Write an expression for the sequence of operations.
Subtract 4 from x, double, and add 8.

Answer:

[tex]a\approx 31[/tex]

[tex]b\approx 72[/tex]

Step-by-step explanation:

Please find the attachment.

We have been given that in triangle ABC, A=25, C=55 and AB=60. We are asked to find the approximate measures of the remaining side lengths of the triangle.

We will use Law of Sines to solve for side lengths of given triangle.

[tex]\frac{\text{sin}(A)}{a}=\frac{\text{sin}(B)}{b}=\frac{\text{sin}(C)}{c}[/tex], where a, b and c are opposite sides corresponding to angles A, b and C respectively.

Upon substituting our given values, we will get:

[tex]\frac{\text{sin}(25)}{a}=\frac{\text{sin}(55)}{60}[/tex]

[tex]a=\frac{60\text{sin}(25)}{\text{sin}(55)}[/tex]

[tex]a=\frac{60*0.422618261741}{0.819152044289}[/tex]

[tex]a=\frac{25.35709570446}{0.819152044289}[/tex]

[tex]a=30.9552980807967304[/tex]

[tex]a\approx 31[/tex]

Therefore, the measure of side 'a' is approximately 31 units.

We can find measure of angle B using angle sum property as:

[tex]m\angle A+m\angle B+m\angle C=180[/tex]

[tex]25+m\angle B+55=180[/tex]

[tex]m\angle B+80=180[/tex]

[tex]m\angle B=100[/tex]

[tex]\frac{\text{sin}(100)}{b}=\frac{\text{sin}(55)}{60}[/tex]

[tex]b=\frac{60\text{sin}(100)}{\text{sin}(55)}[/tex]

[tex]b=\frac{60*0.984807753012}{0.819152044289}[/tex]

[tex]b=\frac{59.08846518072}{0.819152044289}[/tex]

[tex]b=72.1336967815383509[/tex]

[tex]b\approx 72[/tex]

Therefore, the measure of side 'b' is approximately 72 units.

Answer: Zoey needs 168 square feet if carpet squares.

Step-by-step explanation:

Zoey wants to cover her bedroom floor with carpet squares. Each square has an area of 1 square foot.

The formula for determining the area of a rectangle is expressed as

Area = length × width

Her bedroom measures 12 feet by 14 feet. Therefore, the area of her bedroom would be

12 × 14 = 168 square feet.

Therefore, the number of carpet squares that Zoey needs would be

168/1 = 168 square feet

Answer:

  168 squares

Step-by-step explanation:

Each square is 1 foot on a side, so along the 14-foot wall, Zoey will need 14 squares. Altogether, Zoey will need 12 rows of 14 squares, so 12×14 = 168 squares.

Answer:option 1 and option 2 represents a direct variation.

Step-by-step explanation:

A direct variation is one in which, as one variable increases in value, the other variable increases in value. Also as one variable decreases in value, the other variable decreases in value.

Looking at the options,

1) y = 2х

When x = 2, y = 2×2 = 4

When x = 3, y = 2×3 = 6

Therefore, y = 2x is a direct variation.

2) y=x+4

When x = 2, y = 2 + 4 = 6

When x = 3, y = 3 + 4 = 7

Therefore, y = x + 4 is a direct variation.

3) y= 1/2x

When x = 2, y = 1/2×2 = 1/4 = 0.25

When x = 3, y = 1/2×3 = 1/6 = 0.17

Therefore, y = 1/2x is not a direct variation.

4) y = 3/x

When x = 2, y = 3/2 = 1.5

When x = 3, y = 3/3 = 1

Therefore, y = 3/x is not a direct variation.

Using the process of Gaussian elimination, the system of linear equations is rewritten in the form of a matrix. It is then transformed into the Row-Echelon form, which helps determine possible solutions. The solution for this particular system of equations is x1 = 2, x2 = 2, and x3 = 1.

To solve this system of linear equations, you can use a process called

Gaussian elimination

. You start by rewriting the system in augmented matrix. Thus, the system

2x1 + x2 − 2x3 = 4

4x1 + 2x3 = 10

−4x1 + 5x2 − 17x3 = −15

becomes the matrix

[2 1 -2 4]

[4 0 2 10]

[-4 5 -17 -15]

The next step is to convert this matrix into the Row-Echelon form. Once you have a matrix in Row-Echelon form, you can easily see if there are any solutions by looking at the location of the zeros. If there is a row with all zeros on the left and non-zero terms on the right, then there is no solution. If there are infinite many solutions, its row will end with zeros. In this case, the solution is x1 = 2, x2 = 2, and x3 = 1.

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The survey question 'Do you eat at least five servings a day of fruits and vegetables?' is designed to collect categorical data, as it classifies respondents into groups based on their affirmative or negative answer, rather than providing a numerical value.

The question "Do you eat at least five servings a day of fruits and vegetables?" is designed to collect categorical data. This is because the answers to the question will classify respondents into different categories, specifically those who do eat at least five servings of fruits and vegetables per day and those who do not. As such, the data obtained will be qualitative in nature, allowing us to compare and organize individuals based on their dietary habits.

For a more comprehensive understanding, let's compare data types. In contrast to categorical data, a quantitative variable is numeric and can be measured or counted. It can further be subdivided into discrete or continuous data. Quantitative discrete data involve counts of items or occurrences (e.g., the number of classes you take per school year), while quantitative continuous data involve measurements that can take on any value within a given range (e.g., the weights of soups measured in ounces).

Returning to the student's survey question about fruit and vegetable consumption, it is evident that the data collected does not involve counting or measuring numerical values, but rather involves placing respondents into categories based on their dietary habits. Therefore, the variable in question is indeed categorical.

Answer:

B.

Step-by-step explanation:

Find the total profit.

P=7300-5500

P=1800

Since each shovel makes up 50 of the profit.

50N=1800

N=1800%2F50

N=36

36 shovels were sold.

Answer:

School collected 273 coats in second month.

Step-by-step explanation:

Given

Total number of coats collected in 2 months = 542

Number of coats collected in first month = 269

We need to find the number of coat school collected in second month.

Solution:

Now we can say that;

Total number of coats collected in 2 months is sum of Number of coats collected in first month and Number of coats collected in Second month.

Also We can say that;

Number of coats collected in Second month is equal to Total number of coats collected in 2 months minus Number of coats collected in first month.

framing in equation form we get;

Number of coats collected in Second month = [tex]542-269 =273[/tex]

Hence School collected 273 coats in second month.

Answer:

a. is symmetric but not reflexive and transitive

b. is reflexive and transitive but not symmetric

c. is reflexive, symmetric and transitive

Step-by-step explanation:

The cases are missing in the question.

Let the cases be as follows:

a. R = {(1, 3), (3, 1), (2, 2)}

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)}

c. R = ∅

R is defined on the set {1, 2, 3}

a.  R = {(1, 3), (3, 1), (2, 2)} is

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)} is

is reflexive because (1, 1), (2, 2), (3, 3) is in R

is not symmetric because for (1,2) (2,1) is not in R

is transitive becaue for (1,1) and (1,2) we have (1,2) in R

c. R = ∅ is

reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.

Answer:12 baskets must be sold in order to break even

Step-by-step explanation:

The materials needed by the fundraiser team to make each basket cost $3.75.

Let x represent the number of baskets that the team made and also sold. if they spent $75 to advertise their site, then the total cost for x baskets would be

3.75x + 75

The baskets are being sold for $10 each. It means that the total revenue would be

10x

In order to break even, the revenue must be equal to total cost. Therefore,

10x = 3.75x + 75

10x - 3.75x = 75

6.25x = 75

x = 75/6/25 = 12

Answer:

A

Step-by-step explanation:

The linear model can be assessed by the checking the independent variables having power 1 which shows the linear relationship between x and y. For example, as in the option B, C and D, the power of Xi's is one. Whereas the non linear model has the power for independent variables greater than 1. For example, as in option  A the model is a quadratic model because X associated with β2 has a power of a 2.

Thus the nonlinear model can be expressed as

Y = β0 + β1X + (β2X)2 + ε.

Answer: The other dimension can be expressed as

(50 - 2L)/2

Step-by-step explanation: First and foremost, we would let the other dimension be represented by B. Then, the perimeter of a rectangle is measured as L+L+B+B or better put;

Perimeter = 2L + 2B

Where L is the measurement of the longer side and B is the measurement of the shorter side.

In this case the perimeter of the rectangle measures 50m, and this can now be written as

50 = 2L + 2B

Subtract 2L from both sides of the equation

50 - 2L = 2L - 2L + 2B

50 -2L = 2B

Divide both sides of the equation by 2

(50 - 2L)/2 = B

Answer:

  25-L

Step-by-step explanation:

Let W represent the other side length. The perimeter (P) of the rectangle is ...

  P = 2(W+L)

Solving for W, we get ...

  P/2 = W+L

  P/2 -L = W

Filling in the given value for P, we find ...

  W = 50/2 -L = 25 -L

The other dimension is (25-L) meters.

Answer:

a.

Option 1

Option 2

b. Sample statistics

Step-by-step explanation:

a.

The set that includes the list of all possible individual in the interested area of study is termed as population while the portion or a part of population is termed as sample. The given scenario indicates that population is all elderly people from which 949 elderly people are selected for study and so, 949 elderly people are included in sample. So, option 1 and option 2 are correct for indicated scenario.

b.

The percentages 22% and 17% are calculated from 949 elderly people that are indicated as sample. Hence, the percentages 22% and 17% are the measure of sample and so, they represents sample statistics.

Final answer:

The relevant population is all elderly people, and the sample consists of 949 elderly people. The numbers 22% and 17% represent sample statistics.

Explanation:

a. The relevant population is all elderly people, so options 2 and 3 are correct. The sample consists of 949 elderly people, so option 1 is also correct.

b. The numbers 22% and 17% represent sample statistics. The sample statistics are calculated from the data collected in the sample, which in this case is the proportion of elderly persons with depression who went on to develop dementia.

The answer is

[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

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

Here's how I got that answer:

Start with the piecewise definition for y = |x|.

[tex]g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}[/tex]

Everywhere you see an 'x', replace it with x+2

[tex]g(x+2) = \begin{cases}x+2  \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}[/tex]

[tex]g(x+2) = \begin{cases}x+2  \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}[/tex]

Now tack on "-4" at the end of each piece so that we shift the function down 4 units

[tex]g(x+2)-4 = \begin{cases}x+2-4  \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}[/tex]

[tex]g(x+2)-4 = \begin{cases}x-2  \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

[tex]f(x) = \begin{cases}x-2  \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.

Answer:

Step-by-step explanation:

The equation used to represent the height of the ball, h in feet after t seconds is expressed as

h = -16t^2 + 48t + 4

A) The height of the ball after 2 seconds would be

h = - 16 × 2² + 48 × 2 + 4

h = - 64 + 96 + 4

h = 36 feet

B)The equation is a quadratic equation. The plot of this equation on a graph would give a parabola whose vertex would be equal to the maximum height travelled by the rocket.

The vertex of the parabola is calculated as follows,

Vertex = -b/2a

From the equation,

a = -16

b = 48

Vertex = - - 48/32= 1.5

So the ball will attain maximum height at 1.5 seconds.

C) The maximum height of the ball would be

h = -16 × 1.5² + 48 × 1.5 + 4

h = - 36 + 72 + 4

h = 40 feet

Final answer:

The ball will be at a height of 64 feet after 2 seconds. It will reach its maximum height of 58 feet after 1.5 seconds.

Explanation:

To solve the problem, we need to use the given quadratic equation for the ball's height h(t) = -16t2 + 48t + 4. This equation models the motion of the ball thrown into the air with an initial upward velocity.

A: Height After 2 Seconds

To find the height of the ball after 2 seconds, we substitute t = 2 into the equation:

h(2) = -16(2)2 + 48(2) + 4

h(2) = -16(4) + 96 + 4 = 64 feet

B: Time to Reach Maximum Height

To determine when the ball reaches its maximum height, we need to find the vertex of the parabola, which occurs at t = -b/(2a) where a=-16 and b=48. So t = -48/(2(-16)) = 1.5 seconds.

C: Ball's Maximum Height

To find the ball's maximum height, we substitute t = 1.5 into the height equation:

h(1.5) = -16(1.5)2 + 48(1.5) + 4

h(1.5) = -16(2.25) + 72 + 4 = 58 feet

Answer:

k = ln (6/5)

Step-by-step explanation:

for

f(x)=A*exp(kx)+B

since f(0)=1, f(1)=2

f(0)= A*exp(k*0)+B = A+B = 1

f(1) = A*exp(k*1)+B =  A*e^k + B = 2

assuming k>0 , the horizontal asymptote H of f(x) is

H= limit f(x) , when x→ (-∞)

when x→ (-∞) , limit f(x) =  limit (A*exp(kx)+B) = A* limit [exp(kx)]+B* limit = A*0 + B = B

since

H= B = (-4)

then

A+B = 1 → A=1-B = 1 -(-4) = 5

then

A*e^k + B = 2

5*e^k + (-4) = 2

k = ln (6/5)    ,

then our assumption is right and k = ln (6/5)  

The value of k is [tex]k=ln(\frac{6}{5} )[/tex].

Given function is,

                           [tex]f(x)=Ae^{kx} +B[/tex]

Substitute [tex]f(0)=1,f(1)=2[/tex] in above equation.

We get,

                       [tex]A+B=1\\\\Ae^{k}+B=2[/tex]

Given that horizontal asymptote of f is -4.

               [tex]\lim_{x \to -\infty} Ae^{kx}+B=-4\\ \\ B=-4[/tex]

So,  [tex]A=1-B=1-(-4)=5[/tex]

Substitute value of A and B.

                [tex]5e^{k}-4=2\\ \\e^{k} =\frac{6}{5}\\ \\k=ln(\frac{6}{5} )[/tex]

Hence, the value of k is [tex]k=ln(\frac{6}{5} )[/tex].

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Answer:

The fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots

Step-by-step explanation:

Given polynomial equation is [tex]X^4-4X^3-2X^2+12X+9=0[/tex]

And also given that 3,-1 and -1 are  the roots of the given polynomial equation

To find the fourth root of the polynomial equation and to solve the fourth root is real :

By synthetic division

_3|   1    -4    -2    12    9

       0    3    -3   -15    -9

___________________

_-1|       1     -1     -5    -3     0

            0     -1      2     3

___________________

            1      -2     -3     0

Therefore x-3 and x+1 is a factor

Therefore 3 and -1 are roots

Now we have the quadratic equation [tex]x^2-2x-3=0[/tex]

[tex](x+1)(x-3)=0[/tex]

Therefore x=-1,3 are the roots

Therefore the fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots.

Final answer:

The fourth root of the polynomial equation X⁴-4X³-2X²+12X+9=0 must be real because a polynomial of degree n has n roots, and since we already have real roots, the remaining root must also be real to have a pair. Upon analyzing, the fourth root is found to be 3.

Explanation:

The student has provided three roots of the fourth-degree polynomial equation X⁴-4X³-2X²+12X+9=0: 3, -1, and -1 (the latter being a repeated root). To determine why the fourth root must also be a real number, we can invoke the fundamental theorem of algebra, which states that a polynomial of degree n will have exactly n roots in the complex number system (including real and complex roots). Given that a polynomial with real coefficients will have complex roots that come in conjugate pairs, and since the known roots are all real, the unknown fourth root must also be real to satisfy the theorem.

Let's find the fourth root. The polynomial can be factored using the known roots:

Therefore, we have the equation (X-3)(X+1)²(X-a)=0, where 'a' is the unknown root. The product of the roots taken one at a time equals the constant term (9) of the polynomial with an alternate sign. This gives us the equation: 3 × -1 × -1 × a = 9. Solving for 'a' yields a=3, which is the fourth root.

Final answer:

To solve for q in the equation 3(q + 4/3) = 23, we distribute the 3, subtract 4 from both sides, and then divide by 3 to find that q is approximately 6.33.

Explanation:

To solve for q in the equation 3(q + \dfrac{4}{3}) = 23, we need to apply some basic algebra principles. First, we distribute the 3 into the parentheses.

3q + 3 \times \dfrac{4}{3} = 23

3q + 4 = 23

Now, subtract 4 from both sides to get 3q alone on one side.

3q = 23 - 4

3q = 19

Last, divide both sides by 3 to solve for q.

q = \dfrac{19}{3}

q = 6.333...

Thus, q is approximately equal to 6.33 when rounded to two decimal places.

The value of q is q= 19 / 3.

Let's solve for q in the equation:

3(q+ 3 / 4)=23

We can solve the equation by distributing the terms, adding/subtracting to both sides, dividing both sides by the same factor, and simplifying.

Steps to solve:

1. Distribute the terms:

3q+4=23

2. Add/subtract to both sides:

3q+4−4=23−4

3q=19

3. Divide both sides by the same factor:

3q / 3 = 19 / 3

4. Simplify:

q= 19 / 3

Therefore, the value of q is q= 19 / 3.

Answer:

No because if the Rays meet at a point other than Q the angles will change

Step-by-step explanation:

Answer:   2+i

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

Work Shown:

-3 + 6i - (-5 - 3i) - 8i

-3 + 6i + 5 + 3i - 8i

(-3+5) + (6i+3i-8i)

2+1i

2+i

The simplified expression is in the form a+bi with a = 2, b = 1.

Answer:

2 + i.

Step-by-step explanation:

-3 + 6i - (-5 - 3i) - 8i       Distribute the negative over the parentheses:

= -3 + 6i + 5 + 3i - 8i

= - 3 + 5 + 6i + 3i - 8i     Now simplify like terms:

= 2 + i.

Answer:

D = A < D < B < C

Step-by-step explanation:

A = 4.6 x 10-4

Can be written as,

= 0.00046

B = 2.4 x 10-3

Can be written in this form,

= 0.0024

C = 3.5 x 105

Is written as,

350000

D = 6.3 x 10-4

Is also written as,

= 0.00063

A and D are in 4 decimal places and therefore from the 3th decimal place 46 is less than 63 so therefore 0.00046 is less than 0.00063.

B is greater than A and D because B which is 0.0024 is in 3 decimal places. C which is 35000 is the greatest because there are no decimal places and it is in tenth thousand.

So therefore,

A < D < B < C

The variable is the average miles per gallon (mpg), which is a quantitative measure. The implied population is all new cars.

(a) The variable in this situation is the average miles per gallon (mpg) for all new cars.

(b) The variable is quantitative, as it deals with a numerical measure, i.e., the number of miles a car can travel per a gallon of fuel.

(c) The implied population would be all new cars in general - though it's specified that this is based on a sample from Consumer Reports, which may not cover every single new car in existence.

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Answer:

24.2%  students received grades with z-scores between 0.15 and 0.85

Step-by-step explanation:

We are given the following in the question:

The grades of a benchmark test for North High School were normally distributed.

WE have to find the percentage of students that  received grades with z-scores between 0.15 and 0.85.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P(score between 0.15 < z < 0.85)

[tex]P(0.15 \leq z \leq 0.85)\\\\= P(z \leq 0.85) - P(z \leq 0.15)\\\\\text{Calculating the value from standard normal z-table}\\\\= 0.802 - 0.560 = 0.242 = 24.2\%[/tex]

24.2%  students received grades with z-scores between 0.15 and 0.85

Answer: This is an reduction.

Step-by-step explanation:

Given : A street is drawn by dilating segment [tex]\overline{FG}[/tex] about center A with a scale factor greater than 0 but less than 1.

Then by using (2.) , we can say that this is an reduction.

The asymptotes are found using the rational function ax^n/ bx^m where n is the degree of the numerator and m is the degree of the denominator.

In the given equation the numerator isn’t raised to any power so n is considered equal to 1. The Demi actor has x raised to the 2nd power so m equals 2.

If n < m then the c axis, y= 0 is the horizontal asymptote.

Also because n is less than m there are no vertical asymptote.

The domain is any real number so the domain would be (-infinity, infinity)

Answer:

The answer to your question is  25 and 28

Step-by-step explanation:

Number 1 = x

number 2 = y

Conditions

1)                    x + y = 53         ------------- l

2)                   x - y =  3            ------------ ll

Solve this system of equations by elimination

                     x + y = 53

                     x - y =  3

                   2x      = 56

Solve for x

                      x = 56/2

                      x = 28

Substitute x in equation 2

                    28 - y = 3

                         - y = -28 + 3

                         - y = -25

                           y = 25

Subtracting a value from the input x shifts the graph that number of units to the right.

The answer would be C.

Answer:

After factorizing the given expression we get the value as [tex]x(3x-1)[/tex].

Step-by-step explanation:

Given:

[tex]3x^2-x[/tex]

We need to factorize the given expression using GCF.

Solution:

[tex]3x^2-x[/tex]

Now GCF means Greatest common factor.

From the given 2 numbers we need to find the greatest common factor.

[tex]3\times x\times x- 1 \times x[/tex]

In the given expression GCF is 'x'.

Hence we can say that;

[tex]x(3x-1)[/tex]

Hence After factorizing the given expression we get the value as [tex]x(3x-1)[/tex].

Answer:

Therefore,

[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]

[tex]\cot B =\dfrac{1}{2}[/tex]

[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]

Step-by-step explanation:

Given:

Right △ABC has its right angle at C,

BC=4 , and AC=8 .

To Find:

Cos A = ?

Cot B = ?

Csc B = ?

Solution:

Right △ABC has its right angle at C, Then by Pythagoras theorem we have

[tex](\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}[/tex]

Substituting the values we get

[tex](AB)^{2}=4^{2}+8^{2}=80\\AB=\sqrt{80}\\AB=4\sqrt{5}[/tex]

Now by Cosine identity

[tex]\cos A = \dfrac{\textrm{side adjacent to angle A}}{Hypotenuse}\\[/tex]

Substituting the values we get

[tex]\cos A = \dfrac{AC}{AB}=\dfrac{8}{4\sqrt{5}}=\dfrac{2}{\sqrt{5}}\\\\Ratinalizing\\\cos A=\dfrac{2\sqrt{5}}{5}[/tex]

[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]

Now by Cot identity

[tex]\cot B = \dfrac{\textrm{side adjacent to angle B}}{\textrm{side opposite to angle B}}[/tex]

Substituting the values we get

[tex]\cot B = \dfrac{BC}{AC}=\dfrac{4}{8}=\dfrac{1}{2}[/tex]

Now by Cosec identity

[tex]\csc B = \dfrac{Hypotenuse}{\textrm{side opposite to angle B}}\\[/tex]

Substituting the values we get

[tex]\csc B = \dfrac{AB}{AC}=\dfrac{4\sqrt{5}}{8}=\dfrac{\sqrt{5}}{2}[/tex]

Therefore,

[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]

[tex]\cot B =\dfrac{1}{2}[/tex]

[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]

Answer:

a) [tex]P(\bar X >4)=P(Z>\frac{4-3}{\frac{3}{\sqrt{50}}}=2.357)[/tex]

[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]

b) [tex]P(\bar X <2.5)=P(Z>\frac{2.5-3}{\frac{3}{\sqrt{50}}}=-1.179)[/tex]

[tex]P(Z<-1.179)=0.1192[/tex]

c)  

[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]

[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the number of people of a population, and for this case we know that:

Where [tex]\mu=3[/tex] and [tex]\sigma=\sqrt{9}=3[/tex]

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And we want to find this probability:

[tex] P(\bar X >4)= P(z> \frac{4-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]

Part b

[tex] P(\bar X <2.5)= P(z> \frac{2.5-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(Z<-1.179)=0.1192[/tex]

Part c

For this case we want this probability:

[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]

Answer: 5 seconds

Step-by-step explanation:

Given : A ball is thrown into the air from the top of a building.

The height, h(t), of the ball above the ground t seconds after it is thrown can be modeled by [tex]h(t) = -16t^2 + 64t + 80[/tex] .

When ball reaches the ground , then its height from ground become zero.

i.e. [tex]h(t) = -16t^2 + 64t + 80=0[/tex]

Divide equation by 16 , we get

[tex]-t^2 + 4t + 5=0\\\\\Rightarrow\ t^2-4t-5=0\\\\\Rightarrow\ t^2+t-5t-5=0\\\\\Rightarrow\ t(t+1)-5(t+1)=0\\\\\RIghtarrow\ (t+1)(t-5)=0 \\\\\Rightarrow\ t= -1 , 5[/tex]

Since time cannot be negative , therefore t= 5

Hence, the ball will take 5 seconds ( after being thrown) to hit the ground.

The ball will hit the ground 5 seconds after being thrown.

Settingh(t) equal to zero gives us the equation:

[tex]\[ -16t^2 + 64t + 80 = 0 \][/tex]

 To solve this quadratic equation, we can factor out the common factor of -16:

[tex]\[ -16(t^2 - 4t - 5) = 0 \][/tex]

 Now we need to factor the quadratic expression inside the parentheses.

We are looking for two numbers that multiply to -5 and add up to -4. These numbers are -5 and +1. So we can write:

[tex]\[ -16(t - 5)(t + 1) = 0 \][/tex]

 Setting each factor equal to zero gives us two possible solutions for t

[tex]\[ t - 5 = 0 \quad \text{or} \quad t + 1 = 0 \][/tex]

 Solving these, we get:

[tex]\[ t = 5 \quad \text{or} \quad t = -1 \][/tex]

Since time cannot be negative, we discard the solution  t = -1

Therefore, the ball will hit the ground 5 seconds after being thrown."