O EQUATIONS AND INEQUALITIESSolving a decimal word problem using a linear equation with th.

You can see in the figure attached that there are two Right triangles.

By definition, Right triangles are those triangles that have an angle that measures 90 degrees.

The larger triangle is the triangle ABC, but you only know the lenght of the side BC, which is:

And for the smaller triangle you only know the side whose lenght is 2.5 meters.

Therefore, since the exercise does not provide any other lenght and it does not provide another angle, you can conclude that the missing values cannot be determine with the given information.

So, the answer is OPTION D.

Given a midline of 4, an amplitude of 3 and a period of 2/3 we are asked to write a sine function.

Explanation

The equation of a sine function is given as

Where A is the amplitude, T is the period and B is the midline of the sine function.

Therefore, we will have;

Answer:

Answer:

Step-by-step explanation:

The domain and range of a linear function is always real numbers (T or F)

It is True. This is because of a couple of reasons.

    1.) You cannot divide by 0.

     2. A negative number cannot have its square root taken.

The range is determined by the domain in a linear function, and thus it must always consist of real numbers.

Given :

WE need to find the value of f(-2)

So, substitute with x = -2

So, the value of f(-2) = 6

ANSWER

1,440

EXPLANATION

We have that 4 boys are competing and also 5 girls are competing. 3 medals are given to the boys and 3 medals are given to the girls.

For the boys, the gold medal can be awarded to one of 4 boys, then the silver medal can be awarded to 3 boys because 1 of them already got the gold medal. Finally, the bronze medal can be awarded to one of 2 boys, since the gold and silver medals are already taken. The number of ways the medals can be given to the boys is,

This situation is similar for the girls, but in this case, there are 5 girls in total,

The total ways the six medals can be given is,

Hence, there are 1,440 ways to give the six medals to the 4 boys and 5 girls.

Answer

ŷ = -0.09x + 74.17

Explanation

For the given data set:

x 245 187 198 189 176 266 210 255

y 50 54 55 78 44 41 51 60

The sum of x = 245 + 187 + 198 + 189 + 176 + 266 + 210 + 255 = 1726

The sum of y = 50 + 54 + 55 + 78 + 44 + 41 + 51 + 60 = 433

Mean x = 1726/8 = 215.75

Mean y = 433/8 = 54.125

Sum of squares (SSx) = 8391.5

Sum of products (SP) = -779.75

(Check the table below of the data for a better understanding).

The regression Equation is given by ŷ = bX + a

b = SP/SSx = -779.75/8391.5 = -0.09292

a = My - bMx = 54.13 - (-0.09 x 215.75) = 74.17279

Therefore, the regression equation for the data set is: ŷ = -0.09292x + 74.17279

The correct answer is ŷ = -0.09x + 74.17

The given points are (5, 1) and (-3, 17).

First, we have to find the slope using the following formula.

Where,

Let's use the coordinates above to find the slope.

The slope is -2.

Now, we use the point-slope formula to find the equation.

Let's use the same coordinates x_1 and y_1, and the slope m = -2.

Now, we solve for y to express the equation in slope-intercept form.

Suplementary angle = 180° - angle 2

is Angle 1,

because Angle 2 + Angle 1 = 180°

Part 2. Vertical angles are

Angles 2 and 5

Given the situation to model the arithmetic and the geometric sequences.

Imagine you are four years old. A rich aunt wants to provide for your future. She has offered to do one of two things.

Option 1: She would give you $1000.50 a year until you are twenty-one.

This option represents the arithmetic sequence

The first term = a = 1000.50

The common difference = d = 1000.50

The general formula will be as follows:

Simplify the expression:

Option 2: She would give you $1 this year, $2 next year, and so on, doubling the amount each year until you were 21.

This option represents the geometric sequence

The first term = a = 1

The common ratio = r = 2/1 = 2

The general formula will be as follows:

Now, we will compare the options:

The first term of both options is when you are four years old that n = 1

you only received money for ten years so, n = 10

So, substitute with n = 10 into both formulas:

So, the answer will be:

For ten years, the option that gives the most money = Option 1

Differentiation - The value of g'(x) = [tex]\frac{1}{10}x^{-9}[/tex].

Apart from integration, differentiation is among the two key ideas in calculus. A technique for determining a function's derivative is differentiation. Mathematicians use a process called differentiation to determine a function's instantaneous rate of change predicated on one of its variables. The most typical illustration is velocity, which is the rate at which a distance changes in relation to time. Finding an antiderivative is the opposite of differentiation. The rate of change of signal with respect to y has been given by dy/dx if x and y are two variables. The general representation of a function's derivative is given by the equation f'(x) = dy/dx, where y = f(x) is any function.

Given that,

G(x) = [tex]\frac{1}{x^{10} }[/tex]

g’(x)=?

g’(x) is the derivative of g(x).

The derivative of [tex]x^{n} = nx^(n-1)[/tex]

[tex]x^{10} = 10x^(10-1)[/tex]

[tex]x^{10}= 10x^9[/tex]

Then,

[tex]\frac{1}{x^{10} }[/tex] = [tex]\frac{1}{10}x^{-9}[/tex]

Hence, The derivative of g(x) is [tex]\frac{1}{x^{10} }[/tex] = [tex]\frac{1}{10}x^{-9}[/tex].

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Given:

To find- the simplification.

Explanation-

We know that the distribution property of multiplication over addition says

Use this property to simplify, and we get

Multiply by opening the bracket, and we get

Now, open the bracket and combine the like terms.

On further solving, we get

Thus, from the distributive property of multiplication over addition, we get v⁴+13v³+37v²-7v-44.

The answer is v⁴ + 13v³ + 37v² - 7v - 44.

Given:

Cut a 10-foot (ft.) long piece of wood into two pieces so that one piece is 2 ft longer than the other.

Required:

Which of the following equations depicts the given situation?

Explanation:

Let 10 feet long piece of wood cut into two pieces of length x(smaller piece) and

x+2 larger piece.

So, the equation will be

x + x + 2 =10

2x + 2=10

Answer:

Option C is correct.

The time that it will take the pipe to fill the pool if the drains left open is 10 hours.

From the information, one inlet pipe can fill an empty pool in 6 hours and a drain can empty the pool in 15 hours.

The information illustrated that the input pipe gills 1/6 if the pool and the drain empties 1/15 in the pool every hour

The required time taken will be:

= 1/6 - 1/15

= 5/30 - 2/30

= 3/30

= 1/10

Therefore, the time taken is 10 hours.

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a) 84.13%

b) 2.28%

c) 15.86%

Given:

the numbers of skateboards produced per day at a certain factory were normally distributed

mean = 20, 500

standard deviation = 55

To find:

a) On what percent of the day did the factories produced 20,555 or fewer?

b) On what percent of the day did the factories produced 20,610 or fewer?

c) On what percent of the day did the factories produced 20445 or fewer?

To determine the answers, we will use the z-score formula and then use the standard normal table to get the equivalence of the z-score

The formula of score is given as:

Answer: D=2/7 R=4/7

Step-by-step explanation: there are 7 parts and D is the 2nd part R is the 4th part.

The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.

What is mean by Rectangle?

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

The rectangle is bounded by the x - axis and the semicircle y = 49 - x².

Since,

The area of rectangle with sides x and y is,

Area = x × y

A = xy

Since, The equation of the semicircle is;

y = 49 - x².

Substitute the values of y in equation (i), we get;

A = x (49 - x²)

A = 49x - x³

Now, Find the derivative and equate into zero,

A' = 49 - 3x²

A' = 0

49 - 3x² = 0

49 = 3x²

x² = 49/3

x = 7/√3

x = 7/1.73

x = 4.04

Hence, y = 49 - x²

y = 49 - (4.04)²

y = 49 - 16.3

y = 32.67

Since, The area is maximum when we can multiply x by y as;

Maximum area = 4.04 x 32.67

Maximum area = 132

Hence, The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.

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To be able to determine which among the coupon gives a lower price, let's determine what is 45% of $73 so that we could compare it with the $30 rebate. The highest amount among the two coupons will give you a lower price.

Let's determinte the 45% of 73:

Coupon A gives you $32.85 dollar off of a $73 Jacket.

Coupon A will give you a lower price compared to Coupon B. The price of the jacket will be $2.85 lesser than using Coupon B.

Considering the normal distribution and the central limit theorem, it is found that:

a) The distribution is: x¯ ~ N(189, 1.91).

b) The probability that the average hand length is less than 191 is of 0.8531 = 85.31%.

c) The first quartile is of 187.7 mm.

d) The assumption is necessary, as the sample size is less than 30.

The z-score of a measure X of a variable that has mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The parameters in this problem are given as follows:

[tex]\mu = 189, \sigma = 7.4, n = 15, s = \frac{7.4}{\sqrt{15}} = 1.91[/tex]

Hence the sampling distribution of sample means is classified as follows:

x¯ ~ N(189, 1.91).

The probability that the average hand length is less than 191 is the p-value of Z when X = 191, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem:

[tex]Z = \frac{X - \mu}{s}[/tex]

Z = (191 - 189)/1.91

Z = 1.05

Z = 1.05 has a p-value of 0.8531, which is the probability.

The first quartile of the distribution is X when Z has a p-value of 0.25, so X when Z = -0.675, hence:

[tex]Z = \frac{X - \mu}{s}[/tex]

-0.675 = (X - 189)/1.91

X - 189 = -0.675 x 1.91

X = 187.7 mm.

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Answer: 1

Step-by-step explanation:

We are given (a + 3) - (a + 2)

To think of this another way, we can distribute the negative sign out into the (a + 2)

(a + 3) -(a) - (2)

Now our expresssion looks like this:

(a + 3) - a - 2

Simplifying, we get

a - a + 3 - 2

The a terms cancel leaving us with

3-2

and that equals

1

Answer:

1

Step-by-step explanation:

1. Rewrite

         : (a+3)-(a+2) = a + 3 - a - 2

2. Subtract

         : 3-2 = 1 ... so now the equation is a + 1 - a

3. Combine like terms

         : a -a = 0 (the a's cancel out) ... now you're left with 1

Since there is nothing left, your answer is 1.

The acute triangle is defined by the condition,

Here, we have,

Thus, the triangle is an obtuse triangle.

Tim should order approximately 6 pizza.

Division in mathematics is the process of dividing an amount into equal parts. For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on. One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division. The other operations are multiplication, addition, and subtraction. Mathematicians use addition, subtraction, multiplication, and division as their four fundamental arithmetic operations. The division is one of these four operations that we employ most frequently in our daily work. It involves dividing a huge group into equally sized smaller units. Divide 25, for instance, by 5.

Given Data

Number of employees = 22

Slice of pizza one should get = 1/4

Dividing 22 by 1/4

[tex]\frac{22}{4}[/tex]

5 and [tex]\frac{1}{2}[/tex]

Tim should order approximately 6 pizza.

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In order to graph g(x) with the graph of f(x), first we need a translation of 3 units to the right, because of the term f(x - 3)

Then, we need a translation of 4 units down, because of the term -4.

So the movements are: translations of 3 units right and 4 units down.

In order to have a linear function, the rate of change needs to be the same in each point

For

(-18,2)=(x1,y1)

(-14,4)=(x2,y2)

for

(-14,4)=(x1,y1)

(-12,5)=(x2,y2)

for

(-12,5)=(x1,y1)

(0,12)=(x2,y2)

as we can see here are the two numbers so we will obtain the equation in order to know the number that needs to be change

therefore if x=0

the number we need to change is 12 and need to be changed for 11

(0,11)

a. The number that needs to be changed in order to create a linear function is 12

b. That number needs to be changed to 11 in order for the function to be linear

The y-intercept is the value of y when x = 0

To identify y-intercept on a graph, we will check for the the value of y when the line crosses the y axis

From the graph, the line crosses the y axis at y = 6

Hence, the y-intercept is 6

To get the slope, we will pick any two points on the line.

Using points (0, 6) and (4, 0)

Applying the slope formula:

NOTE: the slope is negative because it is going from up to down (moving downwards)

The equation of slope in intercept form: y = mx + b

m = slope = -3/2

b = y-intercept = 6

The equation in y-intercept becomes:

As given by the question

There are given that the function:

Now,

To find the critical point, differentiate the given function with respect to x and put the result of function equal to zero

So,

Then,

Now,

To find the y-coordinate, we need to substitute the above value, x = 4, -5, into the function f(x)

So,

First put x = 4 into the given function:

And,

Put x = -5 into the function f(x):

Hence, the critical point is, (4, -304) and (-5, 425).

Now,

To find the local maxima and local minima, we need to find the second derivative of the given function:;

So,

Now,

The put the value from critical point into the above function to check whether it is maxima or minima.

So,

First put x = 4 into above function:

And,

Put x = -5 into the above function

Then,

According to the concept, if f''(x)>0 then it is local minima function and if f''(x)<0, then it is local maxima function

Hence, the given function is local maxima at (-5, 425) and the value is -54 and the given function is local minima at point (4, -304) and the value is 54.

We will have the following:

The equation of a straight line is

y = mx + c

4x + 20y = -180

make 20y the subject of the formula

20y = -180 - 4x

20y = -4x - 180

divide all through by 20

20y/20 = -4x/20 - 180/20

y = -1/5x - 9

The answer is y = -1/5x - 9 where your slope is -1/5 and intercept is -9

we have the equation

using a graphing tool

see the attached figure

From the question the following can be derived:

(a)

Let x cans of juice and y cans of soda be purchased for the dorm. Then the cost of the juice and soda is 0.60x + 0.75y. The equation of all the combinations of juice and soda is 0.60x + 0.75y = 24.

(b)

The cost of exactly 24 cans of juice is $24 * 0.60 = $14.40. After this purchase, the remaining sum of money available is $24 - $14.40 = $9.60. This will suffice to buy 12 cans of soda, leaving a balance of $0.80. Thus. the entire money cannot be spent if exactly 24 cans of juice are purchased.

(c)

Below is a graph of the line 0.6x + 0.75y = 24 or 4x + 5y = 160 is plotted. All possible cimbinations of juice and soda will lie on this line. The x-intercept is 40 and the y-intercept is 32. Since neither of x and y can be negative, hence the lower and upper bounds for x are 0 and 40 and the lower ad upper bounds for y are 0 and 32. Also , x has to be multiple of 5 and y has to be a multiple of 4. As may be observed from the graph, only 9 combinations are possible which are (x, y):

(0, 32), (5, 28), (10, 24), (15, 20), (20, 16), (25, 12), (30, 8), (35, 4), (40, 0).

Graph: