How to find the total surface area of cross section solid factors and how to use phytagoras theorem to determine unknown side then find total surface area

Answer:

To find the area of the trapezoid-shaped playground, we need to use the formula:

Area = (1/2) × (base1 + base2) × height

In the given diagram, the length of the top base is 24 feet, the length of the bottom base is 48 feet, and the height is 52 feet.

So, the area of the playground is:

Area = (1/2) × (24 + 48) × 52

= (1/2) × 72 × 52

= 1,872 square feet

Therefore, the area of the playground is 1,872 square feet.

The closest option to this answer is option B, which is 1,768 square feet. However, the correct answer is actually 1,872 square feet.

Jose is in the quadrant 4 as his coordinates are (-2, -3).

The x-axis and y-axis in a coordinate plane stand in for the horizontal and vertical axes, respectively. Using top right as quadrant I, topmost left as quadrant II, bottom left as quadrant III, and bottom right as quadrant IV, the four quadrants are numbered anticlockwise.

The given coordinates are:

Cameron = (-3, 2)

Neveah = (2, -3),

Jude = (2,3) and

Jose is at (-2, -3).

We must find the point whose x-coordinate is positive and y-coordinate is negative in order to know which point is in quadrant IV.

Hence, Jose is in the quadrant 4 as his coordinates are (-2, -3).

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To write a quadratic function f whose zeros are -2 and 9 , we have to factor the function by (x+2) and (x-9).

A quadratic function is a second-degree mathematical function whose graph is a parabola. The general form of a quadratic function is given by f(x) = ax² + bx + c, where a, b and c are constants and a cannot be equal to zero. The variable x represents the input of the function and f(x) represents the output or result of the function. To find the quadratic function of f we first need to multiply these factors, like this:

So the quadratic function whose zeros are -2 and 9 is:

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The area of the shaded region is 90 cm².

The space filled by the surface of an object or any flat shape can be thought of as the area in terms of geometry. The quantity of unit squares that cover an object's surface when it is closed is known as the area of the object. Inches, millimeters, square feet, and other square measurements are used to determine the area.

The area of shaded region = area of given square - area of all given triangles

Now, Area of given square = side × side

                                             = 12 × 12

                                             = 144 cm²

Similarly, Area of all triangles

             = 4 × (area of small triangle) + 4 × (area of big triangle)

             = 4 × ( 1/2 × 3× 3) + 4 × ( 1/2 × 3 × 6)

             = 18 + 36

             = 54 cm²

∴ The area of shaded region = 144 - 54

                                                = 90 cm²

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As per the concept of integral, the series ∑ne⁻ⁿ converges.

Let's consider the series ∑ne⁻ⁿ. Since each term of the series is positive, the first condition of the Integral Test is satisfied. To check the second condition, we need to determine whether the function f(x) = xe⁻ˣ is decreasing for x ≥ 1.

To do this, we can take the derivative of f(x) with respect to x:

f'(x) = e⁻ˣ - xe⁻ˣ

Setting f'(x) = 0, we get:

e⁻ˣ - xe⁻ˣ = 0

x = 1

So f(x) has a maximum value at x = 1. Since f'(x) < 0 for x ≥ 1, f(x) is decreasing for x ≥ 1.

Now, we can set up the corresponding integral:

∫₁^∞ xe⁻ˣ dx

To evaluate this integral, we can use integration by parts:

u = x, dv = e⁻ˣ dx

du = dx, v = -e⁻ˣ

∫₁^∞ xe⁻ˣ dx = -xe⁻ˣ │₁^∞ + ∫₁^∞ e⁻ˣ dx

= 0 + e⁻ˣ │₁^∞

= 1

Since the integral converges, the series ∑ne⁻ⁿ also converges by the Integral Test.

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Complete Question:

How do you use the Integral Test to determine convergence or divergence of the series: ∑ne⁻ⁿ from n=1 to infinity?

1) The center of mass of the solid bounded by x = y^2 and the planes x = z, z = 0, and x = 1  the center of mass of the solid is (1/3, 2/15, 1/3). 2) The total charge on the disk is 4/3 coulombs. 3) The center of mass of the triangular region is (2/3, 2/3).


Center of Mass = (∫xyzρdV)/(∫ρdV).

Here, V is the volume of the solid. Since the density is constant, we can pull it out of the integral:

Center of Mass = k*(∫xyzdV)/(∫dV).

We can now use the volume formula for the solid which is V = ∫xyzdxdyz. Plugging this in the above formula, we get:

Center of Mass = k*[(∫x∫ydxdyz)/(∫dxdyz)]

Evaluating the integrals, we get the x coordinate of the center of mass to be (1/3), the y coordinate to be (2/15) and the z coordinate to be (1/3). Thus, the center of mass of the solid is (1/3, 2/15, 1/3).

2. To find the total charge Q on the disk x^2 + y^2 ≤ 1 such that the charge density at any point (x, y) is rho(x, y) = x + y + x^2 + y^2 (in coulombs per square meter), we need to use the following formula:

Q = ∫∫rho(x, y)dxdy

Evaluating the integral, we get Q = (1/3) + (1/3) + (1/3) + (1/3) = 4/3. Thus, the total charge on the disk is 4/3 coulombs.

3. To find the center of mass of the triangular region with vertices (0, 0), (2, 0) and (0, 2) if the density is given by rho(x, y) = 1 + x + 2y, we need to use the following formula:

Center of Mass = (∫xyρdA)/(∫ρdA).

Here, A is the area of the triangle. Evaluating the integral, we get the x coordinate of the center of mass to be (2/3) and the y coordinate to be (2/3). Thus, the center of mass of the triangular region is (2/3, 2/3).

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Answer:

11

Step-by-step explanation:

1. The Formula used SA = 2w (l +h) + lh

2. The box is 0.37125 inch deep.

3. The SA of box with lid is 294.7972 inch².

The area is the territory covered by a shape or figure, whereas the perimeter is the distance covered by the shape's outside boundary. The unit of area is the square unit or unit², while the unit of perimeter is the unit.

Given:

length = 13.2 inch

Height = 10.5 inch

And, Surface Area used = 295.02 inch²

Using the Formula

SA = 2lw + 2wh + lh

SA = 2w (l +h) + lh

295. 02 = 2w ( 13.2 + 10.5) + (13.2)(10.5)

147.51 = w x 24 + 138.6

8.91 = 24w

w = 0.37125 inch

SA of box with lid

= 2( lw+ wh + lh)

= 2( 13.2 x 10.5 + 10.5 x 0.37125 + 0.37125 x 13.2)

= 2(138.6 + 3.8981 + 4.9005)

= 2 x 147.3986

= 294.7972 inch²

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Answer: the circles diameter is 14 meters

Step-by-step explanation:                                {Formula for circumference

2TUR = 43.96                                                      of a circle : L 27UR}

2x3 . 14r = 43.96

r=7

d=2r=2x7=14                                                         {d=2r}  

Answer:

The circumference of a circle can be calculated using the formula:

C = 2πr

where C is the circumference, π is the mathematical constant pi (approximately 3.14), and r is the radius of the circle.

To find the diameter of the circle, we can use the formula:

d = 2r

where d is the diameter and r is the radius.

Given that the circumference of the circle is 43.96 meters, we can use the formula for circumference to solve for the radius:

C = 2πr

43.96 = 2 × 3.14 × r

r = 43.96 / (2 × 3.14)

r = 6.998 meters

Now we can use the formula for diameter to find the diameter:

d = 2r

d = 2 × 6.998

d = 13.996 meters

Therefore, the diameter of the circle is approximately 13.996 meters.

Step-by-step explanation:

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The minimum total cost for the gas station to store, ship, and purchase 22,032 gallons of diesel for one year is $199,152.87.

The gas station has a steady annual demand for 22,032 gallons of diesel. The cost to store 1 gallon for 1 year is $9, the cost to ship each order of diesel is $34, and the cost to purchase each gallon is $19. To calculate the total cost of storing, shipping, and purchasing the diesel for one year, we can use the following formula:

Total cost = (storage cost per gallon x annual demand) + (shipping cost per order x number of orders) + (purchase cost per gallon x annual demand)

To find the number of orders, we can divide the annual demand by the number of gallons per order. In this case, the number of gallons per order is not given, so we will use the variable "x" to represent it:

Number of orders = 22,032 / x

Plugging this back into the formula, we get:

Total cost = (9 x 22,032) + (34 x 22,032 / x) + (19 x 22,032)

Simplifying, we get:

Total cost = 198,288 + (748,288 / x)

To minimize the total cost, we can take the derivative of the total cost with respect to x and set it equal to zero:

d(Total cost) / dx = -748,288 / x^2 = 0

Solving for x, we get:

x = sqrt(748,288 / 0) = 865.12

Therefore, the gas station should order 865.12 gallons of diesel per order to minimize the total cost. The minimum total cost is:

Total cost = 198,288 + (748,288 / 865.12) = $198,288 + $864.87 = $199,152.87

The minimum total cost for the gas station to store, ship, and purchase 22,032 gallons of diesel for one year is $199,152.87.

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The interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually.

To calculate the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually, we will use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years

Plugging in the given values:

A = 18,000(1 + 0.05/2)^(2*5)

A = 18,000(1.025)^10

A = 23,386.28

To find the interest earned, we subtract the initial investment from the final amount:

Interest earned = A - P

Interest earned = 23,386.28 - 18,000

Interest earned = $5,386.28

Therefore, the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually is $5,386.28.

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The equation for the exponential function is f(x) = (4.5)((0.5/4.5)-1/2)x. The equation for an exponential function is f(x) = abx, where a and b are constants.

To determine the values of a and b, we can use the two points given in the question, (1,4.5) and (-1,0.5).

Let's substitute the point (1,4.5) into the equation.

f(1) = a*b1
4.5 = a*b

Now let's substitute the point (-1,0.5) into the equation.

f(-1) = a*b-1
0.5 = a*b-1

We can now solve for a and b.

a = 4.5 / b
b-1 = 0.5 / a
b-1 = 0.5 / (4.5/b)
b-1 = 0.5b/4.5
b-2 = 0.5/4.5
b = (0.5/4.5)-1/2

Thus, the equation for the exponential function is f(x) = (4.5)((0.5/4.5)-1/2)x

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The answer of Caroline is 19.33 years old and Cameron is 53.67 years old

Let's start by assigning variables to represent Caroline's age and Cameron's age. We'll use C for Caroline's age and M for Cameron's age.

According to the problem, the sum of their ages today is 73, so we can write the equation:

C + M = 73

Five years ago, Cameron was 2 times older than Caroline. so we can write the equation:

M - 5 = 2(C - 5)

Now we can use the first equation to solve for one of the variables. Let's solve for C:

C = 73 - M

Next, we can substitute this value of C into the second equation:

M - 5 = 2(73 - M - 5)

Simplifying the equation gives us:

M - 5 = 146 - 2M - 10

3M = 161

M = 53.67

Now we can use this value of M to find C:

C = 73 - 53.67

C = 19.33

So Caroline is 19.33 years old and Cameron is 53.67 years old.

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A student has 8 different choices when choosing a pair of pants and a shirt.

Without using intricate calculations, the likelihood of an event occurring is shown using a probability tree diagram. It shows every consequence that an event might have. A probability tree serves the aim of listing all potential outcomes of an event and calculating the likelihood of each one. A probability tree diagram can be used to indicate conditional probabilities or to show a sequence of independent occurrences.

The student can choose from four shirts and two pairs of pants (black or tan) (yellow, red, green, or white). We multiply the number of options for pants by the number of options for a shirt to get the total number of options:

2 choices for pants × 4 choices for a shirt = 8 total outfit choices

Therefore, a student has 8 different choices when choosing a pair of pants and a shirt.

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Answer:

Molly is correct in stating that the two functions are not inverses of each other.

To be inverses of each other, two functions must satisfy the property that when they are composed in either order, they result in the identity function, which is represented by f(x) = x.

In this case, we have:

f(g(x)) = (x + 5) + 5 = x + 10

g(f(x)) = (x + 5) + 5 = x + 10

Since both compositions of the functions result in x + 10, which is not equal to x, the two functions are not inverses of each other.

Step-by-step explanation:

Answer:

Molly is correct. The fact that $f(g(x)) = g(f(x)) = x+5$ indicates that the two functions, $f$ and $g$, are symmetric about the line $y=x$, which means that they are not inverses of each other.

To determine whether two functions are inverses of each other, we need to show that their composition results in the identity function. That is, if $f(x)$ and $g(x)$ are two functions, then $f(g(x)) = g(f(x)) = x$ for all $x$ in the domain of $f$ and $g$.

In this case, we see that $f(g(x)) = g(f(x)) = x+5$, which is not the identity function. Therefore, the two functions are not inverses of each other.

After using the properties of logarithms to rewrite and simplify the logarithmic expression  ln((e⁸)/7) simplifies to 1.

What are natural logarithms?

Natural logarithms are a type of logarithm that uses the number e as its base. The natural logarithm of a positive number x (written as ln(x)) is the exponent to which e must be raised to get x. In other words, ln(x) represents the power to which e must be raised to obtain x.

The number e is a mathematical constant that is approximately equal to 2.71828. It is a special number that appears in many areas of mathematics, science, and engineering. Natural logarithms have a variety of applications in fields such as calculus, probability theory, and statistics.

Some properties of natural logarithms include:

ln(1) = 0

ln(e) = 1

ln(xy) = ln(x) + ln(y) for any positive numbers x and y

ln(x/y) = ln(x) - ln(y) for any positive numbers x and y

ln(xᵃ) = a ln(x) for any positive number x and any real number a

Natural logarithms can be evaluated using a calculator or by using the properties of logarithms to simplify expressions. They are commonly used in mathematical and scientific calculations that involve exponential growth or decay.

We can use the property of logarithms that states: log(aᵇ) = b log(a) for any base a and any real number b.

Using this property, we can rewrite ln((e⁸)/7) as:

ln((e⁸)/(e⁷))

[tex]= ln(e^{(8-7)})[/tex]

= ln(e)

= 1

Therefore, ln((e⁸)/7) simplifies to 1.

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Answering the question, we may state that According to the graph, from function largest y-intercept to smallest y-intercept, we have:

[tex]f(x) = 5 f(x) = 2 f(x) + 1 f(x) = 1/2 f(x) = 1/5 (x)[/tex]

Mathematicians investigate the relationships between numbers, equations, and related structures, as well as the locations of forms and possible placements for these items. A set of inputs and their corresponding outputs are referred to as a "function" in this context. If each input results in a single, unique output, the relationship between the inputs and outputs is known as a function. Each function has its own domain, codomain, or scope. A common way to denote functions is with the letter f. (x). is an x for entry. One-to-one capabilities, so multiple capabilities, in capabilities, and on functions are the four main categories of accessible functions.

According to the graph, from largest y-intercept to smallest y-intercept, we have:

[tex]f(x) = 5 f(x) = 2 f(x) + 1 f(x) = 1/2 f(x) = 1/5 (x)[/tex]

As a result, the sequence is:

[tex]f(x) = 5^(x) (highest y-intercept) (highest y-intercept)[/tex]

[tex]f(x) = 2^(x) + 1 f(x) = 1/2^ (x)[/tex]

[tex]f(x) = 1/5^(x) (lowest y-intercept) (lowest y-intercept)[/tex]

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The coordinates (2,1) will be on graph but (1,3) is not on graph.

What is a coordinate?

A coordinate is a set of two or more numbers or variables that identify the position of a point, line, or plane in a space of a given dimension. Coordinates are used to pinpoint a particular location, such as a specific point on a map or a specific point in a mathematical equation.

This means that for every 1.5 cups of dried fruit, there is 1 pound of granola. The graph of this proportional relationship would be a line that goes through the origin and has a slope of 1.5. For the point (2,1), the x-coordinate (2) is exactly 1.5 times the y-coordinate (1). This means that if you used 2 cups of dried fruit, you would get 1 pound of granola. Therefore, this point would be on the graph of the proportional relationship, so the answer is Yes. However, for the point (1,3), the x-coordinate (1) is not 1.5 times the y-coordinate (3). This means that if you used 1 cup of dried fruit, you would not get 3 pounds of granola.

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Answer: x=6, y=8

Using elimination means that we eliminate one of the variables in both equations which help find out the other variable, which we then substitute to find the whole equation.

    3x-4y=-14

+   3x+2y=-2

=>-4y=-14

+   2y=-2

=>-2y=-16

=>y=8

So to find x,

3x-4(8)=-14

=3x=-14+32

=x=18/3 so x=6

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Answer:

Starting with the given equation:

√x³ - 100 = 5

Adding 100 to both sides:

√x³ = 105

Squaring both sides:

x³ = 11025

Taking the cube root of both sides:

x = 15

Therefore, the exact solution to the given equation is x = 15.

Note that no inverse functions were needed to solve this equation

Step-by-step explanation:

Answer:

Step-by-step explanation:

Im not sure I can't see the picture.

Answer: Hence, the width of the path is 4 meters.

First we should find the area of the pool. Our width is 14 and our length is 18. In order to find the area, we must multiply these two numbers. 14 times 18 is 252. This means that the area of the pool is 252.

Since we know that one side of the pool is 18 and one side is 14, we can write out the equation below.

572= (18+2x)(14+2x)

The first thing we should do is combine like-terms.Our equation would now be as below.

x^2+16-80=0

Next we would take our quadratic equation- x=-b+-√b2-4ac/2a and fill it in with the numbers that we had in the first equation, our equation would be like below.  A=1   B=16   C=-80

x=-16+-√16^2-4* 1 * -80 /2 * 1

When then simplify our equation leaving us with x=-16+-√16^2-4*1(-80)/2 * 1, which broken down is x=-16+-√+-24/2.

Lastly we must subtract and add to receive our two answers.

One side is 4 and the other is -20. Since the width of the walk cannot not be a negative number, we know that the width of the path is 4 meters.

Hence, the width of the path is 4 meters.

I hope this helped & Good Luck <3!!!

400 children and 600 adults bought tickets to the circus.

Let's call the number of children who bought tickets "C" and the number of adults who bought tickets "A". We can set up two equations to represent the information given in the question:

C + A = 1000 (the total number of people who bought tickets)
15C + 40A = 30000 (the total gate revenue)

We can use the first equation to solve for one of the variables in terms of the other. For example, we can solve for C in terms of A:

C = 1000 - A

Now we can substitute this equation into the second equation to solve for A:

15(1000 - A) + 40A = 30000
15000 - 15A + 40A = 30000
25A = 15000
A = 600

So 600 adults bought tickets. We can use the first equation to find the number of children who bought tickets:

C + 600 = 1000
C = 400

So 400 children bought tickets. Therefore, the answer is that 400 children and 600 adults bought tickets to the circus.

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The required, graph of the function f(x) = √[x+3]-1 is a curve that starts at (-3,-1) and extends to the right, increasing in value but at a decreasing rate due to the square root function.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
The graph of the function f(x) = √[x+3]-1 is a curve that starts at the point (-3,-1) and extends to the right indefinitely. The square root function √x has a domain of x ≥ 0, so in this case, the domain of the function is x ≥ -3.

The graph is always above the x-axis, as the square root function can only output non-negative values. The graph also approaches but never touches the horizontal line y = 0 as x increases without bound, since the -1 term in the function only shifts the graph downward by one unit.

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(a) The possible number of positive real zeros of P(x) can be determined using Descartes' Rule of Signs. The Rule of Signs states that a polynomial with real coefficients can have at most as many positive real zeros as its leading coefficient has sign changes in the coefficients. This means that the polynomial has at most 3 positive real zeros.

(b) Descartes' rule of signs can also be used to prove that 1-2 cannot be a zero of P(x). According to the Rule of Signs, the number of positive real zeros of P(x) is limited to 3. However, if 1-2 is a zero of P(x), then the polynomial would have 4 sign changes, which is not possible. Therefore, 1-2 cannot be a zero of P(x).

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If we consider the number line with rational number marked on it, then the number of parts to be present between 1 and 2 and -1 and -2 will be one part each.

A number line is a pictorial representation or drawing of numbers in which there are equal intervals between each number and is used to represent the real numbers on it. Real numbers are those numbers which may be positive or negative integers, rational numbers or irrational numbers. In general, a quantity which can be expressed as an infinite decimal expansion is called as a real number.

If we draw a number line and mark the points as 1, 2, 3,...,∞ and -∞,..., -3, -2, -1 on the right and left hand side of 0, then the intervals between each real number is equal to 1. Hence the number of parts between 1 and 2 is one part and -1 and -2 is one part. However, these parts may be subdivided into more parts to get more fine value (smaller value).

This value will be a fractional value between 1 and 2 or -1 and -2. Hence parts between two numbers on a number line can be infinity, but for sake of simplicity, one is taken in general case.

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Therefore , the solution of the given problem of unitary method  comes out to be the height of the table in the scale model because we lack any measurements to base our calculations on.

Divide the measures of just this microsecond portion by two in order to complete the task using the unitary variable technique. Briefly stated, the characterised by a group and colour subgroups are both removed from the unit method when a wanted item is present. For example, 40 pens subset with a changeable price would cost Rupees ($1.01). It's possible that one country will have total influence over the approach taken to accomplish this. Almost every living creature has a distinctive quality.

Here,

We could use the scale factor to determine the height of the table in the scale model if we knew the measurements of the actual table and the scale model. The scale factor is the ratio of the actual object's dimensions to the scale model's dimensions. For instance, if the scale factor is 1:12 and the actual table is 48 inches tall, the scale model table would be 12 inches tall.

12 times 48 inches, or 4 inches,

However, we are unable to calculate the height of the table in the scale model because we lack any measurements to base our calculations on.

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The y-intercept of f(x)=−3(x−2)(x+1)(x+4)  is 0,the y-intercept of f(x)=x2(x−1)3(x−4)  is 0.

f(x)=−3(x−2)(x+1)(x+4)End Behavior:  The end behavior of this function is that as x approaches negative infinity, the function approaches negative infinity, and as x approaches positive infinity, the function approaches positive infinity.

Model Degree: This is a cubic function, as the highest degree of the function is 3. Leading Coefficient: The leading coefficient of this function is -3. End Behavior (using limit notation):

lim x→-∞ f(x) = -∞
lim x→+∞ f(x) = +∞
y-intercept: The y-intercept of this function is 0.

f(x)=x2(x−1)3(x−4)End Behavior: The end behavior of this function is that as x approaches negative infinity, the function approaches negative infinity, and as x approaches positive infinity, the function approaches positive infinity.

Model Degree: This is a quartic function, as the highest degree of the function is 4. Leading Coefficient: The leading coefficient of this function is 1. End Behavior (using limit notation):
lim x→-∞ f(x) = -∞
lim x→+∞ f(x) = +∞

y-intercept: The y-intercept of this function is 0.

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Each student gets 1/3 of the juice.

What is Division?

One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division. The additional operations are multiplication, addition, and subtraction.

When no more full chunks of the size of the second number can be allocated during the computation of the quotient, the division with remainder or Euclidean division of two natural numbers yields an integer quotient, which is the number of times the second number is entirely contained in the first number, and a remainder, which is the portion of the first number that remains.

Each student gets 1/3 of the juice.

Equation: 1/3 x 5 = 5/3 or 1 2/3 liters per student.

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To find the zeros, set the function equal to 0.  In other words, solve f(x)=0.

             f(x) = 0

x^2 +6x + 8 = 0

 (x+4)(x +2) = 0

x = -4 and x = -2

The vertex will happen when x = -b/2a:

    [tex]x=\dfrac{-6}{2(1)} = -3[/tex]
Then use this x-value of –3 to find the y-value of the vertex:

   y = (-3)^2 + 6(-3) + 8 = 9-18+8 = -1

The vertex is (-3,-1).

Side note: The x-value of the vertex can also be found by finding the average of the two zeros:

    (-4 + (-2)) / 2 = -6/2 = -3