Practice & Problem Solving
Identify the domain and range

{(2, 1), (4, 2), (6, 3), (8, 4), (10,5)}

Answer:

Step-by-step explanation:

On solving the provided function, we can say that altitude of the given isosceles triangle is 6.63 unit.

A triangle with two sides that have the same length is said to be isosceles. Let's carry out a little exercise to better comprehend this. Fold the corners of a rectangular piece of paper in half.

An isosceles triangle has the following characteristics: two equal sides and two equal angles. The two equal sides are referred to as the legs of the triangle, and the angle between them as the vertex angle or apex angle.T he base and the apex angle are divided by the perpendicular from the apex angle. The isosceles triangle is divided into two congruent triangles by the perpendicular, which is sometimes referred to as its line of symmetry.

[tex]12 ^2=10^2 +(a/2)^2\\ 144 -100 = a^2\\ 44 = a^2\\ a^2 =44 ; a =\sqrt{44} =6.63\\[/tex]

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On the basis of their sides, we may similarly determine the areas of various shapes like rectangles, parallelograms, triangles, and any polygon.

The space the thing occupies is known as the area. Any shape can call this area home. We solely take into account the side's length when calculating the square's area. A square's area is equal to the sum of its squared sides since all of its sides are equal.

Similar to triangles, we can determine the area of any polygon based on its sides, as well as the areas of other forms like rectangles, parallelograms, and triangles. The radius of the surface, or the separation of the outer line from the axis for curved surface objects, is used to compute the area of the surface.

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Answer:

Step-by-step explanation:

Paul's Pizzeria: y = 15x

Leo's Pizzeria: y = 12x

Therefore, Paul's Pizzeria takes in more money per pizza <3

The area of the flag is 32 in²  

The right-angled triangle that has two equal sides is known as the Isosceles right triangle. In other words, if an Isosceles triangle has two equal sides then the triangle is said to be an Isosceles right triangle.  

The formula for the Area of an Isosceles right triangle is given by  

Where x is the equal sides of the triangle.

Here we have

The equal side of a right-angle triangled Flag is 8 inches

=> As we know Area right-angle triangle =  x²/2      

Area of the flag = 8²/2

= 64/2 = 32 inch²    

Therefore,

The area of the flag is 32 in²  

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The expression given in question is not a polynomial because it could not satisfy the criteria for polynomial.

The algebraic expression that contains two or more terms having different power of the same variables along with subtraction, addition or multiplication sign is termed as polynomial.

such as x² - 4x+7

In the above expression, there are different power of same variable x,

or x³+ 3x²+x-5

The terms are separated by either addition or subtraction sign.

We know that a polynomial expression contains different terms in where there is different power of same variables. The highest power is called the degree of polynomial.

As example, 4x³+2x²-3x+2

we have highest power 3 so this is called 3-degree polynomial.

We stated earlier that an expression can be said a polynomial whenever two or more terms which contain numbers and variables along with addition or subtraction operator. From the question we have a term 3xA² that contain a number and two variable x and A. There is no addition or subtraction sign. So, the term could not satisfy the criteria for polynomial. It is called monomial.

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Each corresponding side of the two triangles is equal, we can conclude that △abc ≅ △dbc.

Required:

1. Show that angles ∠A and ∠D are equal.

2. Show that angles ∠B and ∠D are equal.

3. Show that sides AB and BD are equal.

Not Required:

1. Show that angles ∠C and ∠D are equal.

2. Show that sides AC and CD are equal.

To show that △abc ≅ △dbc,

we need to use the Side-Side-Side (SSS) Congruence Theorem. This theorem states that if three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.

Therefore, to show that △abc ≅ △dbc, we have to show that each corresponding side of the two triangles is equal. Additionally, we have to show that the angles opposite of the corresponding sides are equal. This can be done using the Angle-Angle-Side (AAS) Congruence Theorem. This theorem states that if two angles and the side between them in one triangle are equal to the same two angles and the side between them in another triangle, then the two triangles are congruent.

To show that △abc ≅ △dbc, we must show that angles ∠A and ∠D are equal, angles ∠B and ∠D are equal, and sides AB and BD are equal. The other steps (showing that angles ∠C and ∠D are equal and sides AC and CD are equal) are not required, since they are implied by the equal angles and sides already established.

Once all of these equalities have been established, we can conclude that △abc ≅ △dbc.

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Answer:

5040 ways

Step-by-step explanation:

We need to place 3 particular books so that they are always together.

let 3 books are one books

we have 5 books

these can be arranged in ⁵P5 = 5! = 12O

total books = 3 × 2 = 6

Required no. of books are 120×6 = 720.

Answer:

8.5 mph

Step-by-step explanation:

First find how far Hobbit traveled in 4 hrs (8AM - Noon)

3 mph x 4 hr = 12 miles

Now find Gandalf's speed:

Galdalf traveled the same 12 miles that Hobbit did PLUS 5 more miles in 2 hours (10AM -Noon) for a total of 17 miles

Speed = 17 mi / 2 hr = 8.5 mph

When dividing 456.3 by 100, how should the decimal point be moved is:

Decimals are numbers that have two components, a whole number component and a fractional component, which are separated by a decimal point. 12.5 is a decimal number, for instance.

In mathematics, the decimal system—also known as the Hindu-Arabic number system or the Arabic number system—is a positional numeral system that uses 10 as its base and calls for the use of 10 different numbers, including 0, 1, 2, 3, 4, 5, 6, 7, and 8. To indicate decimal fractions, a dot (decimal point) is also necessary.

We use decimals on a daily basis when dealing with things like money, weight, and length. Decimal numbers are used when higher accuracy is required than full numbers can provide.

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Angles are 6 types.

1.Acute Angle: An acute angle is any angle that is less than 90 degrees and is between 0 degrees and 90 degrees.

2.Obtuse Angle: Opposite of an acute angle is an obtuse angle. In other terms, an obtuse angle is one that is larger than 90 degrees and less than 180 degrees. It is the angle that is between 90 degrees and 180 degrees.

3. Right Angle: A right angle is always 90 degrees in length. Acute angles are those that are less than 90 degrees, whereas obtuse angles are those that are higher than 90 degrees.

4. Straight Angle: A straight angle has a measured angle of 180 degrees. Since the angle between its arms is 180 degrees, you can see that it is simply a straight line.

5.Reflex angle: The arms make an acute angle because this measurement is less than 90 degrees. What about the angle on the opposite side, though? What is the name of the larger angle that complements the acute angle? It is referred to as a reflex angle.

A reflex angle is any angle having a dimension higher than 180 degrees but less than 360 degrees (which corresponds to 0 degrees).

6.Complete Rotation: Entire rotation or full angle refers to an angle with a value of 360 degrees. When one of the arms extends completely, it is created.

Angles are 6 types.

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The values of the expressions are P 9, 3 = 504, C 9, 3 = 84, P 8, 8 = 40320 and C 9, 9 = 1

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

(A) P9, 3. (B) C9, 3. (C) P8, 8. (D) C9, 9.

The above expressions are permutation and combination expressions

Using the above as a guide, we have

Expression (A) P9, 3

Apply the permutation formula

P n, r = n!/(n - r)!r!

So, we have

P 9, 3 = 9!/6!

Evaluate

P 9, 3 = 504

Expression (B) C 9, 3

Apply the combination formula

C n, r = n!/(n - r)!r!

So, we have

C 9, 3 = 9!/(6! * 3!)

Evaluate

C 9, 3 = 84

Expression (C) P8, 8

Apply the permutation formula

P n, r = n!/(n - r)!r!

So, we have

P 8, 8 = 8!/0!

Evaluate

P 8, 8 = 40320

Expression (D) C 9, 9

Apply the combination formula

C n, r = n!/(n - r)!r!

So, we have

C 9, 9 = 9!/(9! * 0!)

Evaluate

C 9, 9 = 1

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Part A: The common difference is 4.

Part B: The 4th term is 17 and the 5th term is 21  of this sequence.

What do you mean by arithmetic sequence?

The difference between each term in an arithmetic sequence is a constant. To put it another way, we just add the same value repeatedly, indefinitely.

According to the given question,

We have the given information:

The arithmetic sequence:

5, 9, 13,...

Now, we need to find the common difference and we will find it in this way,

The common difference is the difference between two consecutive terms in a sequence. To find it, we just need to do simple subtraction.

d = t2 - t1

= 9-5 = 4

So, the common difference is 4.

Then, we need to find the 4th and 5th terms of this sequence,

t4 = t3 + d

= 13+4

= 17

t5 = t4 + d

=17 + 4

=21

Therefore, the 4th term is 17 and the 5th term is 21.

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The common difference of this arithmetic sequence is 4 and 4th, 5th term of this arithmetic sequence is 17 and 21.

Each phrase in an arithmetic sequence increases by adding or subtracting a specific constant, k. Each word in a geometric sequence is raised by either multiplying by or dividing by a certain constant, k.

Part A:

The constant difference between consecutive terms of an arithmetic sequence is called the common difference and its is denoted by d.

Therefore, the common difference (d) is equal to 9 - 5 = 4.

The 4th and 5th terms of this sequence are

Part B:

Using Arithmetic progression formula,

aₙ = a + (n- 1)d

where

n = nth term

a (first term = 5

d = common difference = 4

4th term of the sequence

a₄ = a + (n- 1)d

a₄ = 5 + (4 - 1)4

a₄ = 17

5th term of the sequence

a₅ = a + (n- 1)d

a₅ = 5 + (5 - 1)4

a₅ = 21

Hence the 4th and 5th term of this arithmetic sequence is 17 and 21.

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Answer:

soln:

x/y=1/6/1/3

=1/6×3/1

=1/2

Therefore x/y = 1/2

The required perimeter of the given figure will be 60.84 m.

We have been given the figure which is made up of a rectangle and half a circle.

We have to determine the perimeter of the given figure.

As per the given figure, the required solution would be as:

The required perimeter = 2 × (rectangle's length) + rectangle's width + circumference of half a circle

The required perimeter = 2 × 15 + 12 + π × 6

The required perimeter = 30 + 12 + 3.14 × 6

The required perimeter = 30 + 12 + 18.84

Apply the addition operation, and we get

The required perimeter = 60.84 m

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Stem-and-leaf plots contain original data values where histograms do not. Therefore, option B is the correct answer.

The "stem" values are listed down, and the "leaf" values go right (or left) from the stem values.

The "stem" is used to group the scores and each "leaf" shows the individual scores within each group.

The advantage of a stem leaf diagram is it gives a concise representation of data. The advantage of a frequency histogram is, that it is visually strong. Histograms are usually preferable to stem and leaf diagrams in large data sets. The disadvantage of a stem leaf diagram is not visual.

Therefore, option B is the correct answer.

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As a result, the test is significant and we can draw this conclusion that the population mean is less than 20.

A test statistic is a statistic used for testing statistical hypotheses. A test statistic, which is a tally of a data set that can be used to conduct the test and boils the data down to a single figure, is a common way to characterize a hypothesis test.

a) Here, the test statistic is calculated as:

Z= [tex]\frac{p'-p}{\sqrt{(p(1-p)/n} }[/tex]

Z=[tex]\frac{19.4-20}{\sqrt{(20(1-20)/50} }[/tex]

Z= -2.12

Therefore -2.12 is the required test statistic value here.

b) Given that this is a one-tailed test, the following p-value was calculated using standard normal tables: p = P(Z -2.12) = 0.0169

The necessary p-value in this case is therefore 0.0169.

c) The test is significant in this case, and the null hypothesis can be rejected because the p-value is 0.0169 0.05, which is the level of significance. This means that there is enough data to conclude that the mean is less than 20.

d) For a significance level of 0.05, we have the following data from normal standard tables:

P(Z < -1.645) = 0.05

Therefore, the following is the rejection criteria in this case: Reject H0 if Z -1.645.

We can reject the null hypothesis here and draw the same conclusion as in the previous section because the test statistic value is -2.12 -1.645. As a result, the test is significant and we can draw this conclusion that the population mean is less than 20.

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Answer:

Each blueberries serving contain 21 calories or the slope of this is 21

Step-by-step explanation:

The slope is rise/run

We see that when the x increase by 1, the y increase by 21, so the slope is 21.

Answer:  f(3) = -1

Step-by-step explanation:

f(x) = -3 |x-1| +5

f(3), we just need to put 3 in for both the x

f(3) = -3 |3-1| +5

f(3) = -9 + 3 +5

f(3) = -1

Answer:

length x width x height or probably like area x height

Step-by-step explanation:

The height of the poll is 40/3 and 13.3ft.

Let us assume that the Height of an object is h.

Let us assume that the shadow of an object is y.

(As the shadow is always proportional to height .)

Thus y∝h

y = kh

As given

when a person 5 ft. tall has a shadow of 6 ft.

h = 5 ft and y = 6 ft

Put in y = kh

6 = 5k

[tex]k=\frac{6}{5}[/tex]

As given

A telephone pole has a shadow of 16 ft.  

Let us assume that the height of the pole is x.

h = x and y = 16 ft

16 = hk       compare the value of k

[tex]k=\frac{16}{x} \\\\\frac{6}{5} =\frac{16}{x} \\\\x=\frac{16*5}{6} \\\\x=\frac{80}{6} \\\\x=\frac{40}{3} ft[/tex]

0r

x=13.3ft

Therefore, The height of the poll is 13.3ft

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The ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent.

To use a congruence criterion in the given situation, you will need to determine which criterion is most appropriate based on the information provided. There are several congruence criteria that can be used to determine whether two triangles are congruent, including the Side-Side-Side (SSS) criterion, the Side-Angle-Side (SAS) criterion, the Angle-Side-Angle (ASA) criterion, and the Angle-Angle-Side (AAS) criterion.

In this case, the ASA criterion might be the most appropriate, as it states that two triangles are congruent if two angles and the included side are congruent. Given that ∠A=∠C=90∘ and AE=CB, you can use the ASA criterion to conclude that triangles AEB and CED are congruent.

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The question is -

Which congruence theorem can be used to prove EB = BD, AE = CB, ∠ A = ∠ C = 90°?

Answer:

Step-by-step explanation:

it's a rectangular box  with dimensions, 17.7 x 6.8 x 6.7 = 806.412

806.4 [tex]ft^{3}[/tex]   to the nearest 10th  

Answer:

A, B

Step-by-step explanation:

Given the system of equations, we should first solve for x and y so we can answer this question:

equation 1: x + y = 4, equation 2: x - y = 2 ⇒ add both equations together

x + y + x - y = 4 + 2 ⇒ combine like terms

2x + 0y = 6 ⇒ divide both sides by 2

x = 3 ⇒ plug into the x + y = 4 equation to solve for y

3 + y = 4 ⇒ subtract 3 from both sides

y = 4 - 3 = 1

Since we know y is the speed of the water current and x is the canoe's still water speed, we know that the water current's speed is 1 mph and the canoe's still water speed is 3 mph

This corresponds with choices A and B

The z-score is 0 and the probability that the sample mean will be greater than 59.8 is 0.5000

From the question, the given parameters about the distribution are

The z-score

The z-score of the data value is calculated using the following formula

z = (x - mean value)/standard deviation

Substitute the given parameters in the above equation

z = (59.8 - 59.8)/18.0

Evaluate

z = 0

The probability

The probability that the sample mean will be greater than 59.8 is then calculated as:

P(x > 59.8) = P(z > 0)

From the z table of probabilities, we have;

P(x > 59.8) = 0.5

Hence, the probability is 0.5000

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Devon's monthly payment will be about 416.6

To make for the life of the loan.

What is interest ?

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. The three types of interest include simple (regular) interest, accrued interest, and compounding interest.

Have given,

He considered to take lone about 75,000 for a 15 year period,

In 1 year he will be paid [tex]\frac{75,000}{15}[/tex] = 5000

Then in 1 month he will be paid [tex]\frac{5000}{12}[/tex] = 416.6

Devon's monthly payment will be about 416.6

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The characteristic polynomial is P(λ) = λ² - 12λ + 27.

The real eigenvalues of the matrix are 3, 9. Answer A.

The full question is in the attachment. The characteristic polynomial from a matrix is P(λ) = |A - λI|

A - λ

[tex]A \:-\: \lambda I \:=\: \left(\begin{array}{cc}6&3\\3&6\end{array}\right) \:-\: \lambda \left(\begin{array}{cc}1&0\\0&1\end{array}\right)[/tex]

[tex]A \:-\: \lambda I \:=\: \left(\begin{array}{cc}6&3\\3&6\end{array}\right) \:-\: \left(\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right)[/tex]

[tex]A \:-\: \lambda I \:=\: \left(\begin{array}{cc}6 \:-\: \lambda&3\\3&6 \:-\: \lambda\end{array}\right)[/tex]

If matrix P [tex]\left(\begin{array}{cc}a&b\\c&d\end{array}\right)[/tex] than determinant of matrix P = |P| = ad - bc

P(λ) = |A - λI

P(λ) = (6 - λ) (6 - λ) - (3 × 3)

P(λ) = 36 - 12λ + λ² - 9

P(λ) = λ² - 12λ + 27

The eigenvalues

P(λ) = 0

λ² - 12λ + 27 = 0

(λ - 3) (λ - 9) = 0

λ - 3 = 0    or   λ - 9 = 0

λ₁ = 3               λ₂ = 9

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The rule for transformed function W(x ) = V(12x) is, W(x) = 1728 x³ - 864 x² and the value of W(2) is 10368 in³.

Transforming a function means that the curve representing the graph "moves left, right, up and down" or "stretches " or "mirrors". For example, the graph of the function f(x) = x² + 3 can be obtained by simply shifting the graph of g(x) = x² up by 3 units. We have , function of Volume of box in cubic inches, V(x) = x³- 6 x² --(1)

also, an another function, W(x) = V(12x) repreent the volume (in cubic feet ) when x is meaured in inches. Now , understand the problem,

We are a function V whose inputs are in inches and whose outputs are in cubic inches . We are provide an another function W whose inputs are in inches and whose outputs are in cubic feet., W(x) = V(12x) makes sense because there are 12 inche in 1 feet. We are asked to write a rule for W and interpret the output for an input. Write the transformed function W(x) ,

W(x) = V(12x)

=> W(x) = (12x)³ - 6 (12x)²

=> W(x) = 1728 x³- 6× 144 x²

=> W(x) = 1728 x³ - 864 x²

Now, calculate W(2)

W(2) = 1728 (2)³ - 864 (2)²

=> W(2) = 13824 - 3456

=> W(2) = 10368 in³

When x is 2 feet , the volume of the is 8100 cubic inches.

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Complete question:

The function V(x)=x3−6x2 repreent the volume (in cubic inche) of the box hown. The function W(x)=V(12x) repreent the volume (in cubic feet) when x i meaured in inches. Write a rule for W. Then find W(2)