_______ 2. Find the length of PQ

A. 4.5π units
B. 9π units
C. 54π units
D. 12π units

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Let's solve ~

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {a}^{2} - 64 }{ {a}^{2} - 10a + 24} \sdot \cfrac{ {a}^{2} - 12a + 36 }{ {a}^{2} + 4a - 32} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{( {a}^{} + 8)(a - 8) }{ {a}^{2} - 6a - 4a+ 24} \sdot \cfrac{ {a}^{2} - 6a - 6a + 36 }{ {a}^{2} + 8a - 4a- 32} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{( {a}^{} + 8)(a - 8) }{ {a}^{} (a - 6) - 4(a - 6)} \sdot \cfrac{ {a}^{} (a - 6) - 6(a - 6) }{ {a}^{}(a + 8) - 4(a + 8)} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{( {a}^{} + 8)(a - 8) }{(a - 6) (a -4)} \sdot \cfrac{ (a - 6) (a - 6) }{ {}^{}(a - 4)(a + 8)} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{(a - 8) }{(a -4)} \sdot \cfrac{ (a - 6) }{ {}^{}(a - 4)} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{(a - 8)(a - 6) }{(a -4) {}^{2} } [/tex]

Or [ in expanded form ]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {a}^{2} - 8a - 6a + 48 }{ {a}^{2} - 8a + 16 } [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {a}^{2} -14a + 48 }{ {a}^{2} - 8a + 16 } [/tex]

Answer:

[tex]\dfrac{(a-8)(a-6)}{(a-4)^2}[/tex]

Step-by-step explanation:

Given expression:

[tex]\dfrac{a^2-64}{a^2-10a+24} \cdot \dfrac{a^2-12a+36}{a^2+4a-32}[/tex]

Factor the numerator and denominator of both fractions:

[tex]\textsf{Apply the Difference of Two Squares formula} \:\:\:x^2-y^2=(x-y)(x+y):[/tex]

[tex]\begin{aligned} a^2-64 & =a^2+8^2 \\ & =(a-8)(a+8)\end{aligned}[/tex]

[tex]\begin{aligned}a^2-10a+24 & =a^2-4a-6a+24\\& = a(a-4)-6(a-4)\\ & = (a-6)(a-4) \end{aligned}[/tex]

[tex]\begin{aligned}a^2-12a+36 & =a^2-6a-6a+36\\& = a(a-6)-6(a-6)\\ & = (a-6)(a-6) \end{aligned}[/tex]

[tex]\begin{aligned}a^2+4a-32 & =a^2+8a-4a-32\\& = a(a+8)-4(a+8)\\ & = (a-4)(a+8) \end{aligned}[/tex]

Therefore:

[tex]\dfrac{(a-8)(a+8)}{(a-6)(a-4)} \cdot \dfrac{(a-6)(a-6)}{(a-4)(a+8)}[/tex]

[tex]\textsf{Apply the fraction rule}: \quad \dfrac{a}{b} \cdot \dfrac{c}{d}=\dfrac{ac}{bd}[/tex]

[tex]\dfrac{(a-8)(a+8)(a-6)(a-6)}{(a-6)(a-4)(a-4)(a+8)}[/tex]

Cancel the common factors (a + 8) and (a - 6):

[tex]\dfrac{(a-8)(a-6)}{(a-4)(a-4)}[/tex]

Simplify the numerator:

[tex]\dfrac{(a-8)(a-6)}{(a-4)^2}[/tex]

Answer: 2

Step-by-step explanation:

Each of the two sides of the equation is set equal to y

Step-by-step explanation:

g(x) is also called y. the functional result value for an input value x is shown in the y direction above or below that x value on the x axis.

so, when it says g(x) = c, the solution is to find the values of x for which b the function g delivers c as result.

now, we look at the graphic.

for example, g(x) = 0 (c = 0) has actually 3 solutions, as the function crosses the x-axis (and that means y or g(x) = 0) 3 times. so, there are 3 different values of x that create g(x) = 0.

but we are looking for a c that has only one solution (one value of x to create that result).

c = 1 ?

let's imagine a horizontal line through y = 1.

we see, it cuts the function also 3 times. 3 solutions.

c = -1 ?

a similar horizontal line through y = -1 cuts the function also 3 times. 3 solutions

but c = 3 !

a horizontal line through y = 3 stays above all the waves of the function and then cuts the function only one time at the very right. 1 solution !

and that is therefore our answer.

The value of P(4<X≤12) is 0.45

The complete question is added as an attachment

To calculate P(4<X≤12), we use the following equation

P(4<X≤12) = P(4<X≤8) + P(8<X≤12)

From the histogram, we have:

P(4<X≤8) = 0.1

P(8<X≤12) = 0.35

So, we have:

P(4<X≤12) = 0.1 + 0.35

Evaluate

P(4<X≤12) = 0.45

Hence, the value of P(4<X≤12) is 0.45

Read more about probability at:

https://brainly.com/question/10444337

#SPJ1

Answer:

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

We need to make a stem and leaf plot

Rearrange the data in the ascending order:

Group the data as follows:

The first two digits form a stem and the last digits form a leaf.

Make a plot:

       Stem   |  Leaf

[tex]direct \: variation \: lines \: pass \: through \: the \: origin[/tex]

By adding fractions, we conclude that the correct option is the third one:

"one and two sevenths tubes"

We know that:

The total amount of paint will be given by the direct addition of these two fractions, notice that the denominator is the same in both cases, so we can just add the numerators.

5/7 + 4/7 = (5 + 4)/7 = 9/7

We can rewrite this as:

9/7 = 7/7 + 2/7 = 1 + 2/7

Thus, the correct option is the third one

"one and two sevenths tubes"

If you want to learn more about fractions:

https://brainly.com/question/11562149

#SPJ1

The curve pass through the y-axis at the coordinate point (0, 10) showing that the y-intercept of the function is (0, 10)

The graph of a polynomial function is a smooth continuous curve. The point where the curve intersects the x-axis is the zero of the polynomial.

A polynomial is also known to have a degree of 3 and above. Hence the given polynomial has a leading degree of 3 and expressed as:

f(x)=−x^3+x^2+9x−9

The graph of the function is as plotted below. From the function, you can see that the curve pass through the y-axis at the coordinate point (0, 10) showing that the y-intercept of the function is (0, 10)

Learn more on polynomial graph here: https://brainly.com/question/10918240

#SPJ1

Values of c and d make the equation true are c=6, d=2

Equations

                                [tex]\sqrt[3]{162x^{c}y^{5} } = 3x^{2} y^{3} \sqrt[3]{6y^{d} }[/tex]

                               [tex](\sqrt[3]{162x^{c}y^{5} })^{3} = (3x^{2} y^{3} \sqrt[3]{6y^{d} })^{3}[/tex]

                                 [tex]{162x^{c}y^{5} } = (3x^{2} y^{3} \sqrt[3]{6y^{d} })^{3}[/tex]

                                  [tex]{162x^{c}y^{5} } = 27x^{6} y^{3} ({6y^{d})[/tex]

                                     [tex]{x^{c}y^{5} } = x^{6} y^{3} ({y^{d})[/tex]

                                     [tex]{x^{c}y^{5} } = x^{6} y^{3+d}[/tex]

                                   c = 6

                                   d = 2

To learn more about equations from the given link

https://brainly.com/question/14751707

#SPJ4

The first five terms of the sequence with the given nth term are 1, 3/2, 3/2, 9/8, 27/40.

In this question,

The nth term of the sequence is [tex]a_n=\frac{3^{(n-1)} }{n!}[/tex]

Now substitute the values of n = 1,2,3,4,5

For n = 1,

⇒ [tex]a_1=\frac{3^{(1-1)} }{1!}[/tex]

⇒ [tex]a_1=\frac{3^{0} }{1}[/tex]

⇒ a₁ = 1

For n = 2,

⇒ [tex]a_2=\frac{3^{(2-1)} }{2!}[/tex]

⇒ [tex]a_2=\frac{3^{(1)} }{(1)(2)}[/tex]

⇒ [tex]a_2=\frac{3}{2}[/tex]

For n = 3,

⇒ [tex]a_3=\frac{3^{(3-1)} }{3!}[/tex]

⇒ [tex]a_3=\frac{3^{(2)} }{(1)(2)(3)}[/tex]

⇒ [tex]a_3=\frac{9}{6}[/tex]

⇒ [tex]a_3=\frac{3}{2}[/tex]

For n = 4,

⇒ [tex]a_4=\frac{3^{(4-1)} }{4!}[/tex]

⇒ [tex]a_4=\frac{3^{(3)} }{(1)(2)(3)(4)}[/tex]

⇒ [tex]a_4=\frac{27}{24}[/tex]

⇒ [tex]a_4=\frac{9}{8}[/tex]

For n = 5,

⇒ [tex]a_5=\frac{3^{(5-1)} }{5!}[/tex]

⇒ [tex]a_5=\frac{3^{(4)} }{(1)(2)(3)(4)(5)}[/tex]

⇒ [tex]a_5=\frac{81}{120}[/tex]

⇒ [tex]a_5=\frac{27}{40}[/tex]

Hence we can conclude that the first five terms of the sequence with the given nth term are 1, 3/2, 3/2, 9/8, 27/40.

Learn more about sequence here

https://brainly.com/question/14147167

#SPJ4

Answer:

[tex]y = \sqrt{x+25} + 5[/tex]

Step-by-step explanation:

Swap x and y

x = y² – 10y

Add 25 to each side to complete the square

x + 25 = y² – 10y + 25

Factor

x + 25 = (y – 5)²

Take the square root of each side

[tex]\sqrt{x+25}[/tex] = y – 5

Add 5 to both sides

[tex]\sqrt{x+25}[/tex] + 5 = y

Answer:

1. Given.

2. <5 is supplementary to <4.

2. Angles 5 and 4 are consecutive angles, and by the consecutive angles theorem, the two are supplementary.

3. m<4 + m<5 = 180

3. By the definition of supplementary angles, m<4 + m<5 = 180.

4. m<2 = m<4

4. Angles 2 and 4 are corresponding angles, and by the corresponding angles postulate, the two are congruent. By the definition of congruent angles, m<2 = m<4.

5. m<5 + m<2 = 180

5. Substitution property of equality.

The probability of Maria selecting 2 questions on Science is 1/ 5

From the information given, we have that there are;

The total questions = 2 + 4 + 4 = 10 questions

We have that Maria picks two questions at random

The probability of her selecting two questions on Science is given as

P(2 science) = number of Science questions/ total number of questions

P(2 science) = 2/ 10

P(2 science) = 1/ 5

The probability of picking two questions on Science is 1/5

Thus, the probability of Maria selecting 2 questions on Science is 1/ 5

Learn more about probability here:

https://brainly.com/question/24756209

#SPJ1

Answer:

1/45

Step-by-step explanation:

Answer:

  Order of exponentiation can be swapped here, and the fractional power written as a root.

Step-by-step explanation:

The commutative property of multiplication and the rules of exponents apply.

The rule is ...

  (a^b)^c = a^(bc)

The given expression can be rearranged to ...

  [tex]\left(3^\frac{1}{5}\right)^2=3^{\frac{1}{5}\cdot2}=3^{2\cdot\frac{1}{5}}=\left(3^2\right)^\frac{1}{5}\\\\=9^\frac{1}{5}=\boxed{\sqrt[5]{9}}[/tex]

A number line ranges from minus 5 to 5 with increment of 1 unit. Left, Point F is plotted on the number line at minus 3. Right, point G is plotted on the number line at 1.50, then the distance between F and G is 4.50.

Calculating the Distance between F and G:

It is given that, on a number line,

Point F is located at -3

Point G is at 1.50

Since, 1.5 > (-3) and each number is at a 1 unit difference form the adjacent number on the number line,

The difference between F and G = 1.50 - ( -3 )

This implies that the distance between F and G = 1.50 + 3

= 4.50 units

The numbers lying between the points F and G on the number line are: -2, -1, 0, and 1

Learn more about a number line here:

https://brainly.com/question/13189025

#SPJ1

Answer: 4 1/2

Step-by-step explanation:

Got it right

The mean of the data set A is 4.58 while the mean of data set B is 5.

An equation is an expression that shows the relationship between two or more numbers and variables.

The mean of a set of numbers is the average of that group of number. It is the ratio of the sum of the data to the total number of data.

Mean of data set A = [(3 * 1) + (4 * 3) + (5 * 2) + (6 * 2) + (7 * 3) + (8 * 1)] / (1 + 3 + 2 + 2 + 3 + 1) = 4.58

Mean of data set B = [(4 * 2) + (5 * 2) + (6 * 2)] / ( 2 + 2 + 2) = 5

The mean of the data set A is 4.58 while the mean of data set B is 5.

Find out more on equation at: https://brainly.com/question/2972832

#SPJ1

By finding some points on the curve and then connecting them.

Here we have the rational function:

t = d/s

Where t represents time, d distance, and s speed.

We assume that d is a fixed value, then our variables are time and speed.

On the examples, you can see that the value of d is:

d = 616.5km, then we just need to graph the equation:

t = (616.5km)/s

To do so, you can evaluate the function in different values of s, like the points the problem asks to graph, and then connect all these points with a curve.

Doing that, you should get a graph like the one below:

If you want to learn more about rational functions:

https://brainly.com/question/1851758

#SPJ1

Answer:

  (-∞, ∞)

Step-by-step explanation:

Unless there are restrictions on the domain, the range of any odd-degree polynomial function is "all real numbers."

The given function is a linear function, degree 1. This is an odd degree, so the range of the function is "all real numbers."

  -∞ < f(x) < ∞

  (-∞, ∞) . . . . . in interval notation

The element value c₂₄ = -12 and d₁₃ = 14 if C = 2A + 3B and A and B are the matrix.

To find the element:

As we have given two matrices A and B:

Therefore, the element value c₂₄ = -12 and d₁₃ = 14 if C = 2A + 3B and A and B are the matrix.

Know more about matrix here:

https://brainly.com/question/24079385

#SPJ4

C) Company B: Using linear regression models from both data sets, she determines that it pays about $10 more.

What is linear regression?

Linear regression analysis is used to predict the value of a variable based on the value of another variable.

The variable you want to predict is called the dependent variable.

The variable you are using to predict the other variable's value is called the independent variable.

Learn more about linear regression here: https://brainly.in/question/14924595

Amelia should work for (C) company B: using linear regression models from both data sets, she determines that it pays about $10 more.

Therefore, Amelia should work for (C) company B: using linear regression models from both data sets, she determines that it pays about $10 more.

Know more about linear regression models here:

https://brainly.com/question/25987747

#SPJ4

The complete question is given below:

If Amelia wants to make the maximum amount of money working only 22 hours per week, which company should she work for? explain your answer.

A) Company A: Using linear regression models from both data sets, she determines that it pays about $14 more.

B) Company A: Using quadratic regression models from both data sets, she determines that it pays about $5 more.

C) Company B: Using linear regression models from both data sets, she determines that it pays about $10 more.

D) Company B: Using exponential regression models from both data sets, she determines that it pays about $8 more.

Answer:

The value of x will be 2.5cm.

Step-by-step explanation:

Volume = Length*Width*Height

where,

[tex]v = l \times w \times h \: ...substitute \: values \\ 120cm {}^{3} = 12cm \times w \times 4cm \\ 120cm {}^{3} = 48cm {}^{2} \times w \: ...divide \: both \: \\ sides \:by \: 48cm {}^{2} \\ \frac{120cm {}^{3} }{48cm {}^{2} } = w \\ w = 2.5cm[/tex]

Answer:

Step-by-step explanation:

x+3y=1

[tex]g(x)=5cos((\pi )/(2)x-(3\pi )/(2))-2[/tex]

generate by:  Amplitude:5    Period:4

                      Phase shift:(3 to the right)    Vertical shift:-2

x=3,g(x)= 3

x=4,g(x)= -2

x=5,g(x)= -5

x=6,g(x)= -2

x=7,g(x)= 3

the graph is like cos(x)

learn more about trigonometric graphs here:

https://brainly.com/question/18265536

#SPJ1

The equation of the line of best fit, using the least square method is v = 2.19t + 6.7, making the 2nd option as the right choice.

In the question, we are asked to find the equation of the line of BEST FIT for the given scatter plot.

We let the time be the independent variable x, and velocity is the dependent variable y.

We find the line of best fit using the least square method.

First, we calculate the mean:

Now, we find the slope of the line of best fit, using the formula:

m = {∑(x - μ(x))(y - μ(y))}/{∑(x - μ(x))²}.

Taking value from the tables, which is attached, we get:

m = 239.35/109.2683333 = 2.190479096.

Now, we calculate the y-intercept of the line of best fit, using the formula:

b = μ(y) - m.μ(x),

or, b = 20.5 - 2.190479096*6.283333 = 6.736489681.

Now, the equation of the line of best fit, in the form of y = mx + b, can be written as, y = 2.190479096x + 6.736489681, which on approximation, and making y as v, the velocity, and x as t, the time, we get: v = 2.19t + 6.7.

Thus, the equation of the line of best fit, using the least square method is v = 2.19t + 6.7, making the 2nd option as the right choice.

Learn more about the least square method at

https://brainly.com/question/15088393

#SPJ1

Answer:

7/10

Step-by-step explanation:

take 3, 2, 1 from the equation and your left with 7 numbers. Add those together you get 10. 7/10 is the probability that you will draw a number greater than 3

After 16 days, the total homework questions that have been assigned would be 80 questions.

Ratio expresses the equivalent relationship between two numbers. It shows the frequency of the number of times that one value is contained within other value(s).

A ratio table can be described as table that shows equivalent values between two numbers.

The first step is to determine the ratio between the number of days and the total number of questions.

Ratio = money saved / money earned

8/10 =  total number of questions / number of days

55 / 11 = 5

Now, multiply the ratio gotten in the previous step by the number of days

Total number of homework questions = number of x ratio

5 x 16 = 80 questions

To learn more about ratios, please check: https://brainly.com/question/25927869

#SPJ1

Pick 2 coordinates:

I choose first & last one (0,15) & (8-9)

Change in y is 24

(15--9 = 24)

Change in x is 8

(0 - 8) = - 8

24/-8 = - 3

So, slope = - 3

Hope this helps!

Answer:

-3

Step-by-step explanation:

The formula to find the slope is (change in y)/(change in x). In other words, rise over run.

If we look at the data, we can see that the x and y value changes.

x value: 0 --> 2 --> 4 --> 6 --> 8

y value: 15 --> 9 --> 3 --> -3 --> -9

When y changed from 15 to 9, the x changed from 0 to 2.

y: 15 --> 9

x: 0 --> 2

There was a difference of -6 in the y and a difference of 2 in the x.

At the top, I wrote that the slope was (change in y)/(change in x)

With -6 and 2 we found, we can say that the slope is -6/2.

If we simplify this answer, we end up with -3.

So -3 is the slope of the data.

By the use of the completing the square method, the maximum height is 3 ±√29/4 + 9 m.

The maximum height has been modeled by the use of the equation f(x)= -4x^2 +24x -29. Now we know that f(x)= h so we can write;

h = -4x^2 +24x -29

Now we have;

0 = -4x^2 +24x -29

-4x^2 +24x -29= 0

x^2 - 6x = -29/4

(x - 3)^2 = -29/4 + 3^2

(x - 3)^2 =  -29/4 + 9

x = 3 ±√29/4 + 9 m

Learn more about completing the square:https://brainly.com/question/4822356

#SPJ1