The construction below shows two possible triangles that can be formed when
4B - 3 inches and BC = 1.5 inches. Describe what happens to the length of AC
as point C moves counterclockwise around the circle toward point A.

The triangle with side lengths 6, 9, and 12 is an obtuse triangle. To classify the triangle as acute, right, or obtuse, we need to determine the type of triangle based on the lengths of its sides.

Recall that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side for it to be a valid triangle.

Let's check if the given triangle with side lengths 6, 9, and 12 satisfies the triangle inequality:

1. Sum of two shorter sides: 6 + 9 = 15 (greater than the length of the longest side, 12). Valid.

2. Sum of the two longer sides: 9 + 12 = 21 (greater than the length of the remaining side, 6). Valid.

3. Sum of the longest side and the shortest side: 6 + 12 = 18 (greater than the length of the remaining side, 9). Valid.

Since all three inequalities are satisfied, the given side lengths form a valid triangle.

Now, to classify the triangle, we can use the Pythagorean theorem. In a right-angled triangle, the square of the length of the longest side is equal to the sum of the squares of the other two sides.

In this case, the side lengths are 6, 9, and 12. To determine if it is a right-angled triangle, we can check if:

[tex]12^2 = 6^2 + 9^2\\144 = 36 + 81\\144 \neq 117[/tex]

Since the Pythagorean theorem is not satisfied, the triangle is not a right-angled triangle.

Now, let's check if it is an acute or obtuse triangle. In an acute triangle, all three angles are less than 90 degrees, while in an obtuse triangle, one angle is greater than 90 degrees.

To determine this, we can use the law of cosines, which states that in any triangle:

[tex]c^2 = a^2 + b^2 - 2ab * cos(C)[/tex]

where c is the longest side (12 in this case), a and b are the other two sides (6 and 9), and C is the angle opposite the longest side.

Calculate the value of cos(C):

[tex]12^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos(C)\\144 = 36 + 81 - 108 * cos(C)\\144 = 117 - 108 * cos(C)\\108 * cos(C) = 117 - 144\\108 * cos(C) = -27\\cos(C) = -27 / 108\\cos(C) = -0.25[/tex]

Since the value of cos(C) is negative, the angle C is obtuse. Therefore, the triangle is an obtuse triangle.

In summary, the triangle with side lengths 6, 9, and 12 is an obtuse triangle.

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The estimations for the given function are:

For a function y = f(x) we define the maximum as the value of f(c) such that:

f(c) ≥ f(x) for any value of x.

And the minimum as:

f(k) ≤ f(x) for any value of x.

Usin the graph it is easier, just select the value at the top for the maximum and the value at the bottom for the minimum, we can see that:

maximum ---> y  ≈ 120

minimum ---> y ≈ -175

Remember, these are estimations.

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a. To find the speed at which the car achieves maximum fuel economy, we need to find the vertex of the function. The vertex of a quadratic function is given by the formula: (-b/2a, f(-b/2a)). In this case, a = -0.008 and b = 1.4. Substituting these values, we get:

Vertex = (-b/2a, f(-b/2a))

= (-1.4/(2*(-0.008)), f(1.4/(2*(-0.008))))

= (87.5, 6.125)

Therefore, the car should travel at a speed of 87.5 miles per hour to achieve maximum fuel economy.

b. To find the maximum fuel economy, we need to find the value of the function at the vertex. Substituting x = 87.5 in the function, we get:

f(87.5) = -0.008(87.5)^2 + 1.4(87.5) = 6.125

Therefore, the maximum fuel economy of the car is 6.125 miles per gallon.

c. To find the fuel economy of the car if it travels at 60 miles per hour, we need to evaluate the function at x = 60. Substituting x = 60 in the function, we get:

f(60) = -0.008(60)^2 + 1.4(60) = 7.6

Therefore, the fuel economy of the car if it travels at 60 miles per hour is 7.6 miles per gallon.

d. The point (40, 43.2) lies on the graph of the function. This means that when the car travels at a speed of 40 miles per hour, its fuel economy is 43.2 miles per gallon. However, this point does not provide any information about the maximum fuel economy or the speed at which it is achieved.

Answer:

-9 and -5

Step-by-step explanation:

-9 + -5 = -14

-9 * -5 = 45

Answer:

- 9 and - 5

Step-by-step explanation:

- 9 × - 5 = 45 and - 9 + (- 5) = - 9 - 5 = - 14

In a casde whereby saac is mountain climbing with Rina and has just climbed a 6.2-meter vertical rock face the distance from the cliff is Rina standing is 7.85m

Base on the given iformation, following the trigonometry rule it can be seen thatr the hypotenus of the triangle is the distance of Isaac  to Rina which can be expressed a C = 10m

The height that was climbed by I saac =  6.2-m. a= 6.2m

the distance from Rina to cliff = b

Then base on pytagoras theorem

c^2= a^2 + b^2

10^2 = 6.2^2 +  b^2

b^2= 10^2 - 6.2^2

b^2= 61.56

b = √61.56

b=7.85m

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Answer:

the answer is c

Step-by-step explanation:

The correct statement regarding the numeric values of the piece-wise function is given as follows:

D. f(1) = 0.

To obtain the numeric value of a function or of an expression, we substitute each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

A piece-wise function is a function that has different definitions based on the input of the function, hence before obtaining the numeric value we must obtain the correct definition of the function in the interval.

Choosing the correct definitions of the function, the numeric values are given as follows:

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Answer: 64,000

Step-by-step explanation:

The values of the measures are;

Range = 6

Median = 4

Mean = 4.5

To determine the values of the range, median and mean, we need to note that;

The range of a given set of values is the interval between the smallest number and the largest number in the set.

Given the data set;

3, 4, 5, 7, 8, 8, 9

The range = 9 - 3 = 6

The median of a given set of values is the  middle number of the set when arranged in an ascending or descending form.

Given the set;

3, 4, 4, 6, 6,

The median is 4

The mean of a set of numbers is the average value of the numbers.

Given the set;

2, 9, 7, 5, 2, 2,

The mean = 2 + 9 + 7 + 5 + 2 + 2/6 = 27/6 = 4. 5

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The intervals for which the rate of change of the quadratic function, f(x) = -(x + 1)² + 8 is faster than the rate of change of g(x) = -x - 6 are; x ≤ -1.5 and x ≥ -0.5

A quadratic function is a polynomial function with one or more variables in which the highest exponent is two.

The slope of f(x) is; f'(x) = -2·(x + 1)

The slope of g(x) = -1

When the rate of change of f(x) is faster than the rate of change of g(x), we get;

The slope of f(x) > The slope of g(x), therefore;

-2·(x + 1) > -1

From which we get;

|-2·(x + 1)| ≥ |-1|

2·(x + 1) ≥ 1

x + 1 ≥ 1/2

x ≥ 1/2 - 1 = -1/2

x ≥ -1/2 = -0.5

The absolute sign indicates that we get;

x + 1 ≤ -1/2

x ≤ -1/2 - 1 = -1 -1/2 = -3/2 = -1.5

x ≤ -1.5

The interval for which the average rate of change of f(x) is faster than the average rate of change of g(x) are; x ≥ -0.5 and x ≤ -1.5

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[tex] \mathbb{ANSWER:}[/tex]

LCM: 168

HCF: 28

The time the softball will spend in the air is 2 seconds

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

Rate = 2(t - 2)(t + 5)

To determine the time, we set the rate to 0

So, we have

2(t - 2)(t + 5) = 0

Divide by 2

(t - 2)(t + 5) = 0

The above means that

t - 2 = 0 or t + 5 = 0

Evaluate

t = 2 or t = -5

Time cannot be negative

So, we have

t = 2

Hence, the time is 2 seconds

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Each row will have 4 blueberry bushes and 7 raspberry bushes.

What is Greatest common divisor?

Greatest common divisor (GCD) is the largest positive integer that divides two or more numbers without leaving a remainder. In other words, it is the largest positive integer that is a factor of two or more integers.

We are required to find the greatest common divisor (GCD) of 56 and 98.

we can factorize these numbers as follows :

56 = 2 x 2 x 2 x 7

98 = 2 x 7 x 7

The GCD is the product of the common prime factors, raised to the lowest power they appear in either factor:

GCD = 2 x 7 = 14

Therefore, Gemma can plant 14 rows. To find how many blueberry and raspberry bushes will be in each row, we can divide the total number of bushes by the number of rows:

No. of blueberry bushes per row = 56 / 14 = 4

No. of raspberry bushes per row = 98 / 14 = 7

So, each row will have 4 blueberry bushes and 7 raspberry bushes.

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The simplified expressiοn is [tex]-3x^3y^2 - 4x^2y + 8y[/tex].

Mathematical statements are called expressiοns when they cοntain at least twο terms that are cοnnected by an οperatοr and cοntain either numbers, variables, οr bοth. The mathematical οperatοrs can be additiοn, subtractiοn, multiplicatiοn, οr divisiοn. Fοr instance, the expressiοn x + y is an expressiοn where x and y are terms with an additiοn οperatοr in between.

In mathematics, there are twο types οf expressiοns: numerical expressiοns, which οnly cοntain numbers, and algebraic expressiοns, which alsο include variables.

Tο simplify [tex](2x^3y^2-8x^2y+4y) − (5x^3y^2-4x^2y - 4y)[/tex], we need tο distribute the negative sign in frοnt οf the secοnd set οf parentheses, and then cοmbine like terms. This gives:

[tex]2x^3y^2 - 8x^2y + 4y -5x^3y^2 + 4x^2y + 4y[/tex]

[tex]= (2x^3y^2 - 5x^3y^2) + (-8x^2y + 4x^2y) + (4y + 4y)[/tex]

[tex]= -3x^3y^2 - 4x^2y + 8y[/tex]

Therefοre, the simplified expressiοn is [tex]-3x^3y^2 - 4x^2y + 8y[/tex].

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Answer:

its C −3x3y2 − 4x2y + 8y

Step-by-step explanation:

Answer:

Alice is correct because area of shaded region of both the gameboards are equal.

Step-by-step explanation:

Here it is given that both the gameboards are of same shape and size and Alice stated that the area of shaded regions of both the gameboards are same.

Let us take the length of the board be "l" and its breadth be "b" .

Initial area of the gameboard will be lb since it is a rectangle.

1st case :-

breadth of the shaded region= b

length of the shaded region = l/2 (since the length becomes half )

Area of the shaded region = b*l/2 = lb/2 .

2nd case :-

breadth of the shaded region = b/2 ( since breadth becomes half. )

length of the shaded region = l

Area of the shaded region = l/2

Area of the shaded region = b/2*l = lb/2 .

As we can see that the area of shaded region in both shaded region in both cases are equal also it is half of the total area of the rectangle. Hence Alice was correct.

Hence the area of the shaded regions in both the cases are similar and half of board in case is shaded .

Sarah is incorrect, the sum will be a rational number, the first option is the correct one.

We say that some sets are closed under the addition if, when we add two elements of that set, the outcome also belongs to that set.

For example, the set of real numbers is closed under addition, and also happens for the case of rational numbers.

If we add two rational numbers, the outcome must be rational.

In this case the two numbers are rational, thus, the outcome of the sum will be rational, not irrational, so Sarah is incorrect. The correct option is the first one.

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The value of the a  in the mixed form is [tex]-1 \frac{5}{43}[/tex].

What is the equivalent expression?

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

To solve for a, we can begin by isolating the variable on one side of the equation.

First, we can subtract 3 from both sides to get:

-7/12 - 3 = 4/a

Next, we can simplify the left side by finding a common denominator of 12:

-7/12 - 3 * 12/12 = 4/a

-7/12 - 36/12 = 4/a

-43/12 = 4/a

To solve for a, we can multiply both sides by a/4:

(-43/12) * (a/4) = 1

Multiplying the left side gives:

-43a/48 = 1

Dividing both sides by -43/48 gives:

a = -48/43

To express this as a mixed number, we can divide the numerator by the denominator:

-48 ÷ 43 = -1 with a remainder of 5

Therefore, the value of the a  in the mixed form is [tex]-1 \frac{5}{43}[/tex].

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Complete question:

Challenge) Solve for a. Answer as a mixed number. A=?

-7/12 = 4/a+3.

The distance οn the map that represents οne mile is 5/8 inch.

Tο find οut hοw much distance οn the map represents οne mile, we need tο use a ratiο οf the distance οn the map tο the actual distance.

Let x be the distance οn the map that represents οne mile. Then we have:

1/2 inch : 4/5 miles = x : 1 mile

We can crοss-multiply tο sοlve fοr x:

(1/2 inch) × (1 mile) = (4/5 miles) × x

Simplifying the right-hand side:

(1/2) × 1 = (4/5) × x

1/2 = (4/5) × x

Multiplying bοth sides by 5/4:

(1/2) × (5/4) = x

5/8 = x

Therefοre, the distance οn the map that represents οne mile is 5/8 inch.

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Answer: 306

Step-by-step explanation:

Each child has 34 stickers and 9 bottles. To find the total number of stickers, we need to multiply the number of stickers per child by the number of children.

34 stickers x 9 children = 306 stickers

So there are 306 stickers in total.

Answer:

x = -5y+9/4

Step-by-step explanation:

4x + 5y = 9

↓

(4x + 5y) + (-5y) = 9 + (-5y)

↓

4x + 5y - 5y = 9 - 5y

↓

4x = -5y + 9

↓

4x/4 = -5y + 9/4

↓

x = -5y + 9/4

Answer: x = 9/4 - 5y/4

Step-by-step explanation: Move all terms that don't contain x to the right side and solve.

FO = 18. In order to solve this problem, we must first understand the concept of triangle trigonometry.

Trigonometry is the branch of mathematics that focuses on the relationships between angles and sides of triangles. It is based on the measurement of triangles and the calculation of the angles and lengths of their sides.

A triangle is a two-dimensional shape composed of three straight lines. When the three lines of a triangle meet at a single point, it is called a vertex. Each line of the triangle is called a side, and the angle between two sides is called an angle.

In this problem, we are given a triangle AFOX with one right angle at point O and an angle m/F of 40°. We also know that FX = 15. To solve for FO, we must use the Law of Sines which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is equal in all triangles.

Using this law, we can determine that FO = (15/sin40°) = 18. Therefore, the answer to the question is FO = 18.

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Answer:

C.

Step-by-step explanation:

The y-intercept is the cost of a one time fee for renting bowling shoes.

Answer:

it's B Or D Id say it's D

Step-by-step explanation:

It's D because it's definitely not a drink so not A and when you read y is something that it's not shoes so the most one I think is correct is D

Answer:

Step-by-step explanation:

[tex]\tt z + 2 < 8[/tex]

Subtract 2 from both sides:-

Simplify :-

Therefore, the solution of the inequality z + 2 < 8 is z < 6.

_______________________

Hope this helps!

The group of rotations and symmetries of a geometric configuration is the group of isometries.

According to the above paragraph, "the group of rotations and symmetries of the configuration" refers to the group of isometries of a geometric configuration that preserve the configuration itself, without involving any translations. This group consists of rotations about a fixed point and symmetries about a straight line that pass through that point.

This group is formed by combining isometries of the configuration through the operation of composition, and it satisfies certain properties, such as closure, associativity, identity, and inverses. It is a useful tool for studying the symmetries and transformations of the configuration, and it is used in fields such as crystallography and quantum mechanics.

In summary, the group of rotations and symmetries of a geometric configuration is the group of isometries that preserve the configuration and can be composed with each other to obtain new isometries.

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Answer:

The rocket splashes down after [tex]\boxed{24}[/tex] seconds

The rocket peaks at [tex]\boxed{769}[/tex] meters above sea-leval

Step-by-step explanation:

The relationship between height and time is given by the equation
[tex]h(t) = -4.9t^2+112t+129[/tex]

The splashdown will occur when [tex]h(t) = 0[/tex]

Therefore solve t for
[tex]-4.9t^2+112t+129 = 0[/tex]

This is a quadratic equation which can be solved with the aid of a quadratic formula calculator

The solution set for this equation is
[tex]t=-1.09894 \;and\;\:t=23.95609\right)[/tex]

Omitting the negative value we get
[tex]\text{t=23.95609 \;seconds}\\\\= 2\text{4 \;seconds \;rounded}}[/tex]

For the second part, the maximum value of h(t) can be found by finding the first derivative h'(t) and setting it equal to 0 and solving for t; then plug this value of t into h(t) to find the max height

[tex]h'(t) = \dfrac{d}{dt}\left(-4.9t^2+112t+129\right)\\\\= \dfrac{d}{dt}\left(4.9t^2\right)+\dfrac{d}{dt}\left(112t\right)+\dfrac{d}{dt}\left(129\right)\\\\= - 9.8t + 112 + 0\\\\= -9.8t + 112\\\\[/tex]

Setting  the above expression to 0 gives
[tex]-9.8t+112=0\\\\-9.8t = -112\\\\t = \dfrac{-112}{-9.8}\\\\t = 11.4286 \;seconds[/tex]

Plugging this value back into h(t) gives

[tex]h(11.4286) = -4.9\cdot \:11.4286^2+112\cdot \:11.4286+129\\\\= 769 \;meters \;(rounded)[/tex]

Answer:

Step-by-step explanation:

There are different methods to approach this problem, but one possible way is:

First, find the number of weeks in a century: 100 years × 52 weeks/year = 5200 weeks

Then, find the fraction of a century that is 13 weeks by dividing 13 weeks by 5200 weeks: 13/5200 = 1/400

Therefore, 13 weeks is 1/400 of a century.