Mr. Salinas told his class that 20% of their
test is open-ended. There are 10 open-ended
questions. How many total questions are on
the test?

The values are given below:

Given :  S is a geometric sequence

(√x + 1) , 1 and (√x -1) are first three terms of the sequence

To Find : Value of x

Recall that a geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.

a , ar , ar² , ... , arⁿ⁻¹

The nth term of a geometric sequence with the first term a and the common ratio r is given by:   aₙ = arⁿ⁻¹

(√x + 1) , 1 and (√x -1) are first three terms of the sequence

=> r =  1/(√x + 1) = (√x -1) /1

1/(√x + 1) = (√x -1) /1

=> 1 = (√x + 1)  (√x -1)

=> 1 = x - 1

=> x = 2

Hence x =2

a = (√2 + 1)

r = 1/(√2 + 1)

5th term of the sequence = ar⁴

=  (√2 + 1)  (1/(√2 + 1))⁴

=  (√2 + 1) /(3 + 2√2)²

=  (√2 + 1) /(17 + 12√2)

= (√2 + 1)(17 - 12√2)  /(17 + 12√2) (17 - 12√2)

=   -7 + 5√2

a = 5 , b = - 7

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The dimension of the square metal sheet is x = 4 ft

Dimensions are the measure of a physical property of an object such as length, width or height.

Since an open box is to be constructed from a square sheet of metal with dimensions x feet by x feet by removing a square of side 1 foot from each corner and turning up the edges.

The volume V of the box is V(x) = (x - 2)².

To find the dimensions of the sheet metal needed to make a box that will hold 4 cubic feet by solving the equation V(x) = 4, we make V(x) = 4 and solve the equation.

So,  V(x) = (x - 2)².

(x - 2)² = 4

Taking square roots of both sides, we have that

√(x - 2)² = ±√4

x - 2 = ±2

x = 2 ± 2

x = 2 - 2 or x = 2 + 2

x = 0 or x = 4

Since the dimension cannot be 0, x = 4

So, the dimension of the sheet is x = 4 ft

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The rate at which Mandy used up her bags of dog food is 2/3 bags per day

What is unit rate?

A unit rate is the cost for only one of anything. This is expressed as a ratio with a denominator of 1. For instance, if you covered 70 yards in 10 seconds, you did so at an average speed of 7 yards per second. Although both of the ratios—70 yards in 10 seconds and 7 yards in one second—are rates, only the latter is a unit rate.

Slope = change in y /change in x

Here,   Slope = change in bags of food/ /change in days

form the graph = 4/6  = 4  bags per 6 days

4/6 simplifies to 2/3

=>  2/3 bags per day

The rate at which Mandy used up her bags of dog food is 2/3 bags per day

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Answer:

a. The fare for a 9-mile journey =  £ 10.6

Step-by-step explanation:

Fare = £2.50 + £0.90 x miles

Fare(9) = £2.50 + £0.90 x ( 9 miles) = £2.50 +  £ 8.1 =  £ 10.6

a. The fare for a 9-mile journey =  £ 10.6

The number that completes the square is 12.25

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

x^2 + 7x + 27 = 0

Take the coefficient of x

So, we have

k = 7

Divide by 2

k/2 = 3.5

Take the square of both sides

Square = 12.25

This means that the number to add is 12.25

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2) The equation of tangent line is,

⇒ y = 324x log 3  + 84 - 648 log 3

3) And, The value of x is,

⇒ x = nπ ± π/6

What is mean by Function?

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

2) The equation of curve is,

y = 3ˣ² + 3

At x = 2

y = 3⁴ + 3

y = 84

Hence, The value of point = (3, 84)

Now, Find dy/dx as;

y = 3ˣ² + 3

dy/dx = 3ˣ² log 3 × 2x

dy/dx = 2x × 3ˣ² log 3

At x = 2

dy/dx = 2 × 2 × 81 × log 3

m = dy/dx = 324 log 3

We know that;

The slope intercept form of equation of line is,

y = mx + c

y = 324x log 3  + c

At point (x, y) = (2, 84)

84 = 324 × 2 log 3 + c

⇒ c = 84 - 648 log 3

Hence, The equation of tangent line is,

⇒ y = 324x log 3  + 84 - 648 log 3

3) We know that;

Horizontal tangent will have a slope of 0.

Hence, The derivative represent the rate of change of a function.

The equation of graph is,

y = 4x - 3 tan x

y' = 4 - 3 sec²x = 0

sec²x = 4/3

sec²x = (2/√3)²

x = nπ ± π/6

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The solution of the system x + y ≥ 75 and 2.5x + 10.5y ≤ 800  of the equation is

The system of equation x + y ≥ 75 and 2.5x + 10.5y ≤ 800 will have a solution that is as written in the equation.

This is done by substituting each value

Using (250, 10)

250 + 10 ≥ 75

260 ≥ 75 correct

2.5x + 10.5y ≤ 800

2.5 * 250 + 10.5 * 10 ≤ 800

730 ≤ 800 correct

since the the values are correct for the equations,  (250, 10) is the solution

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The number of positive results for all persons with TB will be D. 423

Probability simply means the chance that a particular thing or event will happen. It is the occurence of likely events. It is simply the area of mathematics that deals with the numerical estimates of the chance that an event will occur or that a particular statement is true

Rate of incidence of TB is 450 per 100,000

Column Totals of "Has TB" = (450 / 100,000) * Grand Total = (450 / 100,000) * 100,000 = 450

If a person does have TB, then the DNA test will be positive 94% of the time.

Therefore, P = Number of positive results for all persons with TB:

= 0.94 * 450

= 423

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Answer: If the new or compared value is P% more than the reference value, you can calculate the reference value as follows:

Reference value = New or compared value / (1 + P/100)

For example, if the new value is 120% of the reference value, then P = 20 and the reference value can be calculated as:

Reference value = New value / (1 + 20/100) = New value / 1.2

If the new or compared value is P% less than the reference value, you can calculate the reference value as follows:

Reference value = New or compared value / (1 - P/100)

For example, if the new value is 80% of the reference value, then P = 20 and the reference value can be calculated as:

Reference value = New value / (1 - 20/100) = New value / 0.8

Step-by-step explanation:

The solution is, y = 2x + 5+4 when graph would shift by 4 unit.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

Given  : y = 2x + 5 .

To find :

If you wanted to shift the graph up what would be equation .

Solution : We have given that y = 2x + 5 .

BY the transformation rule ;

if f(x) +k then graph would shift up by k units .

Then to shift the graph of y = 2x + 5 up we need to add something in 5

Like   y = 2x + 5+4

Then graph would shift by 4 unit.

Therefore,   y = 2x + 5+4 graph would shift by 4 unit.

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The image of the polygon after the rotation is added as an attachment

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

(1, 0), (2, 0), (0, 2) and (2, 2)

The location of point G is

G = (3, -1)

To rotate about point G, we start by subtracting the points from G

So, we have

[(1, 0), (2, 0), (0, 2) and (2, 2)] - (3, -1)

This gives

(-2, 1), (-1, 1), (-3, 3) and (-1, 3)

The rule is 90 degrees clockwise rotation

This is represented as

(x, y) ⇒ (y, -x)

So, we have

(-2, 1), (-1, 1), (-3, 3) and (-1, 3) ⇒ (1, 2), (1, 1), (3, 3) and (3, 1)

Next, we plot (1, 2), (1, 1), (3, 3) and (3, 1)

See attachment for the image of the transformation

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If x represents the number of apples purchased and y represents the number of mangos we can set up the following:

Kayden has $20, so can spend up to this amount:

He must buy a minimum of 16 apples and mangos altogether:

Also, he must buy no less than 16 apples:

We also need to consider y can't be negative, hence add one more inequality:

Plot all four inequalities and see what is the zone of intersection.

The solution is the set of whole coordinates for x and y within the triangle with vertices at (16, 0), (16, 4) and (26.67, 0).

One possible solution is (20, 2) another one (24, 1).

See attached.

The coterminal angle is 304°, which lies in the fourth quadrant with a reference angle of 44°.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

To find the coterminal angle of 664°, we can add or subtract any multiple of 360° to it.

664° - 360° = 304°

The coterminal angle of 664° is 304°.

To determine the quadrant that 664° lies in, we can divide it by 90° and take the quotient:

664° ÷ 90° = 7 with remainder 34

As remainder is 34 and it is less than 90, the angle lies in the fourth quadrant.

To find the reference angle of 664°

we can subtract the nearest multiple of 90°:

664° - 6(90°) = 44°

The reference angle of 664° is 44°.

Therefore, the coterminal angle is 304°, which lies in the fourth quadrant with a reference angle of 44°.

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The terminal point of p(x,y) on the unit circle is (-2, -1).

The unit circle is a circle with a radius of 1, centered at the origin (0, 0) on a Cartesian coordinate system. It is used to illustrate relationships between angles and their corresponding trigonometric functions. The unit circle is also used to explain concepts in trigonometry such as the sine and cosine functions, the Pythagorean theorem, and the definitions of the six trigonometric functions. It is especially useful in solving problems involving angles, such as finding the values of trigonometric functions given the angle in radians.

Step 1: Find the angle t in its standard form.

t = -4π/3 + 2π = -2π/3

Step 2: Calculate the coordinates using the unit circle formula.

x = cos(-2π/3)

x = -1/2

y = sin(-2π/3)

y = -√3/2

Step 3: The terminal point of p(x,y) on the unit circle is (-2, -1).

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The width of the arch at floor level is 10 feet.

In mathematics, a function is a relationship between two sets of objects, where each object in the first set (called the domain) is paired with exactly one object in the second set (called the range). In other words, a function is a rule that assigns a unique output value to each input value.

Functions can be described in several ways, including algebraic expressions, tables of values, graphs, and verbal descriptions. They can be used to model real-world phenomena, solve equations, and make predictions based on data.

The input values of a function are typically denoted by the variable x, while the output values are denoted by the variable y. The notation f(x) is often used to represent a function, where f is the name of the function and x is the input value.

Functions can be classified into different types based on their properties, such as linear, quadratic, exponential, trigonometric, and logarithmic functions. They can also be combined and manipulated using operations such as addition, subtraction, multiplication, division, composition, and inversion.

One of the key concepts in calculus is the derivative, which is the rate at which the output of a function changes with respect to its input. The derivative is used to calculate slopes, rates of change, and optimization problems in a wide range of applications.

Overall, functions are a fundamental concept in mathematics and have many important applications in science, engineering, economics, and other fields.

To find the width of the arch at floor level, we need to find the x-intercepts of the function y = -1/3(x-5)(x+5). At the x-intercepts, y is equal to zero, so we can set the equation equal to zero and solve for x:

0 = -1/3(x-5)(x+5)

There are two solutions to this equation: x = -5 and x = 5. These are the x-coordinates of the points where the arch intersects the x-axis.

The distance between these two points is the width of the arch at floor level. We can calculate this distance by subtracting the smaller x-value from the larger x-value:

Width = 5 - (-5) = 10 feet

Therefore, the width of the arch at floor level is 10 feet.

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The given function can also be represented as,

[tex]f(x) =3x^2+6x-24[/tex]  

and

[tex]f(x) =3(x+4)(x-2)[/tex]

A relationship between several inputs and outputs is called a function. Simply described, a function is a relation between inputs in which each input is coupled to exactly one output. There is a range, codomain, and domain for each function. A function is usually referred to as f(x), where x is the input . Typically, a function is written as y = f. (x).

Given function is ,

[tex]f(x) =3(x+1)^2-27[/tex]

This function can also be expressed as,

[tex]f(x) =3(x+1)^2-27\\\\f(x) =3(x^2+1+2x)-27\\\\f(x) =3x^2+3+6x-27\\\\f(x) =3x^2+6x-24\\\\f(x) =3(x^2+2x-8)\\\\f(x) =3(x^2+(4-2)x-8)\\\\f(x) =3(x^2+4x-2x-8)\\\\f(x) =3(x+4)(x-2)[/tex]

So, the given function can also be stated as,

[tex]f(x) =3x^2+6x-24[/tex]  

and

[tex]f(x) =3(x+4)(x-2)[/tex]

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The equation of the line in point slope form is y + 6 = 3 (x + 3), and the line in slope intercept form is y = 3x + 3.

Slope of a line is the ratio of the change in y coordinates to the change in the x coordinates of two points given.

We have the points on the line given.

(-3, -6) and (2, 9)

Slope of the line = (9 - -6) / (2 - -3) = 15/5 = 3

Equation of a line in point slope form is,

y - y₁ = m (x - x₁)

where (x₁, y₁) is a point on the line and m is the slope.

Substituting a point (-3, -6) and the slope = 3, we get,

y - -6 = 3 (x - -3)

y + 6 = 3 (x + 3), which is the point slope form.

Equation of a line in slope intercept form is,

y = mx + c

where m is the slope and c is the y intercept.

Substituting m = 3 and (x, y) = (2, 9),

9 = (3 × 2) + c

c = 9 - 6 = 3

Equation in slope intercept form becomes,

y = 3x + 3

Hence the point slope form and slope intercept form of the given line are y + 6 = 3 (x + 3) and y = 3x + 3 respectively.

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it takes 24 minutes for the rock climber to climb 288 feet.

How to determine?

The correct equation to represent this situation is:

$$\frac{720}{60}m=288$$

Here, we first convert the rate from feet per hour to feet per minute by dividing by 60. Then, we set up the equation where the product of the rate and the time in minutes equals the distance climbed.

Simplifying this equation, we get:

$$12m=288$$

Dividing both sides by 12, we get:

$$m=24$$

Therefore, it takes 24 minutes for the rock climber to climb 288 feet.

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Answer:

In the given triangle.

Base= 3

Perpendicular=4

Hypotenuse=5

Step-by-step explanation:

CosB = B/H

=3/5

Sin B = P/H

= 4/5

Tan B = P/B

= 4/3

The maximum time that tony can park his car at the garage if he wants to pay less than $8 is 5.6 hours

Let's start by setting up an inequality that represents the situation:

Total cost for parking x hours <= $8

We know that the first hour costs $2.75, and each additional hour or part thereof costs $1.25. So the cost for x hours can be expressed as:

Cost = $2.75 + $1.25 × (x - 1)

Note that we subtract 1 from x because the first hour is already accounted for in the flat fee.

Now we can substitute this expression into the inequality:

$2.75 + $1.25 × (x - 1) <= $8

Simplifying this inequality, we get:

$1.25x <= $7

Dividing both sides by $1.25, we get:

x <= 5.6

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the value of the definite integral of (2x²2 + x - 1)/(x - 2) from 1 to 2 is -3.

How to solve it?

To solve this integral, we can use partial fraction decomposition. We first write:

(2x²2 + x - 1)/(x - 2) = A + B/(x - 2)

Multiplying both sides by (x - 2), we get:

2x²2 + x - 1 = A(x - 2) + B

Substituting x = 2, we get:

7A + B = 7

Substituting x = 1, we get:

-A + B = -2

Solving these equations simultaneously, we get:

A = -3 and B = 10

So we have:

(2x²2 + x - 1)/(x - 2) = -3 + 10/(x - 2)

Now we can integrate this expression:

∫[-3 + 10/(x - 2)] dx from 1 to 2

= [-3x + 10 ln|x - 2|] from 1 to 2

= [-3(2) + 10 ln|2 - 2|] - [-3(1) + 10 ln|1 - 2|]

Since ln|0| is undefined, we know that the natural logarithm term evaluates to zero. Therefore, we get:

= -6 + 3 + 0

= -3

So the value of the definite integral of (2x²2 + x - 1)/(x - 2) from 1 to 2 is -3.

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Answer:

Step-by-step explanation:

For this kind of problem, all you need to know is one formula:

The Distance Formula

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

in this case,

[tex]x_2=1[/tex], [tex]x_1=0[/tex], [tex]y_2=-3[/tex], [tex]y_1=-10[/tex]

[tex]D = \sqrt{(1-0)^2+(-3--10)^2}[/tex]

[tex]D = \sqrt{1^2+7^2}\\D = \sqrt{1+49}[/tex]

[tex]D = \sqrt{50}\\D = \sqrt{25*2} D = 5\sqrt{2}[/tex]

If u and v be linearly independent unit vectors then  the set of all possible values of u⋅v is zero.

Vectors are used in science to describe anything that has both a direction and a magnitude.

Since u and v are unit vectors, their magnitudes are both 1.

|u + v|² = (u + v)⋅(u + v) = u⋅u + 2u⋅v + v⋅v

= 1 + 2u⋅v + 1

= 2 + 2u⋅v.

On the other hand, we also have:

|u + v|² = (u + v)⋅(u + v) = u⋅u + 2u⋅v + v⋅v = 2,

since u and v are linearly independent and hence orthogonal.

Equating the two expressions for |u + v|² we get:

2 + 2u⋅v = 2,

u⋅v = 0.

Therefore, the only possible value for u⋅v is 0.

Hence,  u and v be linearly independent unit vectors then  the set of all possible values of u⋅v is zero.

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A quadratic equation in standard form to represent the data in the table is as follows;

y = 0.5x² - 4x + 9

In order to determine a quadratic equation in standard form for the line of best fit that models the data points described above, we would use a scatter plot (graphing calculator).

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display a quadratic equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the data set, a quadratic equation for the line of best fit is given by:

y = 0.5x² - 4x + 9

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Trains make up approximately 49.04% of the total journeys across all modes of transportation.

We must divide the total number of train trips by the total number of trips made using all modes of transportation, then multiply the result by 100 to get the percentage that trains make up of the total.

There have been 2320 train trips overall, and there have been 4730 trips overall using all forms of transportation.

Trains therefore account for the following proportion of the total:

(2320 / 4730) x 100% = 49.04%

Train travel thus accounts for roughly 49.04 percent of all travels made using all modes of transportation.

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Answer:

9/2

Step-by-step explanation:

Answer:Il y a 2 configurations de Thalès La première donne BC/CE = AC/CD = 1/2

la seconde donne BC/CE = CF/CA => CF = CA/2 = 3/2

Step-by-step explanation: