in this case, 0.5 for a heads and 0.5 for a tails) and multiplying it by the number of trials (in this case, 5). This gives us a probability of 0.53, or about 0.03125.

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Answer:

[tex]\sf 1. \quad \dfrac{3}{8}[/tex]

[tex]\sf 2. \quad \dfrac{7}{24}[/tex]

Step-by-step explanation:

Spinner 1 is divided into 6 congruent sections.

There are 3 red sections, 2 blue sections and 1 yellow section.

Therefore, the probability of spinning each of the three colors is:

    [tex]\bullet \quad \sf P(R_1)=\dfrac{3}{6}=\dfrac{1}{2}[/tex]

    [tex]\bullet \quad \sf P(B_1)=\dfrac{2}{6}=\dfrac{1}{3}[/tex]

    [tex]\bullet \quad \sf P(Y_1)=\dfrac{1}{6}[/tex]

Spinner 2 is divided into 3 sections of differing sizes.

The red section is half of the spinner. The blue and yellow sections are quarters of the spinner.

Therefore, the probability of spinning each of the three colors is:

    [tex]\bullet \quad \sf P(R_2)=\dfrac{1}{2}[/tex]

    [tex]\bullet \quad \sf P(B_2)=\dfrac{1}{4}[/tex]

    [tex]\bullet \quad \sf P(Y_2)=\dfrac{1}{4}[/tex]

If each spinner is spun once, then:

The probability that both spinners both show red is:

[tex]\sf P(R_1)\;and\;P(R_2)=\dfrac{1}{2} \times \dfrac{1}{2}=\dfrac{1}{4}[/tex]

The probability that both spinners both show blue is:

[tex]\sf P(B_1)\;and\;P(B_2)=\dfrac{1}{3} \times \dfrac{1}{4}=\dfrac{1}{12}[/tex]

The probability that both spinners both show yellow is:

[tex]\sf P(Y_1)\;and\;P(Y_2)=\dfrac{1}{6} \times \dfrac{1}{4}=\dfrac{1}{24}[/tex]

Therefore, the probability that both spinners show the same colour is:

[tex]\sf P(R_1\;\&\;R_2)\;or\;P(B_1\;\&\;B_2)\;or\;P(Y_1\;\&\;Y_2)=\dfrac{1}{4}+\dfrac{1}{12}+\dfrac{1}{24}=\dfrac{9}{24}=\dfrac{3}{8}[/tex]

If each spinner is spun once, the probability of getting a red from spinner 1 and a blue from spinner 2 is:

[tex]\sf P(R_1)\;and\;P(B_2)=\dfrac{1}{2} \times \dfrac{1}{4}=\dfrac{1}{8}[/tex]

The probability of getting a blue from spinner 1 and a red from spinner 2 is:

[tex]\sf P(B_1)\;and\;P(R_2)=\dfrac{1}{3} \times \dfrac{1}{2}=\dfrac{1}{6}[/tex]

Therefore, the probability of getting a red-blue combination is:

[tex]\sf P(R_1\;\&\;B_2)\;or\;P(B_1\;\&\;R_2)=\dfrac{1}{8}+\dfrac{1}{6}=\dfrac{7}{24}[/tex]

Step-by-step explanation:

Here's how you can use partial fraction decomposition to write (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) as the sum of simpler terms:

Factor the denominator to simplify the expression: x^2 + x - 2 = (x - 1)(x + 2)

Write the partial fraction decomposition of the integrand as the sum of simpler terms: (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = A/(x - 1) + B/(x + 2) + Cx + D, where A, B, C, and D are constants.

Multiply both sides of the equation by (x^2 + x - 2) to find the values of A, B, C, and D: x^3 + 2x^2 + 3x + 4 = A(x + 2) + B(x - 1) + (Cx + D)(x^2 + x - 2)

Substitute x = 1 and x = -2 into the equation to find two equations for A, B, C, and D:

x = 1: 4 + 2 + 3 + 4 = 9 = A(-2) + B + (C + D)(-1)

x = -2: -8 - 4 + 6 - 8 = -16 = A(1) + B(-2) + (C - 2D)(4)

Solve the system of equations to find the values of A, B, C, and D:

A = 9/3, B = 5, C = -11/3, D = 7/3

So the partial fraction decomposition of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) is (9/3)/(x - 1) + (5)/(x + 2) - (11/3)x + (7/3).

The number of unique visitors to library in a month with following probabilities: P0X9)-04, P0X19)-07, P(0 X 29)-0.8, P(0≤x≤39)=0.9, P(0sX<49)-1 are;

a. P(X=50) = 0, since X can't be greater than 49 (according to the given information) and the probability of X being exactly 49 is 1.

b. P(10≤X≤19) = P(X=10) + P(X=11) + ... + P(X=19) = 0.07 - 0.04 = 0.03

c. P(20≤X≤29) = P(X=20) + P(X=21) + ... + P(X=29) = 0.8 - 0.07 = 0.73

d. P(X>39) = 1 - P(0≤X≤39) = 1 - 0.9 = 0.1

I determined the probabilities in the way I did by using the cumulative distribution function of a discrete random variable.

The cumulative distribution function (CDF) of a discrete random variable X is defined as:

CDF(x) = P(X≤x) = ∑P(X=x'), for all x' such that x' ≤ x

So, for each of the given ranges, I calculated the cumulative probability up to that range using the CDF definition and the individual probabilities of X being exactly each value within the range.

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The approximate volume of the astroid is 2.57 * 10^9 m^3

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

Radius = 8.5 × 10^2 m.

Using the formula 4/3 π r^3 for the volume, we get

Volume = 4/3 * (22/7) * (8.5 × 10^2)^3

Evaluate

Volume = 2.57 * 10^9 m^3

Hence, the volume is 2.57 * 10^9 m^3

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The required simplified value of the measure of the multiplication factor x in the circle is 2.25.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

here,
Arc VW = 12x + 25 °
∠VXW = 52°

According to the question,
Arc VW = ∠VXW
12x + 25 = 52
x = 52 - 25 / 12
x = 2.25

Thus, the required simplified value of the measure of the multiplication factor x in the circle is 2.25.

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Answer:

Step-by-step explanation:

The function h(x) = (f o g)(x) means that h is the composition of functions f and g, where g(x) is applied first and then the result is passed to f.

Without further information, it is impossible to determine the specific functions f(x) and g(x). The composition of functions is unique only when both f and g are specified.

The coordinates of image point is (15, 20)

A dilation is a transformation that changes the size of a figure but not its shape.

In a dilation with respect to the origin, a point (x, y) is transformed to a point (kx, ky) where k is a scale factor.

The transformation is given as

T<0, 1>D5

Mathematically, this

(x, y) = 5(x, y + 1)

So the image point is

(x, y) = 5(3, 3 + 1)

(x, y) = (15, 20)

So the image point of is (15, 20)

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The graph of function y = 1/2 (x - 3) - 10 is shown in figure.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The equation of function is,

⇒ y = 1/2 (x - 3) - 10

Now, The equation of function is,

⇒ y = 1/2 (x - 3) - 10

Clearly, This represent the equation of line.

Hence, By using tool;

The graph of function y = 1/2 (x - 3) - 10 is shown in figure.

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The equation that compares dollar amounts to be received or paid at different dates is the Time Value of Money (TVM) equation.

The time value of money(TVM) equation enables us to compare dollar amounts to be received or paid at various dates. This equation takes into account the concept of the opportunity cost of money and accounts for the changes in the value of money over time due to inflation and other factors. The equation calculates the present value of a future amount, or the future value of a present amount, based on an interest rate. This helps in making financial decisions by comparing the value of money at different points in time.

The equation used to compare dollar amounts received or paid at different time periods is called time value of money. This equation takes into account the effect of inflation, investment return, and the time period for which the money is being held, to calculate the present or future value of a sum of money. It helps to determine the equivalent value of money at a certain point in time, considering the opportunity cost of not having access to the money.

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The complete question is:

The formula that enables us to compare the amount of money that will be received or paid at various dates is the

Time value money.

Taylor rule.

interest-rate parity equation.

yield curve.

The distance between the two points S( -5, 7) to V(- 2, 4) is 3.5 units.

The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by,

[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex].

Given, We have to find the distance between the points S(- 5, 7) to V(-2, 4).

Therefore, [tex]|SV| =\sqrt{(- 2 + 5)^2 + (4 - 7)^2}[/tex].

[tex]|SV| =\sqrt{(3)^2 + (-3)^2}[/tex].

[tex]|SV| =\sqrt{9 + 9}[/tex].

|SV| = √18.

|SV| = 2√3.

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The given of the system of equation is has  No solution, parallel lines.

To give a geometric description of the given system of equations.

The geometric description of a system of equations in 2 variables mean the system of equations will represent the number of lines equal to the number of equations in the system given.

i.e.

Number of planes = Number of variables

Number of lines = Number of equations in the system.

Here, we are given 2 variables and 2 equation in each system.

So, they can be represented in the xy-coordinates plane.

And the number of solutions to the system depends on the following condition.

Let the system of equations be:

A1x+B1y+C1=0

A2x+B2Y+C2=0

1. One solution:

There will be one solution to the system of equations,  If we have:

[tex]\frac{A_1}{A_2} not\ equal to \ B_1/B_2[/tex]

2. Infinitely Many Solutions: (Identical lines in the system)

the ratio of A1 and A2 equal to ratio of B1 and B2 and equal toC1 and C2

3. No Solution:(Parallel lines)

the ratio of A1 and A2 equal to ratio of B1 and B2 but not equal ltoC1 and C2

Now, let us discuss the system of equations one by one:

a) A1=2, B1=-4 , C1=-12

A2=-3 , B2= 6 ,C2 = 15

Therefore, no solution i.e. parallel lines.

The complete question is-

Give a geometric description of the following system of equations.a. 2x−4y=12 −3x+6y=−15

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The vertical asymptote and horizontal asymptote of the rational function will be x = 3 and y = -2, respectively.

An asymptote is a line that constantly reaches a given curve but does not touch at an infinite distance.

The rational function is given below.

f(x) = - 2x / (x - 3)

The vertical asymptote is given as,

x - 3 = 0

x = 3

Then the horizontal asymptote is given as,

y = lim x → ∞ f(x)

y = lim x → ∞ - 2x / (x - 3)

y = lim x → ∞ - 2

y = - 2

The vertical asymptote and horizontal asymptote of the rational function will be x = 3 and y = -2, respectively.

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The approximations using the trapezoidal rule, midpoint rule, and Simpson's rule are 0.436904, 0.439881, and 0.439363, respectively.

(a) Trapezoidal Rule:

The trapezoidal rule approximates the definite integral as the sum of areas of trapezoids under the curve, connecting the function values at evenly spaced points within the interval of integration.

The formula for the trapezoidal rule is given by:

∫_a^b f(x) dx ≈ (b - a) * [f(a) + f(b)] / 2

For n = 4 subintervals, we can find the height of each trapezoid by evaluating the function at 0, 1/4, 1/2, 3/4, and 1.

∫_0^1 e^(−6x^2)dx ≈ (1 - 0) * [e^(-6 * 0^2) + e^(-6 * 1^2)] / 2 = (1 - 0) * [1 + e^(-6)] / 2 ≈ 0.436904

(b) Midpoint Rule:

The midpoint rule approximates the definite integral as the sum of areas of rectangles under the curve, using the function value at the midpoint of each subinterval as the height of each rectangle.

The formula for the midpoint rule is given by:

∫_a^b f(x) dx ≈ (b - a) * ∑ f(a + (b - a) * i / n) / n, where i = 1, 2, ..., n

For n = 4 subintervals, we can find the midpoints by evaluating the function at 1/8, 3/8, 5/8, and 7/8.

∫_0^1 e^(−6x^2)dx ≈ (1 - 0) * [e^(-6 * 1/8^2) + e^(-6 * 3/8^2) + e^(-6 * 5/8^2) + e^(-6 * 7/8^2)] / 4 ≈ 0.439881

(c) Simpson's Rule:

Simpson's rule approximates the definite integral as the sum of areas of parabolic curves under the curve, connecting the function values at three evenly spaced points within each subinterval.

The formula for Simpson's rule is given by:

∫_a^b f(x) dx ≈ (b - a) * [f(a) + 4 * ∑ f(a + (b - a) * i / (2n)) + 2 * ∑ f(a + (b - a) * i / n) + f(b)] / (6n)

For n = 4 subintervals, we can find the points by evaluating the function at 0, 1/4, 1/2, 3/4, and 1.

∫_0^1 e^(−6x^2)dx ≈ (1 - 0) * [e^(-6 * 0^2) + 4 * (e^(-6 * 1/4^2) + e^(-6 * 3/4^2)) + 2 * (e^(-6 * 1/2^2)) + e^(-6 * 1^2)] / (6 * 4) ≈ 0.439363

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The complete question is :

Given the following integral and value of n, approximate the following integral using the methods indicated (round your answers to six decimal places):

 [tex]\int\limits {e^{(-6x^2)}} \, dx[/tex]   ,n=4.(ps: b =1, a = 0)

(a) Trapezoidal Rule

(b) Midpoint Rule

(c) Simpson's Rule

Answer:

2

Step-by-step explanation:

y inverse 3 in mathematics

Answer:

a. Total Cost Function: The total cost function for Birat tea packaging company is given by: Total Cost = 8,000,000 + 80x + 15x + 10x + 20x = 8,360,000 + 125x

b. Total Revenue Function: The total revenue function for Birat tea packaging company is given by: Total Revenue = 160x

c. Total Profit Function: The total profit function for Birat tea packaging company is given by: Total Profit = Total Revenue - Total Cost = 160x – 8,360,000 - 125x = 7,225,000 - 125x

d. If you were the owner of the company, you would need to sell at least 58,000 Kgs of tea in order to reach neither loss nor profit.

e. If the market is very competitive and there are lots of substitute products, the Birat Tea Packaging Company can use a variety of strategies to stay competitive. These strategies can include offering discounts, loyalty programs, promotional campaigns, product diversification, and improved marketing efforts. The company should also focus on providing superior customer service to build and maintain customer loyalty. Additionally, the company should stay up-to-date on the latest trends in the industry and explore opportunities to collaborate with other companies to gain a competitive advantage.

Step-by-step explanation:

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{16\% of 298000}}{\left( \cfrac{16}{100} \right)298000}\implies 47680[/tex]

Answer:

Can you elaborate more on the question?

Answer:

S₁₀ = - 40

Step-by-step explanation:

given the first term a₁ = 5 and the nth term [tex]a_{n}[/tex] = - 13

then the sum to n terms is

[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] (a₁ + [tex]a_{n}[/tex] )

here n = 10 , then

S₁₀ = [tex]\frac{10}{2}[/tex] (5 + (- 13)) = 5(5 - 13) = 5 × - 8 = - 40

The value of 1+1 in base 10 is 2 and its value in base 2 is 10

Addition in math is a process of combining two or more numbers. Addends are the numbers being added, and the result or the final answer we get after the process is called the sum.

For example if there 20 apples, 30mangoes and 40 cashews in a basket. The total number of fruits in the basket is obtained by adding all the fruits together.

This is means the total number of of fruit = 20+30+40 = 90 fruits.

Therefore 90 fruits are in the basket.

Similarly, the addition of 1 and 1 will give

1+1 = 2. This answer is in base 10.

1+1 can also be in base two and 1+1 in base 2 = 10

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The given expression is not in the form of [tex]ca^{p}b^{q}[/tex], where c, p, and q are numbers.

To simplify simply means to make anything easier. In mathematics, simplifying an equation, fraction, or problem means taking it and making it simpler. Calculations and problem-solving techniques simplify the issue. By eliminating all common factors from the numerator and denominator and putting the fraction in its simplest/lowest form, we can simplify fractions.

Given the equation,[tex]\frac{ab - b}{b^{2} } - b[/tex],

to find the equation in form of  [tex]ca^{p}b^{q}[/tex] where c, p, and q are numbers,

dividing separately,

ab/b² - b/b² - b

using property aⁿ/aˣ = aⁿ⁻ˣ

ab/b² - b/b² - b = ab⁻¹ - b⁻¹ - b

or ab/b² - b/b² - b = (ab - b - b²)b².

so  (ab - b - b²)b² ≠ [tex]ca^{p}b^{q}[/tex]

Hence the equation is not correct.

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The expression  (ab - a)/b² - b in the form [tex]ca^pb^q[/tex] is ab⁻¹.

Some common rules of exponents are

xᵃ×xᵇ = xᵃ⁺ᵇ.

xᵃ/xᵇ = xᵃ⁻ᵇ.

Addition and subtraction of exponents are only possible for the same base value and when the base is different and both have the same exponent we just multiply the bases and write the exponent.

Given, An expression,  (ab - a)/b² - b.

Now, This can be written as,

a(b - 1)/b(b - 1).

Now, (b - 1) will cancel out and we'll have,

a/b.

ab⁻¹.

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Important to detect outliers in data set, Z-score (3 std dev from mean) and IQR methods are commonly used, choosing right method and threshold crucial for accurate data analysis.

it is more than 3 standard deviations away from the mean. This method is known as the Z-score method or the standard deviation method for detecting outliers. It provides a way to quantify how far away a data point is from the mean in terms of standard deviations. However, it is important to note that the 3 standard deviation threshold is just a general guideline and may not always be appropriate. Other methods, such as the interquartile range (IQR) method, can also be used to detect outliers in a data set.

It's essential to detect outliers as they can greatly impact the analysis and interpretation of data, so choosing the appropriate method and threshold for identifying outliers is crucial.

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The coordinates of the vertex of the following parabola is (0,6).

The given equation of the parabola is [tex]& y=-3 x^2+6 \\[/tex]

The point where the parabola and its axis of symmetry intersect is called the vertex of a parabola. It is used to determine the coordinates of the point on the parabola's axis of symmetry where it crosses it.

The standard form of a parabola is [tex]y=ax^{2} +bx+c.[/tex]

The vertex form of a parabola is [tex]y = a(x-h)^{2} + k[/tex].

[tex]$$\begin{aligned}& y=-3 x^2+6 \\& y-6=-3 x^2 \\& y-6=-3(x-0)^2\end{aligned}$$[/tex]

The vertex formula is used to find the vertex of a parabola. The formula to find the vertex is (h, k) = (-b/2a, -D/4a), where D = [tex]b^{2} -4ac[/tex].

Now comparing the above equation by the equation of parabola

⇒[tex]$$y-b=4(x-a)^2 \cdots$$[/tex]

With vertex (a, b)

So a=0 and b=6.

So vertex of above parabola is[tex]$(\mathbf{0}, \mathbf{6})$[/tex].

Therefore, the vertex of the given parabola [tex]$\mathbf{y}=-\mathbf{3} \mathbf{x}^{\mathbf{2}}+\mathbf{6}$ is $(\mathbf{0}, \mathbf{6})$[/tex].

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Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x,y) point. [tex]& y=-3 x^2+6 \\[/tex]

The sides of the rectangle in question number 1 are 4.5 units and 13.5 units. The sides of the rectangle in question number 2 are 6 units and 12 units.

Let's call the width of the rectangle in question number one as "w". Since the length of the rectangle is three times as long as its width, the length can be represented as 3w.

The perimeter of the rectangle is 36 units, so if we add up the lengths of all four sides, we get:

2w + 2(3w) = 36

2w + 6w = 36

8w = 36

w = 4.5

So the width of the rectangle is 4.5 units. Since the length is three times the width, the length is 3 × 4.5 = 13.5 units. So the sides of the rectangle are 4.5 units and 13.5 units.

Let's call the width of the rectangle in question number 2 as "p". Since the length of the rectangle is twice as long as its width, the length can be represented as 2p.

The area of the rectangle is given as 72 square units, so we can use the formula for the area of a rectangle to find the value of p:

A = l × w

72 = 2p × p

72 = 2p²

36 = p²

p = 6

So the width of the rectangle is 6 units. Since the length is twice the width, the length is 2 × 6 = 12 units. So the sides of the rectangle are 6 units and 12 units.

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The measure of {x} is equivalent to 18.

In Euclidean geometry -

an angle is the figure formed by two rays known as the sides of the angle and sharing a common endpoint known as the vertex of the angle.

Given is a circle with sectors.

Now, the measure of ∠MCP = 180°. We can write -

∠MCL + ∠LCA + ∠ACP = 180°

(2x - 4) + 101 + 47 = 180

2x - 4 + 148 = 180

2x + 144 = 180

2x = 36

x = 18

Therefore, the measure of {x} is equivalent to 18.

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