The ages of four children are 14, 12, 15, and 17
Work out the range of the ages of the four
children.

The smallest distance it has to go to get back to the starting tree is [tex]3\sqrt{5}[/tex]when traveling 6 miles to the south before turning 3 miles to the east.

According to Pythagoras's Theorem, the square of the hypotenuse side in a right-angled triangle is equal to the sum of the squares of the other two sides. Perpendicular, Base, and Hypotenuse are the names of this triangle's three sides. The Pythagorean Theorem demonstrates how to calculate the side lengths of a right triangle by adding the areas of three intersecting squares. As the foundation for more intricate trigonometry theories like the Pythagorean theorem's opposite, this theorem is a very helpful tool.

given

by pythagoras theorem :-

shortest distance = [tex]c^{2} = a^{2} + b^{2} \\[/tex]

[tex]c^{2} = 6^{2} + 3^{2} \\c^{2} = 36 + 9\\c^{2} = 45\\c = 3\sqrt{5}[/tex]

The smallest distance it has to go to get back to the starting tree is [tex]3\sqrt{5}[/tex]when traveling 6 miles to the south before turning 3 miles to the east.

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Answer:

b: Two thousand four hundred and forty (2440)

c: 1.33 x 10^27

Step-by-step explanation:

I'm uncertain by what an ordinary number is, but to calculate the answer for part b, you need to divide the diameter of Mercury by 2, which would give the aforementioned answer.

As for part c, you will need to subtract the masses of Jupiter and Saturn:
mass of Jupiter - mass of Saturn = 1.33 ×[tex]10^{27}[/tex]

A  range of values which is not a possible for scale factor (k): k = 0

We know that the scale factor is the number (also called the conversion factor) which is used to change the size of a figure without changing its shape.

It is used to expand or compress the size of an object.

The scale factor formula is:

Scale factor = Dimensions of the transformed shape ÷ Dimensions of the original shape

If the scale factor k > 1, the object is enlarged.

If the scale factor is 0< k <1, the object is compressed.

If the scale factor is k = 1, the shape of object remains the same.

And the scale factor cannot be zero.

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Answer:

2nd choice:  -3x - 5 = 10

Step-by-step explanation:

10 - x = -5

-x = -5 - 10 = -15

-3x - 5 = 10

-3x = 10 + 5

x = 15/-3 = -5

The y-intercept of the sine function is -3

We know that the standard form of the sine function is, y = a sin(bx + c) + d

where, amplitude = |a|

period = 2π/b

phase shift = -c/b

and vertical shift = d

Here, a sine function has an amplitude of 3, a period of π, and a phase shift of π/4

a = 3,

So, 2π/b = π

⇒ b = 2

And phase shift -c/b = π/4

⇒ -c/2 = π/4

⇒ -c = π/2

⇒ c = -π/2

So, a sine function would be,

y = 3 sin(2x + -π/2) + 0

y = 3 sin(2x + -π/2)

When x = 0,

y = 3 sin(0+ -π/2)

y = 3 sin(-π/2)

y = 3 × (-1)

y = -3

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Answer:

-1, 0, 4

Step-by-step explanation:

2f - 8 ≤ 6f + 4

-4f ≤ 12

f ≥ 12/-4  **flip the inequality when you divide by a negative number

f≥ -3

Check all the answers that are ≥ -3

''By taking log'' on both side Mike solve this equation.

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 expression is,

⇒ 37ˣ = 12

Now,

Since, The expression is,

⇒ 37ˣ = 12

Solve this expression as;

⇒ 37ˣ = 12

Taking log both side, we get;

⇒ log 37ˣ = log 12

⇒ x log 37 = log 12

⇒ x = log 12 / log 37

⇒ x = 1.08 / 1.57

⇒ x = 0.69

Thus, The value of x = 0.69

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The actual solution must be approximated to the tenth place.

How do you solve an equation to the nearest tenth?

To round the decimal number to its nearest tenth, look at the hundredth number. If that number is greater than 5, add 1 to the tenth value. If it is less than 5, leave the tenth place value as it is, and remove all the numbers present after the tenth place.

To estimate the tenth place, you have to determine the actual solution and the approximate.

Assume the solution to the system of equations is:

(x, y), (1.25, 1)

1.25 is to the hundredth place.

So, the next thing is to approximate 1.25 to 1.3 (the tenth place).

So, we have:

(x, y), (1.3, 1)

Hence, the actual solution must be approximated to the tenth place.

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Step-by-step explanation:

a. 5a²c-3a²d + 5b²c3b²d l can't this question but l can calculate a. 5a²c-3a²d + 5b²c-3b²d.

I this your question is a little worng.

The partial fraction expansion of (x³ + 4)/ ((x² - 1)(x² + 3x + 2)) is -3/ 4(x + 1) + 5/12(x - 1) + 4/3(x + 2) - 3/ 2(x + 1)². So the coefficients A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2.

A partial fraction expansion is a way of expressing a rational function (a function that can be written as the ratio of two polynomials) as the sum of simpler fractions, each with a numerator that is a constant or a simple polynomial. The process of finding the partial fraction expansion of a rational function is also known as partial fraction decomposition.

The denominator is

(x² - 1)(x² + 3x + 2) = (x² - 1)(x² + x + 2x + 2)

(x² - 1)(x² + 3x + 2) = (x + 1)(x - 1)(x + 1)(x + 2)

So by  partial fraction expansion

(x³ + 4)/ ((x² - 1)(x² + 3x + 2)) = (x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2))

(x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2)) = A/ (x + 1) + B/(x - 1) + C/(x + 2) + D/ (x + 1)²

A(x - 1)(x + 1)(x + 2) + B(x + 1)²(x + 2) + C(x + 1)²(x - 1) + D(x - 1)(x + 2) = x³ + 4

(A + B + C)x³ + (2A + 4B + C + D)x² + (-A + 5B - C + D)x + (-2A + 2B - C - 2D) = x³ + 4

Thus from coefficients of x³, x², x and 1,

A + B + C = 1

2A + 4B + C + D = 0

-A + 5B - C + D = 0

-2A + 2B - C - 2D = 4

Solving

A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2

--The question is not clear, the question is given below--

"Enter (A,B,C,D) in order below if A, B, C, and D are the coefficients of the partial fractions expansion of

[tex]$$12\cdot\frac{x^3 + 4}{(x^2-1)(x^2 +3x+ 2)}[/tex]  "

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The possible values of x for the given triangle ABC are 0.75<x<8.

The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.

Given that, m∠A < m∠B < m∠C.

a) From triangle ABC,

BC<AC<AB

2x+3<6x<5x+8

Now, 2x+3<6x

Subtract 2x on both the sides of an inequality, we get

3<4x

Divide 4 on both the sides of an inequality, we get

0.75<x

Here, 6x<5x+8

Subtract 5x on both the sides of an inequality, we get

x<8

So, possible values of x are 0.75<x<8

b) From triangle ABC,

BC<AC<AB

3x+1<4x+8<7x+2

Now, 3x+1<4x+8

Subtract 3x on both the sides of an inequality, we get

1<x+8

Subtract 8 on both the sides of an inequality, we get

-7<x

Here, 4x+8<7x+2

Subtract 4x on both the sides of an inequality, we get

8<3x+2

Subtract 2 on both the sides of an inequality, we get

6<3x

Divide 3 on both the sides of an inequality, we get

2<x

So, possible values of x are -7<x<2

Therefore, the possible values of x for the given triangle ABC are 0.75<x<8.

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If 1 inch represents 1.5 feet, we can use this conversion factor to find the length of the actual airplane in feet. Thus 12 inches of model airplane represents an actual length of the actual airplane of 18 feet.

Therefore the answer is 18 feet.

To convert the length of the model airplane (12 inches) to feet, we can multiply it by the conversion factor (1 inch = 1.5 feet)

= 12 inches × 1.5 feet/inch

= 18 feet

So the length of the actual airplane is 18 feet.

It is important to pay attention to the units of measurement when performing conversions and make sure to use the correct conversion factor. And also it's a good practice to always check your answer to make sure it makes sense in the context of the problem.

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The volume of the water walking ball is 43.71 m³

Surface Area of a Sphere. A = 4 π r2. The volume of a Sphere. V = (4 ⁄ 3) π r³. Where r is the radius of the circle

Given :

[tex]$V=\frac{1}{6\sqrt{\pi}}S^{3/2}$[/tex] and surface area of water walking ball is  60 m²

putting the values in the equation we get

V= 464.758/10.632

V=43.71

Hence, the volume of the water walking ball is 43.71 m³

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The solution for given differential equation is y = c1e^(-x^3/3) + c2xe^(-x^3/3)

This is a second-order linear differential equation with constant coefficients. To solve this equation, we can use the method of an auxiliary equation, which is 3r^2 + 6r + 1 = 0. This equation has two complex conjugate roots, r = -3 ± i√3. Therefore, the solution of the differential equation is y = c1e^(-x^3/3) + c2xe^(-x^3/3)cos(√3x^2/2) + c3xe^(-x^3/3)sin(√3x^2/2), where c1, c2, and c3 are arbitrary constants.

The method of an auxiliary equation is a popular way to solve second-order linear differential equations with constant coefficients. It involves finding the roots of an auxiliary equation, which is found by setting the coefficient of the second derivative equal to zero.

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Answer:

-2,2

Step-by-step explanation:

No, the sine ratio of an acute angle cannot be greater than 1. The sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Since the side opposite the angle is shorter than the hypotenuse, the ratio will always be less than one.

To what does sin equate?

In contrast to the cosine of an angle, which corresponds to the ratio of the nearby side to the hypotenuse, the sine of an angle is the ratio of the opposite side to the hypotenuse.

It's important to note that if you were to give an angle value greater than 90 degrees, the sine of the angle will be a negative value, but the ratio will still be less than one.

The sine ratio is a value between -1 and 1, and it is impossible for the sine of an acute angle to be greater than 1.

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I think it's answer this .

The correct equation which represent an identity will be;

⇒ 4(5x+2) = 20x+8

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

Given that;

All the equations are,

1) 2x + 1 = 3x + 4

2) 4x - 3 = 2(2x + 7)

3) 6x + 3 = 2x + 10

4) 4(5x + 2) = 20x + 8

Now,

Solve each equation and find an identity as;

1) 2x + 1 = 3x + 4

⇒ 1 - 4 = 3x - 2x

⇒ - 3 = x

⇒ x = -3

Thus, It is not represent an identity.

2) 4x - 3 = 2(2x + 7)

⇒ 4x - 3 = 4x + 14

⇒ - 3 ≠ 14

Thus, It shows no solution.

3) 6x + 3 = 2x + 10

⇒ 6x - 2x = 10 - 3

⇒ 4x = 7

⇒ x = 7/4

Thus, It is not represent an identity.

4) 4(5x + 2) = 20x + 8

⇒ 20x + 8 = 20x + 8

⇒ 20x - 20x = 8 - 8

⇒ 0 = 0

Clearly, '0' is an additive identity.

Thus, This equation shows an identity.

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Please find attached the graph showing the translation transformation of the triangle ΔABC with vertices A(2, 1), B(-1, 2), and C(-2, -1) using the rule (x, y) → (x - 1, y - 3), to produce triangle ΔA'B'C' with vertices A'(1, -2), B'(-2, -1), and C'(-3, -4), created with MS Excel

A translation transformation is one in which the location of the points on the original figure slides by the same amount and in the same direction to form the image.

The vertices of the triangle ΔABC are; A(2, 1), B(-1, 2), and C(-2, -1)

The specified transformation is; (x, y) → (x - 1, y - 3)

Applying the specified transformation, we get;

A(2, 1) ⇒ (x, y) → (x - 1, y - 3) ⇒ A'(2 - 1, 1 - 3) = A'(1, -2)

B(-1, 2) ⇒ (x, y) → (x - 1, y - 3) ⇒ B'(-1 - 1, 2 - 3) = B'(-2, -1)

C(-2, -1) ⇒ (x, y) → (x - 1, y - 3) ⇒ C'(-2 - 1, -1 - 3) = C'(-3, -4)

The vertices of the image ΔA'B'C' of the triangle ΔABC following the transformation, (x, y) → (x - 1, y - 3) are therefore;

A'(1, -2), B'(-2, -1), C'(-3, -4)

Please find attached the graph of the triangle ΔABC and its image ΔA'B'C' after the translation (x, y) → (x - 1, y - 3), created with MS Excel

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[tex]\begin{array}{|c|c|} \cline{1-2}\text{Class} & \text{Frequency}\\\cline{1-2}20-29 & 5\\\cline{1-2}30-39 & 12\\\cline{1-2}40-49 & 9\\\cline{1-2}50-59 & 6\\\cline{1-2}60-69 & 3\\\cline{1-2}\end{array}[/tex]

=====================================================

Explanation:

The given data set is

{37, 60, 35, 38, 59, 34, 56, 42, 55, 48, 39, 33, 36, 35, 43, 52, 39, 64, 41, 35, 20, 31, 29, 45, 32, 54, 66, 44, 47, 45, 27, 24, 48, 28, 55}

It sorts to

{20, 24, 27, 28, 29, 31, 32, 33, 34, 35, 35, 35, 36, 37, 38, 39, 39, 41, 42, 43, 44, 45, 45, 47, 48, 48, 52, 54, 55, 55, 56, 59, 60, 64, 66}

The five values 20,24,27,28,29 are between 20 and 29. This means the first class or bin will have a frequency of 5.

The next interval spans from 30 to 39. There are 12 items in this interval, and those 12 items are this subset {31, 32, 33, 34, 35, 35, 35, 36, 37, 38, 39, 39}. This means we have a frequency of 12 for the interval from 30 to 39.

Keep this process going until you get to the end of the data set.

Perimeter is the total length of all sides of a shape, also known as the circumference.

Perimeter is a measure of the distance around a two-dimensional shape. It is the sum of the lengths of all the sides of a shape. Perimeter is also known as the circumference. When calculating perimeter, you measure the length of each side of the shape and add them together. For example, the perimeter of a square with sides of 4 inches each would be 16 inches. Perimeter is an important concept in geometry, and it can be used to calculate area, the space within a shape or figure. It can also be used to calculate the distance between two points. Knowing the perimeter of a shape can help you solve many problems in math, design, and science.

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The equivalent quadratic equation to  (x - 2)(x + 9) = 0 is x² + 7x - 18 = 0.

A quadratic equation is an algebraic expression in the form of variables and constants.

A quadratic equation has two roots as its degree is two.

We know if α and β are the roots of a quadratic equation ax² + bx + c = 0,

Then we can construct the equation by the factors (x - α)(x - β).

Given, We have to construct a quadratic equation with (x - 2)(x + 9) = 0.

Now, Multiply the terms we have,

x² + 9x - 2x - 18 = 0.

x² + 7x - 18 = 0.

So, Our required quadratic equation is x² + 7x - 18 = 0.

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In the late 19th century, the Russian press used the phrase 2 + 2 = 5 to describe the moral confusion of social decline at the turn of a century, because political violence characterised much of the ideological conflict among proponents of humanist democracy and defenders of tsarist autocracy in Russia

Therefore , the solution to this problem of equation is y = 16.95  cost for 94 minutes.

A mathematical equation is a formula that links two statements by indicating their equivalence with the equal sign (=). A mathematical statement that proves the equality of two mathematical expressions is known as an equation in algebra. For instance, the components 3x + 5 and 14 in the equation 3x + 5 = 14 are separated by an equal sign. The relationship between two sentences on either side of a letter is expressed mathematically. There is frequently only one variable, which doubles as the symbol. say that 2x - 4 Equals 2.

Here,

y = mx + b

(50,13.98)(78,17.06)

slope = (17.06 - 13.98) / (78 - 50) = 3.08 / 28 = 0.11

slope(m) = 0.11

use either of ur points...(50,13.98)...x = 50 and y = 13.98

now sub into the formula and find b, the y int

13.98 = 0.11(50) + b

13.98 = 5.5 + b

13.98 - 5.5 = b

8.48 = b

so our equation is : y = 0.11x + 8.48.......for 94 minutes...sub in 77 for x

y = 0.11(94) + 8.48

y = 8.47 + 8.48

y = 16.95 <=== this is the cost for 94 minutes

Therefore , the solution to this problem of equation is y = 16.95  cost for 94 minutes.

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There is no upper limit if the ratio raises x to a higher power. The limit is 0 if the ratio causes the power of x to be reduced to zero.

A ratio is an expression for an ordered pair of numbers, a and b, where b is not equal to zero and the expression is written as a/b. A ratio is created by combining two ratios. Assuming a ratio of one boy to three girls, this would suggest that 3/4 of the population is female and only 1/4 is male. A part can be an amount, a percentage, or the difference between two or more objects.

Here,

The ratio of, say, the highest-degree words serves as the upper bound as x grows.

There is no upper bound on how much the ratio can raise the power of x. The limit is 0 if the ratio causes the power of x to be reduced to zero.

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Therefore, the height of the ball after four seconds is 448 feet.

Height refers to the vertical distance between something's top and bottom or its base and anything above it. Height is used to describe everything that may be measured vertically, high or low.

Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria.

We need to plug t = 4 into the equation h = -16t^2 + 176t.

h = -16(4^2) + 176(4)

h = -16(16) + 704

h = -256 + 704

h =  448 feet.

So, 448 feet is the height after four seconds.

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Answer:

see explanation

Step-by-step explanation:

if the denominator of a rational expression is zero then the expression will be undefined.

the numerator is the part of the rational expression that makes it zero.

solve the numerators in each to find values of x

1

[tex]\frac{x+6}{x-4}[/tex]

x + 6 = 0 ( subtract 6 from both sides )

x = - 6 ← value that makes expression equal to zero

2

[tex]\frac{(x+4)(x-2)}{x+6}[/tex]

(x + 4)(x - 2) = 0

equate each factor to zero and solve for x

x + 4 = 0 ⇒ x = - 4

x - 2 = 0 ⇒ x = 2

x = - 4 and x = 2 make the expression equal to zero

3

[tex]\frac{2x+10}{3x-12}[/tex]

2x + 10 = 0 ( subtract 10 from both sides )

2x = - 10 ( divide both sides by 2 )

x = - 5 ← value that makes expression equal to zero

Answer:

1)  x = -6

2)  x = -4  and  x = 2

3)  x = -5

Step-by-step explanation:

A rational expression is undefined when the denominator equals zero.

A rational expression equals zero when the numerator equals zero.

Given rational expression:

[tex]\dfrac{x+6}{x-4}[/tex]

Set the numerator to zero and solve for x:

[tex]\implies x+6=0[/tex]

[tex]\implies x=-6[/tex]

Given rational expression:

[tex]\dfrac{(x+4)(x-2)}{x+6}[/tex]

Set the numerator to zero:

[tex]\implies (x+4)(x-2)=0[/tex]

Apply the zero-product property and solve for x:

[tex]\implies x+4=0 \implies x=-4[/tex]

[tex]\implies x-2=0 \implies x=2[/tex]

Given rational expression:

[tex]\dfrac{2x+10}{3x-12}[/tex]

Factor the numerator and denominator:

[tex]\dfrac{2(x+5)}{3(x-4)}[/tex]

Set the numerator to zero and solve for x:

[tex]\implies 2(x+5)=0[/tex]

[tex]\implies x+5=0[/tex]

[tex]\implies x=-5[/tex]

The solution of the function f(x) = 2(x - 3)² is at x = 3 as shown in the graph.

An equation is used to show the relationship between numbers and variables.

Polynomial can be classified based on degree as linear, quadratic, cubic.

Given the function:

f(x) = 2(x - 3)²

The graph of f(x) is plotted. The solution of the graph can be gotten from its x intercept. Hence the solution is at x = 3

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Plotting method is the graphical representation of mathematical expression.

A plot is a graphical representation of a list of data. Besides, a plot is the graphical analysis of mathematical expressions and equations because plot can show the relationship between two or more variables. There are many plotting methods such as hand plotting method, computer aided plotting method and so on.

At first, we need to choose the coordinate system by which we plot the point.

suppose we choose a XY coordinate plane. At first identify the horizontal or X axis and vertical or Y axis.

besides, we need to determine the required value that will be plotted on the plan.

if we have a list of X and Y value then find the value of X on the x axis and next find the value of Y on the Y axis.

we will notice some points for every pair of x and y value. Finally, we will join the points to achieve a required shaped graph.

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