A sample of an element with a half-life of 8 years has a mass of 10 grams after 100 years. What was the mass of the original sample?

the volume of the smallest region E enclosed by the cone and sphere is (64/3)π(1 - no⁴/5), where no is the constant in the equation of the cone Z = - no Ix² + y².

To compute the volume of the smallest region E enclosed by the cone and sphere, we will use spherical coordinates. In spherical coordinates, a point in 3D space is represented by three values: radius (r), polar angle (θ), and azimuthal angle (φ).

First, we need to find the intersection of the cone and sphere. Substituting Z = - no Ix² + y² into the equation of the sphere, we get x² + y² + (- no Ix² + y²)² = 32. Simplifying this equation gives us x² + y² + no²x⁴ - 2no²x²y² + y⁴ = 32. We can rewrite this equation in terms of r, θ, and φ as follows:

r²sin²θ + no²r⁴cos⁴θsin²θ - 2no²r⁴cos²θsin²θ + no²r⁴cos²θsin⁴θ = 32

Simplifying this equation gives us:

r = √(32/(sin²θ + no²cos²θsin²θ))

Next, we need to find the limits of integration for r, θ, and φ. Since the region E is enclosed by the sphere x² + y² + z² = 32, we know that the maximum value of r is 4√2. The minimum value of r is zero. The limits of integration for θ are 0 to π/2, since the cone is pointing downwards in the negative z direction. The limits of integration for φ are 0 to 2π, since the region E is symmetric about the z-axis.

The volume of the region E can be computed using the following integral:

Vol(E) = ∫∫∫ r²sinθ dr dθ dφ

Integrating over the limits of integration for r, θ, and φ, we get:

Vol(E) = ∫₀^(2π) ∫₀^(π/2) ∫₀^(4√2) r²sinθ dr dθ dφ

Evaluating this integral gives us:

Vol(E) = (64/3)π(1 - no⁴/5)

Therefore, the volume of the smallest region E enclosed by the cone and sphere is (64/3)π(1 - no⁴/5), where no is the constant in the equation of the cone Z = - no Ix² + y².
Hi! To compute the volume of the region E enclosed by the cone Z = -√(x² + y²) and the sphere x² + y² + z² = 32 using spherical coordinates, we can set up the triple integral as follows:

Vol(E) = ∫∫∫ ρ² sin(φ) dρ dθ dφ

In spherical coordinates, the cone Z = -√(x² + y²) becomes φ = 3π/4, and the sphere x² + y² + z² = 32 becomes ρ = 4.

The limits of integration are:
- ρ: 0 to 4
- θ: 0 to 2π
- φ: π/2 to 3π/4

So, the triple integral can be written as:

Vol(E) = ∫(ρ=0 to 4) ∫(θ=0 to 2π) ∫(φ=π/2 to 3π/4) ρ² sin(φ) dρ dθ dφ

By calculating this triple integral, we can find the volume of the region E.

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Hi! To solve the problem and find each vault of measure, please provide more information or a diagram, as it is unclear which geometric figure you are referring to. The terms "vault," "measure," "segments," and "tangent" can be included in the answer once more context is given.

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The cost of the stock, including the broker's fee, is $50,454.30.

The cost of the stock can be found by multiplying the number of shares purchased by the price per share. In this case, the investor purchased 500 shares of Exxon-mobil stock at $98.93 per share, so the cost of the stock can be calculated as follows:

Cost of stock = Number of shares × Price per share

Cost of stock = 500 × $98.93

Cost of stock = $49,465

However, the broker charges 2% of the cost of the stock, which is an additional fee that needs to be added to the total cost. To find the broker's fee, we can simply multiply the cost of the stock by 2%:

Broker's fee = 2% × Cost of stock

Broker's fee = 2% × $49,465

Broker's fee = $989.30

Therefore, the total cost of the stock, including the broker's fee, is:

Total cost of stock = Cost of stock + Broker's fee

Total cost of stock = $49,465 + $989.30

Total cost of stock = $50,454.30

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Therefore, the probability that a randomly selected point within the square falls in the red-shaded area is 68%.

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. The probability of an event A is denoted as P(A). To calculate the probability of an event, we divide the number of favorable outcomes by the total number of possible outcomes.

Here,

The area of the red-shaded region is the area of the square minus the area of the white right-angled triangle. The area of the square is the length of one side squared, which is:

Area of square = 5 cm × 5 cm

= 25 cm²

The area of the right-angled triangle is one-half the base times the perpendicular height, which is:

Area of triangle = (1/2) × base × height

= (1/2) × 4 cm × 4 cm

= 8 cm²

Therefore, the area of the red-shaded region is:

Area of red-shaded region = Area of square - Area of triangle

= 25 cm² - 8 cm²

= 17 cm²

To find the probability that a randomly selected point within the square falls in the red-shaded area, we need to divide the area of the red-shaded region by the total area of the square, which is:

Probability = Area of red-shaded region / Area of square

Probability = 17 cm² / 25 cm²

= 0.68 or 68%

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Answer:

Step-by-step explanation:

where are the streets

An expression for the sequence of operations described "Subtract three from the product of seven and eight." is (7 × 8) - 3.

In order to evaluate and solve this expression, we would have to apply the PEMDAS rule, where mathematical operations within the parenthesis (grouping symbols) are first of all evaluated, followed by exponent, and then multiplication or division from the left side of the equation to the right. Lastly, the mathematical operations of addition or subtraction would be performed from left to right.

Based on the information provided, we have the following mathematical expression:

Expression: (7 × 8) - 3

56 - 3

53

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Answer:  x=20  y=43

Step-by-step explanation:

That little symbol in <1 means that the angle is a right angle, which is = to 90°  so

<1 = 90°

133-y = 90        solve for y by subtracting 133 from both sides

-y = -43             divide by -1 on both sides

y=43

Because all 3 angles make a line, which is 180, and you know <1 = 90   then <2+<3=90 as well.

<2+<3=90

22 + x + 48 =90     simplify

70 + x =90              subtract 70 from both sides

x=20

Minimum

-11

Main concepts:

Concept 1: Identify the type of equation

Concept 2: Identify the concavity (opens up/down)

Concept 3: Finding a vertex of a parabola

Concept 1: Identify the type of equation

First, observe that the equation is a polynomial.  This is a type of equation where there may be multiple terms containing an x, where each term with an x is raised to a whole number power, and may be multiplied by a real number.  Additionally, there may be a constant term added (or subtracted).

For our equation, [tex]y=3x^2-12x+1[/tex], the first two terms contain an x, each raised to a whole number power, and are multiplied by a number.  Additionally, there is a constant added to the end of the equation.  Therefore, this is a polynomial.

The largest power of x in a polynomial is called the "degree" of the polynomial.  Since the largest power of x is 2, this is called a second degree polynomial.  Another common name for a second degree polynomial is a quadratic equation.

This quadratic equation is already in what is known as "Standard form" [tex]y=ax^2+bx+c[/tex]

Concept 2: Identify the concavity (opens up/down)

For quadratic equations, the graph of the equation will be a sort of "U" shape" called a parabola.  The parabola either opens up or down depending on the "leading coefficient" in the quadratic equation.

The "leading coefficient" of any polynomial is the constant number that is multiplied to x in the term with the highest power.  In this case, the leading coefficient is 3.

A parabola opens up or down in correspondence with the sign of the leading coefficient.  If the leading coefficient is positive, the parabola opens upward.  If the leading coefficient is negative, the parabola opens downward.

Since the leading coefficient is 3, the parabola for our example opens upward.  The branches of the "U" will go upward forever, without a maximum.  However, the bottom of the "U" will have a minimum value.  We are assigned to find this minimum value (how low it goes).

Concept 3: Finding a vertex of a parabola

To find the vertex of a parabola, with an equation in standard form, there are a few methods, but the most straightforward is to use the vertex formula:

[tex]h=\dfrac{-b}{2a}[/tex]

Where "h" is the x-coordinate of the vertex, and "a" and "b" are the coefficients from the quadratic equation: [tex]y=ax^2+bx+c[/tex]

[tex]h=\dfrac{-(-12)}{2(3)}[/tex]

[tex]h=\dfrac{12}{6}[/tex]

[tex]h=2[/tex]

So, the parabola will have a vertex with an x-coordinate of "2", meaning that the lowest point will be at a position that is 2 units to the right of the origin... however, we still don't know how high that minimum is.  Fortunately, the equation [tex]y=3x^2-12x+1[/tex] itself gives the relationship between any x-value and the y-value that is associated with it.

[tex]y=3x^2-12x+1[/tex]

[tex]y=3(2)^2-12(2)+1[/tex]

[tex]y=3*4+(-12)*2+1[/tex]

[tex]y=12+-24+1[/tex]

[tex]y=-11[/tex]

So, the vertex of the parabola is (2,-11).

The height of the vertex is -11, so the value of the minimum is -11.

Side note: "What is the value of the minimum" is a different question that "where is the minimum at".  The minimum is at 2.  The actual value of the minimum is -11.

Answer:36

Step-by-step explanation:

3 a minute
18 in 6 minutes
36 in 12
3:1

The number of 6-card hands  in which at least one card from each suit is equal to 8,184,220.

Total number of 6-card hands that can be formed from a deck of 60 cards is,

Using combination formula,

C(60, 6) = 50,063,860

Now, subtract the number of 6-card hands that do not contain at least one card from each suit.

There are 5 ways to choose the suit that will be missing from the hand.

Once this suit is chosen, there are 48 cards remaining in the other suits.

Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suit is,

C(48, 6) = 12,271,512

Overcounted the number of hands that are missing more than one suit.

There are C(5, 2) ways to choose 2 suits that will be missing from the hand.

Once these suits are chosen, there are 36 cards remaining in the other 3 suits.

Choose 6 cards from this set, so the number of 6-card hands that do not contain any cards from the chosen suits is,

C(36, 6) = 1,947,792

We cannot have a 6-card hand that is missing more than 2 suits.

3 suits with no cards in the hand, which is not allowed.

Number of 6-card hands that have at least one card from each suit is,

C(60, 6) - 5×C(48, 6) + C(5, 2)×C(36, 6)

=50,063,860 - 5×  12,271,512 + 10 × 1,947,792

= 50,063,860 -61,357,560 + 19,477,920

= 8,184,220

Therefore, there are 8,184,220 of 6-card hands that have at least one card from each suit.

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Based on the graph, it appears that F(x) is a downward-facing parabola that has been shifted horizontally and vertically.

The vertex of the parabola is located at the point (3,-3), so the equation must include (x - 3) and (y + 3). Additionally, since the graph is narrower than the graph of G(x) = x^2, there must be a coefficient that is greater than 1 in front of the squared term.

Looking at the answer choices, we can eliminate options B and D because they have positive coefficients in front of the squared term, which would result in an upward-facing parabola. Option C has a negative coefficient in front of the squared term, which would result in a wider parabola than the graph shown.

Therefore, the correct answer is A, F(x) = 3(x-3)^2 - 3.

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Answer:

10 outcome is the answer

Note that the light is about 9.17 x 10³ times faster than sound. See the explanation below.

To arrive a the above, we only need to simplify Jakes observation as follows

[tex]\frac{1.1 * 10^{9} km/h }{1.2 * 10^{3}km/h }[/tex]

dividing the coefficients

1.1/1.2 = 0.91666666666

0.91666666666 x 10 ⁹⁻³

= 9.1666667 x 10 ⁵

further simplified, we have
9.17 x 10³

Thus, we can summarize that light is about 9.17 x 10³ faster than the speed of sound going by Jake observation.

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Full Question:

Jake noted that speed of light is approximately 1.1 × 10⁹ km/h;

the speed of sound is approximately 1.2 x 10³ km/h; and

11 + 12 = 0.916.

Using Jake's figures how many times faster is light than speed?

Write your answer in standard form.

a. In the geometric series for f(x) to be convergent, x < - 1

b.  When x =  1¹/₂, the sum to infinity of the geometric series is f(x) = -1.4

A geometric series is the sum of terms of a geometric sequence.

a. Given the series f(x) = ∑ₙ = ₁⁰⁰(x + 2)ⁿ, we want to determine the value of x for which f(x) converges.

Now, let the general term of the sequence be Uₙ = (x + 2)ⁿ, to determine the value of x for which the series is convergent, we use the D'alembert ratio test which states that for a series to be convergent, then

Uₙ₊₁/Uₙ < 1.

So, we have that Uₙ₊₁ = (x + 2)ⁿ⁺¹

So, Uₙ₊₁/Uₙ =  (x + 2)ⁿ⁺¹/ (x + 2)ⁿ

= x + 2

For convergence

Uₙ₊₁/Uₙ < 1

So,

x + 2 < 1

x < 1 - 2

x < - 1

So, for f(x) to be convergent, x < - 1

b. To find the value of f(x) when x = 1¹/₂, we proceed as folows

Since  f(x) = ∑ₙ = ₁⁰⁰(x + 2)ⁿ, substituting x =  1¹/₂ = 1.5 into the equation, we have

 f(x) = ∑ₙ = ₁⁰⁰(1.5 + 2)ⁿ

f(x) = ∑ₙ = ₁⁰⁰(3.5)ⁿ

= 3.5 + 3.5² + 3.5³ + ...

Since this is a geometric progression with sum to infinity, we see that the first term is a = 3.5 and the common ratio is r = ar/a = 3.5²/3.5 = 3.5

Since the sum to infinity of a geometric progression is

S₀₀ = a/(1 - r)

So, substituting the values of the variables into the equation, we have that

S₀₀ = a/(1 - r)

S₀₀ = 3.5/(1 - 3.5)

S₀₀ = 3.5/-2.5

S₀₀ = -1.4

So, when x =  1¹/₂, f(x) = -1.4

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Answer:

The numerical values of the circumference and area are equal

Step-by-step explanation:

Circumference: 12.57

Area: 12.57

12.57=12.57

Hope this helps! :)

The oil spill is increasing by approximately 0.58% per hour.

An oil spill is spreading such that its area is given by the exponential function A(t) = 250(1.15)^t, where A is the area in square feet and t is the time that has elapsed in days. To find the percent increase per hour, first convert the time from days to hours by replacing t with (t/24), since there are 24 hours in a day.

A(t) = 250(1.15)^(t/24)

To determine the hourly percent increase, find the growth factor for one hour (t=1) and subtract 1 to get the percentage:

A(1) = 250(1.15)^(1/24) ≈ 250(1.0058)

The growth factor for one hour is approximately 1.0058. To find the percent increase, subtract 1 and multiply by 100:

(1.0058 - 1) × 100 ≈ 0.58%

The oil spill is increasing by approximately 0.58% per hour.

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The weight of the contents in the container is approximately 873 pounds.

A shipping container is in the shape of a right rectangular prism with a length of 7 feet, a width of 14 feet, and a height of 13.5 feet.

To find the volume of the container, we multiply the dimensions: 7 ft × 14 ft × 13.5 ft = 1,323 cubic feet. The container is completely filled with contents that weigh, on average, 0.66 pound per cubic foot.

To find the weight of the contents in the container, we multiply the volume by the average weight: 1,323 ft³ × 0.66 lb/ft³ ≈ 873.18 pounds.

Rounded to the nearest pound, the weight of the contents in the container is approximately 873 pounds.

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Answer:

Step-by-step explanation:

It's given that, Radius of the circle is 9 cm.

We know that Circumference of the circle is calculated as 2πr

where,

Substituting the required values,

Circumference = 2 × 3.14 × 9

= 6.28 × 9

= 56.52 cm

Hence the required circumference of the circle is 56.52 cm

Marcus is incorrect. The percentage increase in the time he spent doing homework this week compared to last week is 10%.

To determine if Marcus is correct, we need to calculate the percentage increase in the time he spent doing homework this week compared to last week.

First, we calculate the difference in hours:

11 hours - 10 hours = 1 hour

Then, we calculate the percentage increase:

(1 hour / 10 hours) x 100% = 10%

Therefore, Marcus is incorrect. The percentage increase in the time he spent doing homework this week compared to last week is 10%, not 110%.

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The sets of side lengths that could form a right triangle are 36-48-60 (option b) and 14-48-50 (option d).

To determine which of these sets of side lengths could form a right triangle, we will use the Pythagorean theorem (a² + b² = c²), where a and b are the shorter sides and c is the hypotenuse. Let's evaluate each option:

a) 4-7-10
Applying the Pythagorean theorem: 4² + 7² = 16 + 49 = 65, which is not equal to 10² (100). So, this set does not form a right triangle.

b) 36-48-60
Applying the Pythagorean theorem: 36² + 48² = 1296 + 2304 = 3600, which is equal to 60² (3600). So, this set does form a right triangle.

c) 6-10-14
Applying the Pythagorean theorem: 6² + 10² = 36 + 100 = 136, which is not equal to 14² (196). So, this set does not form a right triangle.

d) 14-48-50
Applying the Pythagorean theorem: 14² + 48² = 196 + 2304 = 2500, which is equal to 50² (2500). So, this set does form a right triangle.

In conclusion, the sets of side lengths that could form a right triangle are 36-48-60 (option b) and 14-48-50 (option d).

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The two types of transformations that are produced by the matrix include:

We can see that a 90° counter-clockwise rotation followed by a dilation produces a transformation that stretches and rotates the figure.

On the other hand, a 90° counter-clockwise rotation followed by a reflection in the vertical axis produces a transformation that reflects and rotates the figure.

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The answer is not 9 > 6, as that is a comparison between two numbers, but rather g ≤ 6, as it sets a limit on the number of goldfish that can be in the tank.

The correct inequality that represents the situation is g ≤ 6. The problem states that the maximum number of goldfish that can live in a small tank is 6, meaning that the number of goldfish must be less than or equal to 6.

The symbol ≤ represents "less than or equal to", while the symbol > represents "greater than".

Therefore, the answer is not 9 > 6, as that is a comparison between two numbers, but rather g ≤ 6, as it sets a limit on the number of goldfish that can be in the tank.

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Answer:

E. 10

Step-by-step explanation:

To solve the equation x + 50 = 60, we need to isolate x on one side of the equation. Subtracting 50 from both sides, we get:

x + 50 - 50 = 60 - 50

x = 10

Therefore, the solution to the equation is x = 10. Checking the answer choices, we see that E. 10 is the only number that belongs to the solution set. Therefore, the answer is:

E. 10

The rate of appreciation of the classic car is 18% per year.

An exponent is a mathematical operation that indicates how many times a number or expression is multiplied by itself. It is represented by a superscript number that is written to the right and above the base number or expression. The exponent tells us how many times the base is multiplied by itself.

The value V of the classic car appreciates exponentially, and it is represented by the formula:

V = 32,000[tex]1.18^{2}[/tex]

The term [tex]1.18^{t}[/tex] represents the factor by which the value of the car increases each year. If we calculate this factor for one year (t=1), we get:

(1.18)¹= 1.18

This means that the value of the car increases by 18% in the first year. Similarly, if we calculate the factor for two years (t=2), we get:

(1.18)² = 1.39

This means that the value of the car increases by 39% in the first two years (18% in the first year and an additional 21% in the second year).

Therefore, the rate of appreciation of the classic car is 18% per year.

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The simplified form of the expression 3(w+11) / 6w  is w + 11 / 2w .

Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner.

In other words, we have to expand any brackets, next multiply or divide any terms and use the laws of indices if necessary, then collect like terms by adding or subtracting and finally rewrite the expression.

Therefore, let's simplify the expression:

3(w+11) / 6w

Hence, let's divide both the numerator and denominator by 3

Therefore,

3(w+11) / 6w =  w + 11 / 2w

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The price of one cubic yard of sand is $33.75, then the price of one cubic foot of sand should be  $1.25.

To determine the price of one cubic foot of sand, we need to convert cubic yards to cubic feet. One cubic yard is equal to 27 cubic feet (3 feet x 3 feet x 3 feet). Therefore, if the price of one cubic yard of sand is $33.75, then the price of one cubic foot of sand should be $33.75/27 = $1.25.

This makes sense because one cubic yard contains 27 cubic feet. So, if the price of one cubic yard is $33.75, then the price per cubic foot should be 1/27th of that price.

It is important to note that this assumes the price per unit of sand remains constant regardless of the quantity purchased. In reality, bulk purchases may result in a discounted price per unit. Additionally, factors such as transportation costs and demand may also affect the price of sand.

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Answer: Si la emisión de gases de efecto invernadero aumenta en un 5% anual, ¿cuánto aumentará la temperatura global en 20 años?

Solución: Dado que cada

Step-by-step explanation:

Una empresa produce 400 toneladas de dióxido de carbono al año. Si cada tonelada de dióxido de carbono contribuye al calentamiento global en 0.05 grados Celsius, ¿cuál será el aumento de temperatura causado por la empresa en un año?

Solución: 400 x 0.05 = 20 grados Celsius

La temperatura media de la Tierra ha aumentado en 1 grado Celsius desde la era preindustrial.

Si el aumento de temperatura está directamente relacionado con la cantidad de dióxido de carbono en la atmósfera, ¿cuánto dióxido de carbono adicional se ha emitido desde la era preindustrial hasta ahora?

Solución: Dado que cada tonelada de dióxido de carbono contribuye a un aumento de 0.05 grados Celsius, 1 / 0.05 = 20. Por lo tanto, se han emitido 20 veces la cantidad de dióxido de carbono necesario para contribuir a un aumento de 1 grado Celsius.

Una central térmica produce 1000 megavatios de electricidad al día. Si la eficiencia de conversión de la central térmica es del 30%, ¿cuántas toneladas de dióxido de carbono se emiten al día?

Solución: La eficiencia de conversión de la central térmica es del 30%, lo que significa que se pierde el 70% de la energía.

Por lo tanto, la cantidad de energía producida por la central térmica es de 1000 x 0.3 = 300 megavatios. Si cada megavatio produce 0.5 toneladas de dióxido de carbono, entonces la central térmica emite 300 x 0.5 = 150 toneladas de dióxido de carbono al día.

Si se reduce la emisión de dióxido de carbono en un 20%, ¿en qué medida se reducirá el aumento de temperatura global?

Solución: Si se reduce la emisión de dióxido de carbono en un 20%, se reducirá el aumento de temperatura global en un 20% x 0.05 = 0.01 grados Celsius.

Si la temperatura media en una ciudad ha aumentado en 0.5 grados Celsius en los últimos 10 años, ¿cuál es la tasa de aumento de temperatura por año?

Solución: La tasa de aumento de temperatura por año es de 0.5 grados Celsius / 10 años = 0.05 grados Celsius por año.

Si la concentración de dióxido de carbono en la atmósfera es de 400 partes por millón (ppm) y se espera que aumente en un 2% anual, ¿cuál será la concentración de dióxido de carbono en 10 años?

Solución: El aumento anual de la concentración de dióxido de carbono es de 400 x 0.02 = 8 ppm. Por lo tanto, la concentración de dióxido de carbono en 10 años será de 400 + 8 x 10 = 480 ppm.

Si la emisión de gases de efecto invernadero aumenta en un 5% anual, ¿cuánto aumentará la temperatura global en 20 años?

Solución: Dado que cada

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The equation that could represent each of the graphed polynomial function include the following:

First graph: y = x(x + 2)(x - 3)

Second graph: y = x⁴ - 5x² + 4

In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.

Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.

By critically observing the graph of the polynomial function shown in the image attached above, we can logically deduce that the first graph has a zero of multiplicity 1 at x = 2 and zero of multiplicity 1 at x = -3.

Similarly, we can logically deduce that the second graph has a zero of multiplicity 2 at x = 2 and zero of multiplicity 2 at x = -2.

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Answer:

Step-by-step explanation: The vertex is the point at the bottom if the parabola opens up and at the top if it opens at the bottom.