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Answer:

2x² - 20x - 32 = 0

Step-by-step explanation:

Trip upstream:

distance = 10 km

time = t

current = 4 km/h

speed of boat in still water = x

speed upstream: x - 4

speed = distance/time

distance = speed × time

10 = (x - 4) × t

10 = tx - 4t

t(x - 4) = 10

t = 10/(x - 4)

Trip downstream:

Distance = 10 km

time = 2 - t

current = 4 km/h

speed downstream = x + 4

speed = distance/time

distance = speed × time

10 = (x + 4) × (2 - t)

10 = 2x - tx + 8 - 4t

10 = 2x - 10x/(x - 4) + 8 - 40/(x - 4)

10(x - 4) = 2x(x - 4) - 10x + 8(x - 4) - 40

10x - 40 = 2x² - 8x - 10x + 8x - 32 - 40

2x² - 20x - 32 = 0

Answer: i think c or d

Step-by-step explanation:

Answer:

the answer would be D

Step-by-step explanation:

Positive curve graphs will be upward-sloping.

Negative curve graphs will be downward-sloping.

Both graphs in the first image attached below, show curves sloping upward from left to right. As with upward-sloping straight lines, we can say that generally the slope of the curve is positive. While the slope will differ at each point on the curve, it will always be positive.

In the graphs in the second image attached, both of the curves are downward sloping. Straight lines that are downward sloping have negative slopes; curves that are downward sloping also have negative slopes. The slopes change from point to point on a curve, but all of the slopes along these two curves will be negative.

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Straight lines that are downward sloping have negative slopes; curves that are downward sloping also have negative slopes. The slopes change from point to point on a curve, but all of the slopes along these two curves will be negative.

The value of k is -3, if x-1 is a factor of 4x³+3x² −4x + k

Quadratic Equation:

A quadratic equation is a second-order algebraic equation in x. The quadratic equation in standard form is ax2 + bx + c = 0. where a and b are the coefficients, x is the variable, and c is the constant term. The first condition of the quadratic equation is that the coefficients of x² are non-zero terms (a ≠ 0). To write a quadratic equation in standard form, first write the x² term, then the x term, and finally the constant term. The numbers for a, b, and c are generally written as whole numbers, not as fractions or decimals.

According to the Question:

Let P(x) = 4x³+3x² −4x + k

Given,  (x−1) is a factor of P(x)

x−1 = 0

x= 1

∴ p(1) = 4×(1)³+3×(1)² −4×(1)+k = 0

⇒ 4+3−4+k = 0

⇒ k+3=0

⇒ k=−3

Hence, the value of k is −3.

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The formula used to calculate the solving the volume of the rectangular prism is equal to ( length × width × height ).

As given in the question,

Given base of the prism is rectangular in shape ,

Area of the base with rectangular base is given by :

area of the rectangle = length × width

Volume of the rectangular prism = base area × height

⇒Volume of the rectangular prism = ( length × width ) × height

Therefore, the formula required to calculate the volume of the rectangular prism is given by ( length × width × height ).

The above question is incomplete , the complete question is:

What is the formula in solving volume of the rectangular prism?

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Answer:

X=log 3.2/5 log 21

Step-by-step explanation:

X=0.0764094

The complete table for the pie chart is;

Type of Film             Frequency           Angle in °

Action                            4                           72°

Horror                           11                            198

Comedy                         5                            90

From the given table used to form the pie chart, we see that;

Type of Film             Frequency           Angle in °

Action                            4                           72°

Horror                           11                            

Comedy                         5

We know that the angles in a circle is equal to 360 degrees. Thus;

Angle composed of horror and comedy = 360 - 72 = 288°

Thus;

Angle for horror film = (11/(4 + 11 + 5)) * 360°

= (11/20) * 360°

= 198°

Thus;

Angle for comedy = 288° - 198° = 90°

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Using basic geometry and area formula for calculation of length, Apply fencing on only 3 side , The answer is 109.54 feet.

A subfield of mathematics called geometry examines the dimensions, placements, angles, and sizes of objects.

A plane shape, as opposed to a square, with four straight sides and four right angles, particularly one with uneven neighboring sides.

Area of rectangle is:

A= l x b

area=750 square feet.

for attaining minimum area , these area must be fitted in square,

so, [tex]l = \sqrt {750}[/tex]

=27.38 feet

since, one side is taken by barn, l should be extended 4 times, 2 on 1 side and 2 on sides each.

Total length of fence = 4 * 27.38

= 109.54 feet

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Answer:

64, 2

Step-by-step explanation:

Numbers approach negative infinity as they move left on the number line and positive infinity as they move right on the number line. Negative infinity is always less than any number, positive infinity is always greater than any number.

Main Difference:

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The rational root theorem is the required to represents rational root or zeros or the solution of the given polynomial function with example: g(x) = 4x² - 4x + 1.

As given in the question,

Therefore, the rational root theorem is the required to find the rational root or the solution or the zero of the polynomial function.

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Answer:

Step-by-step explanation:

Let's call the two numbers x and y, where x is the smaller number and y is the larger number. We are given that x and y differ by 3, which means y - x = 3. We are also given that the product of x and y is 88, which means x * y = 88. We can solve this system of equations by substituting the second equation into the first equation:

y - x = 3

y = x + 3

Substituting this expression for y into the second equation, we get:

x * (x + 3) = 88

x^2 + 3x - 88 = 0

This is a quadratic equation that we can solve using the quadratic formula:

x = (-3 +/- sqrt(3^2 - 41(-88))) / (2*1)

= (-3 +/- sqrt(9 + 352)) / 2

= (-3 +/- sqrt(361)) / 2

= (-3 +/- 19) / 2

= -11/2 or 8/2

Therefore, the two numbers are x = -11/2 and y = 8/2, or x = -5.5 and y = 4. These are the solutions to the original system of equations.

The space occupied by the flat surface of a cylinder's base, also known as the base area, is quantifiable.

Given,

Base area of cylinder;

The amount of space that a flat surface at the base of a cylinder occupies is known as the base area. Since the cylinder's base is a circle, the base area of the cylinder is actually an area of a circle.

Formula for base area;

The cylinder's base area is calculated by multiplying the square of its radius by π.

Then,

"πr²" is the formula for the cylinder's base area with radius r.

That is,

Base Area of a Cylinder = (π × radius²) square units.

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Complete question is here;

What is the base area B of cylinder?

The value of the game would be 13.33 dollars.

What is probability?

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a proposition is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Value of the game = E(x)

                               [tex]= [(60\times\frac{5}{6}) + (-10\times\frac{1}{6})]\\\\=50 -1.66\\\\= 48.33[/tex]

P(Plain surface) = 2/6 and

P(Not plain surface) = 1 - 2/6 = 2/3

Let x: Value won/lost in the game

Value of the game

                          [tex]= [(60\times\frac{2}{6}) + (-10\times\frac{2}{3})]\\\\=20 -6.66\\\\= 13.33[/tex]

Hence, the value of the game would be 13.33 dollars.

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Answer:

No

Step-by-step explanation:

5+10<17

Using the concept of proportions, the 20 feet pole will cast an 8ft shadow

Proportion is a mathematical comparison between two numbers.  According to proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.

In this problem, we can find the length of the shadow casted if we write a proportional relationship between the two object.

50 / 20 = 20 / x

Cross multiply both sides and solve for x

50 × x = 20 × 20

50x = 400

x = 400 / 50

x = 8

The shadow  is 8ft

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All equilateral triangles are similar.

In an equilateral triangle, all the lengths of the sides are equal.

In such a case, each of the interior angles will have a measure of 60°

Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle.

A property of equilateral triangles includes that all of their angles are equal to 60°

The AAA Triangle Similarity Postulate asserts that if all of the angles in a triangle are equal to the corresponding angles in another triangle, then those two triangles are equal.

Since every equilateral triangle’s angles are, 60°every equilateral triangle is similar to one another due to this AAA Postulate.

=> For congruence of triangles, we need equal sides for two equilateral triangles, but here it has said nothing about sides of both triangles in Congruent. So, this is incorrect. Here, it can be said that all equilateral triangles are similar.

Hence, all equilateral triangles are similar.

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Any equation that does not contain a discontinuity, defined as a point where the graph of the equation has a sudden break or jump, is considered continuous. Examples of continuous equations include linear equations, polynomial equations, and exponential equations.

Continuity is an important concept in mathematics and related disciplines. A continuous equation is defined as any equation that does not contain a discontinuity, which is a point where the graph of the equation has a sudden break or jump. Discontinuous equations are equations with discontinuities that cause the graph of the equation to have sudden jumps or breaks. Linear equations, polynomial equations, and exponential equations are all examples of continuous equations, since they do not contain any discontinuities. Continuous equations are important because they are used in many areas of mathematics, such as calculus, to solve problems and understand the behavior of certain function

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x (x - 1) (x + 1) = [tex]x^{3}[/tex]- x

Distribute

1. (x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x

2. Distribute

   (x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x

    [tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

3. Multiply by 1

   [tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

4. Distribute

  [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

  [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x

5. Distribute

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x = [tex]x^{3 }[/tex] - x

6. Combine like terms

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x =  [tex]x^{3 }[/tex]- x

   [tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x

7. Move terms to the left side

   [tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x

  [tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0

8. Distribute

 [tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0

 [tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0

9. Combine like terms

  [tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0

   - x + x = 0

10. Combine like terms

    - x + x = 0

     0 = 0

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The graph of quadratic inequality when plotted represents the parabola.

The quadratic inequality will have a range of solutions, unlike the quadratic equation.

The quadratic inequality could be solved using factorization method for example

The quadratic inequality x²+6x+5 ≤0 can be solved by factorizing but instead of two solutions, there are a range of solutions.

For given equation we need to calculate value of x such that the graph is less than zero.

Hence solution to this inequality can be written as

−5≤x≤−1.

And the graph of given quadratic inequality represents the parabola.

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The expression representing the height of the box is the difference between the length and width of the box, which is equal to x subtracting two, or x – 1.

The expression representing the height of the box can be determined by subtracting the width of the box from the length. The length of the box is x 6, and the width is x – 2. To calculate the height, we subtract the width from the length: x – (x – 2) = x – x + 2 = 2. Therefore, the expression representing the height of the box is x – 1. This indicates that the height of the box is equal to the length of the box minus one. This calculation can be applied to any rectangular prism to calculate the height.The volume of the box is represented by the function f(x) = x3 5x2 – 8x – 12. To calculate the volume, we can substitute the length and width into the function:

f(x) = (x – 6)(x – 2)(x – 1) = x3 – 8x2 + 11x – 12.

Therefore, the volume of the box is x3 – 8x2 + 11x – 12.

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Answer:

3,4,5,6,7.

Step-by-step explanation:

if yk, yk

Either Pythagorean Theorem or the Law of Cosines is used to solve for y in a triangle, you can use either of these equations.

The Pythagorean Theorem states that the sum of the squares of the two shorter sides of the triangle is equal to the square of the longest side. This is written as:

                                            a2 + b2 = c2,

where a and b are the lengths of the two shorter sides, and c is the length of the longest side. In this equation, y is the length of the longest side.

The Law of Cosines states that:

                                          a2 = b2 + c2 - 2bc cos A

where A is the angle opposite side a. In this equation, y is side a.

Therefore, to solve for y in a triangle, you can use either the Pythagorean Theorem or the Law of Cosines, depending on the information given.

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Considering the given data which include the domain and the range the possible value of n is

The possible value of n will be n such a way that both the domain and the range are not altered

for a domain of 1, 2, 3, 4, 5, 6, 7, 8 using n, 3n will have a range of

1, 6, 9, 12, 15, 18, 21, 22, 24,

The number 24 is outside the range which is 3, 4, 5, 6, 7, 8,,,,,,,20, 22

n = 8 gave the value of 24 hence 8 cannot be the value of n

n is 1, 2, 3, 4, 5, 6, and 7

written in the form: 1 ≤ n ≤ 7

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Sheila measured the water level in the swimming pool every week for 4 weeks and found that the water level decreased by 8 millimeters in the first week, 5 millimeters in the second week, 12 millimeters in the third week and 5 millimeters in the fourth week. The net result was a loss of 10 millimeters of water over the 4 week period.

Sheila wanted to track the changes in the water level of a swimming pool over time. She measured the level every week for 4 weeks. In the first week, the water level dropped by 8 millimeters. In the second week, the level dropped by 5 millimeters. In the third week, the level dropped by 12 millimeters. Finally, in the fourth week, the level dropped by 5 millimeters. The net result of these measurements was a loss of 10 millimeters of water over the course of the 4 week period. This indicates that the hot and dry spell had a negative effect on the water level in the pool.

Total Water Loss = 8 millimeters + 5 millimeters + 12 millimeters + 5 millimeters = 30 millimeters

Total Water Gain = 0 millimeters

Net Loss = Total Water Loss - Total Water Gain = 30 millimeters - 0 millimeters = -10 millimeters

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The given content is an example of sweepstakes.

What is a sweepstake?

A sweepstake is a type of contest in which a prize or prizes are given to the winner or winners. Sweepstakes began as a type of lottery tied to the sale of a product. Sweepstakes became strictly "No purchase necessary to enter or win" and "A purchase will not increase your chances of winning" under these laws, especially because many sweepstakes companies skirted the law by stating only "no purchase necessary to enter," removing the consideration (one of the three legally required elements of gambling) to prevent sweepstakes abuse.

Based on the above definition,

we can conclude that the given question is an example of a sweepstake.

Hence, this is an example of sweepstakes.

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We simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.

The collection of all potential inputs for a function is its domain.

For instance, the domain of f(x)=x² and g(x)=1/x are all real integers with the exception of x=0.

The set of numbers x for which a function has a valid definition is known as its domain.

A closed interval is referred to as a function's domain if it is specified over the interval a ≤ x ≤ b.

In order to identify the values of the independent variable x and acquire the domain, we simply solve the equation y = f(x).

Simply put, x=g(y) will calculate the function's range after we identify g(y).

Therefore, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.

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The quadratic equations are

a) x² + 5x - 6 = 0  ( the roots are x = -6 , x = 1 )

b) x² - 5x + 6 = 0  ( the roots are x = 2 , x = 3 )

c) x² + 2x - 3 = 0  ( the roots are x = -3 , x = 1 )

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation in the standard form is ax² + bx + c = 0

Now , the value of A is

a)

Let the quadratic equation be x² + 5x - 6 = 0

Now , on simplifying the equation , we get

x² + 5x - 6 = 0

x² - x + 6x - 6 = 0

Taking the common factors in the equation , we get

x ( x - 1 ) + 6 ( x - 1 ) = 0

On factorizing the equation , we get

( x + 6 ) ( x - 1 ) = 0

So , ( x + 6 ) = 0 or ( x - 1 ) = 0

And , x = -6 or x = 1

So , the roots of the quadratic equations are x = -6 or x = 1

b)

Let the quadratic equation be x² - 5x + 6 = 0

Now , on simplifying the equation , we get

x² + 5x + 6 = 0

x² + 2x + 3x - 6 = 0

Taking the common factors in the equation , we get

x ( x + 2 ) + 3 ( x + 2 ) = 0

On factorizing the equation , we get

( x + 3 ) ( x + 2 ) = 0

So , ( x + 3 ) = 0 or ( x + 2 ) = 0

And , x = -3 or x = -2

So , the roots of the quadratic equations are x = -2 or x = -3

c)

Let the quadratic equation be x² + 2x - 3 = 0

Now , on simplifying the equation , we get

x² + 2x - 3 = 0

x² - x + 3x - 3 = 0

Taking the common factors in the equation , we get

x ( x - 1 ) + 3 ( x - 1 ) = 0

On factorizing the equation , we get

( x + 3 ) ( x - 1 ) = 0

So , ( x + 3 ) = 0 or ( x - 1 ) = 0

And , x = -3 or x = 1

So , the roots of the quadratic equations are x = 1 or x = -3

Hence , the quadratic equations are solved

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The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle. So it was proved.

The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle.

If angle ad, side ac-df, and angle cf are present, then ABCDEF is true.

To support: By the angle sum property, ABC DEF A+B+C=180° ______ (1)

In DEF D+E+F=180°___________ (2) (By angle sum property)

From (1) and (2), A+B+C= D+E+F A+E+F= D+E+F

(Given B=E and C=F) A=D ____________________ (3)

Currently, in ABC and DEF A=D (from (3))

AC = DF (Given)

C=F (Give (AAS congruency)

So it was proved.

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