Word Problem #2
On one night, a scientist needs to determine the
distance she is away from the International Space
Station. At the specific time she is determining this the
space station distance they are both on the same line of
longitude 77' E. Furthermore, she is on a latitude of 29°
N and the space station is orbiting just above a latitude
of 61.4° N. In short, the central angle between the two
is 32.4°. If the Earth's radius is 3959 miles and the
space station orbits 205 miles above the surface of the
Earth, then how far is the scientist away from the space
station?
205 mi
3959 miles
324 CE
A
3959 miles

Answer:

3,5,7,9

Step-by-step explanation:

Let x be an odd integer

given that, consider four consecutive odd integers, which are

x , (x +2), (x+4), (x+6)

Now we have to determine these odd integers such that five times the second is equal to one more than eight times the smaller

5(x+2) = 1 + 8x

5x + 10 = 1 + 8x

10 - 1 = 8x - 5x

9 = 3x

x = 3

The real roots of the quadratic equations are

x = -6 and x = 1

What is Quadratic Equation?

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the quadratic equation be represented as A

Now , the value of A is

x² + 5x - 6 = 0   be equation (1)

On simplifying the equation , we get

x² - 1x + 6x - 6 = 0

Taking the common factors in the equation , we get

x ( x - 1 ) + 6 ( x - 1 ) = 0

On factorizing the equation , we get

( x + 6 ) ( x - 1 ) = 0

So , we have two real solutions to the equation ,

( x + 6 ) = 0  and  ( x - 1 ) = 0

So ,

when ( x + 6 ) = 0

Subtracting 6 on both sides of the equation , we get

x = -6

And ,

when ( x - 1 ) = 0

Adding 1 on both sides of the equation , we get

x = 1

Therefore , the values of x are -6 and 1

Hence , the real roots of the quadratic equations are -6 and 1

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A number line is a horizontal line that has equally spread number increments. The numbers included on the line will determine how the number on the line can be answered.

A number line is a pictorial representation of numbers on a straight line. It's a reference for comparing and ordering numbers.

Steps to add/subtract on a number line:

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Answer: (16 + -4)

Step-by-step explanation:

You have to swap the negative numbers with positives and positive numbers for negative.

The SSS postulate states that "if all the three sides of a triangle are equal to the sides of another triangle then the given triangles are congruent" is proved.

Assume that the lengths of all three sides of the triangles ABC and DEF are known.

The diagram is attached at the end of the solution.

Pick a side from triangle ABC. Find the side with the same length as triangle DEF. For example, we pick [tex]\overline{AB}[/tex] and find that [tex]\overline{D E}[/tex] is equivalent in length to [tex]\overline{AB}[/tex].

Next, write a proof statement about our conclusion and the reason for it. For the reason clause, we state that the two sides are given, since no other work was involved besides reading the lengths of the sides of the diagram. The three dots in the upside-down triangle means "because".

[tex]$$\overline{AB} \cong \overline{DE}$$[/tex]

Choose the other side from triangle ABC. Find the side with the same length as triangle DEF.

We choose the second side [tex]\overline {BC}[/tex] and find from the diagram that [tex]\overline{EF}[/tex] has the same length. The proof statement for this step is:

[tex]$$\overline{BC} \cong \overline{EF}$$[/tex].

Step 3: Pick the last side from triangle ABC.  From triangle DEF, identify the side that is the same length. [tex]\overline{AC}[/tex] and [tex]\overline{DF}[/tex] have the same length. We add another proof statement.

[tex]$$\overline{AC} \cong \overline{DF}$$[/tex]

Step 4: We've discovered that the two triangles have three pairs of congruent sides. That means, by the SSS congruence theorem, triangles ABC and DEF are congruent.

Add a proof statement for this conclusion. The list of the proof statements looks like this.

[tex]$\overline{AB} \cong \overline{DE}[/tex] because [tex]\overline{AB}$[/tex] and [tex]$\overline{DE}$[/tex] are given.

[tex]$\overline{BC} \cong \overline{EF}[/tex] because [tex]$\overline{BC}$[/tex] and [tex]$\overline{EF}$[/tex] are given.

[tex]$\overline{AC} \cong \overline{DF}[/tex] because [tex]$\overline{AC}$[/tex] and [tex]$\overline{DF}$[/tex] are given.

[tex]$\triangle A B C \cong \triangle D E F$[/tex] by SSS congruence theorem.

We have proved that the SSS congruence theorem holds for triangles ABC and DEF.

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The features of the sinusoidal function are given as follows:

The general definition of the sine function in this problem is given as follows:

y = asin(b(x+c))+d.

The parameters of the general function are explained as follows:

The equation in this problem is given as follows:

y=−1/2sin(3πx−5)−2​.

From the vertical shift, the midline is given as follows:

y = -2.

The period is calculated as follows:

p = 2π/3π = 2/3.

The phase shift and the amplitude are given, respectively, by:

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The iron shark has a top speed of 2.4 miles per hour faster than the regression line prediction.

ŷ = 0.2143 (100) + 28.17

ŷ = 49.6

Now, we have to subtract y from ŷ

That is,

52 - 49.6 = 2.4

Hence, 2.4 miles per hour

One or more independent (predictor) variables and one or more dependent (criterion) variables are related in regression analysis, a statistical method. A predicted value for the criterion is obtained from the analysis as a result of a linear combination of the predictors. Your independent variable and your dependent variable are related, as seen by the regression line.

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The salary of an employee with 6 years of experience, based on the line of best fit, is B. $ 58,587

First, find the slope of the line of best fit as:

= ( Y2 - Y1) / ( X2 - X1 )

= ( 57, 126 - 52, 000 ) / ( 5 - 0 )

= 1, 025.2

The y - intercept can be found by substituting the values of y, x, and the slope into the linear function of y = mx + c. The point to be used is (5, 57, 126) :

57, 126 = 1, 025.2 ( 5 ) + c

c = 52, 000

The line of best fit is:

y = 1, 025.2x + 52, 000

If an employee with 6 years experience, their salary would be:

= 1, 025.2x + 52, 000

= 1, 025.2 (6 ) + 52, 000

= $ 58,587

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Answer:

⭐Congruent is another word for "same"

The entire numbers, which consist of positive integer numbers and zero, are the number without any fractions. It is denoted by the letter "W," and the set of numbers is "0, 1, 2, 3, 4, 5, 6, 7, 8, 9,...." So, the first five whole numbers are 0 through 4, respectively.

An integer is either zero, a positive natural number, or a negative integer denoted by a minus sign. The inverse additive of the matching positive integers is represented by the negative numbers. The blackboard bold  Z boldface Z is a common way to represent the set of integers in mathematical notation.

Whole numbers refer to the entire set of natural numbers, including "0." Here are some examples: 0, 11, 25, 36, 999, 1200, etc.

The entire numbers, which consist of positive integer numbers and zero, are the number without any fractions.

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The surface area of the cuboid is given as follows:

147.68 cm².

A cuboid is equivalent to a rectangular prism, having dimensions given as follows:

The volume of the cuboid is given by the multiplication of the base area and the height, as follows:

V = Ab x h.

In which:

Ab = l x w.

Considering the base area and the volume of the cuboid, the height is obtained as follows:

120 = 30 x h

h = 120/30

h = 4 cm.

A cuboid has a square base, hence the length = width are obtained as follows:

l² = 30

l = square root of 30

l = w = 5.48 cm

Hence the surface area of the cuboid is obtained as follows:

S = 2(lw + wh + lh)

S = 2(30 + 4 x 5.48 + 4 x 5.48)

S = 147.68 cm².

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The x-intercept is 19.

The equation of the straight line has its slope and given point.

If we have a non-vertical line that passes through any point(x1, y1) and has a gradient m. then general point (x, y) must satisfy the equation

y-y₁ = m(x-x₁)

Where x₁ - x coordinate; y₁ - y coordinate

m - slope

Given the equation as;

Y=10x/3+19

Then;

the slope m = 10/3

The intercept is 19.

Hence, the x  intercept is 19.

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In a grouped data,

it is not possible to find the median for the given observation by looking at the cumulative frequencies.

The middle value of the given data will be in some class interval. So, it is necessary to find the value inside the class interval that divides the whole distribution into two halves.

In this scenario, we have to find the median class.

To find the median class, we have to find the cumulative frequencies of all the classes and n/2.

After that, locate the class whose cumulative frequency is greater than (nearest to) n/2. The class is called the median class.

After finding the median class, use the below formula to find the median value.

[tex]Median = l +(\frac{\frac{n}{2}- cf}{f} )*h[/tex]

Where

l is the lower limit of the median class

n is the number of observations

f is the frequency of median class

h is the class size

cf is the cumulative frequency of class preceding the median class.

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Two triangles with three congruent sides are referred to as side-side-side (SSS) triangles. The measures are identical, or congruent, which means they are precisely the same. All three sides of SSS triangles will be equal to their corresponding sides in the second triangle, however individual triangle sides are not always equal to one another.

Theorems of Triangle Congruence

One of four triangle congruence theorems, the SSS theorem is unique in that it excludes the need of an angle. The following three are listed:

Two sides and the angle separating them are congruent, or side angle side (SAS).

Angle Side Angle (ASA): Congruence of two angles and the side joining them.

The three angles are all congruent (AAA).

The 3 benefits of SSS theorem are as follow,

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Your instrument's pads, finish, springs, and other metal components can all be damaged if you let dirt, moisture, and other debris build up inside of it. It can also corrode strings and other metal components. These faults might cause expensive long-term consequences and are not easily fixed.

Clean the body, head stock, tuners, and fret board using a damp cloth dipped in vinegar, making careful to follow up with a dry towel. Keep in mind that too much moisture ruins wooden instruments. To keep your case odor-free, wait 30 minutes before storing it.

Due to the fact that these solutions often offer the best material compatibility profile and effective filth removal, neutral or detergent solutions with a pH close to neutral are frequently employed for instrument cleaning. Sometimes neutral pH solutions with proteases added to them help remove organic debris.

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We also know that angle 1 is-congruent-to angle 4 and angle 5 is-congruent-to angle 8 by the vertically opposite angles Therefore, angle 1 is-congruent-to angle 8 by the substitution property.

How to find angles when parallel lines are cut by a transversal?

When parallel lines are cut by a transversal, angle relationships are formed. This include corresponding angles, alternate angles , vertical angles etc.

Therefore, line e and f are parallel lines cut by the the transversal g.

∠1 ≅ ∠4(vertically opposite angles)

Hence,

∠4 ≅ ∠8(corresponding angles)

Since, ∠1 ≅ ∠4

Then, by substitution,

∠1 ≅ ∠8 (transitive property)

∠5 ≅ ∠8 (vertically opposite angles)

Therefore, ∠1 ≅ ∠8 by the transitive property.

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Answer:

5 1/3 books

Step-by-step explanation:

1 1/3 = 4/3

So, she read

4/3 book = 1/2 day

8/3 book = 1 day

16/3 book = 2 days

16/3 = 5 1/3 books

So, he read 5 1/3 books in two days at this rate

Answer:

9. 33% Increase

10. 14 games

11. $3.6

12. $10.80

13. 630

14. 6.3

For the given expression 3x² + 5x = - 4, the value of the discriminant is equal to -23.

As given in the question,

Given expression is written as :

3x² + 5x = - 4

⇒ 3x² + 5x + 4 = 0

General formula of the discriminant for the given quadratic function

ax² + bx + c = 0 is written as :

Discriminant is written as  'D'

D = b² - 4ac

Here b = 5 , a = 3 and c = 4

Substitute the value in the discriminant formula we have,

D = 5² - 4(3)(4)

⇒D = 25 - 48

⇒ D = -23

Therefore , the value of the discriminant for the given quadratic expression 3x² + 5x + 4 = 0 is equal to -23.

The above question is incomplete ,the complete question is :

What is the value of discriminant 3x² + 5x = -4 ?

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Let x represent the precursor to 3. Preimages of 3 are 13 and 15, respectively. the collection of all domain components that correspond to a certain subset of the codomain;

Mages and pre-images: what are they?

On geometry, new shapes can be produced by shifting, scaling, and other geometrical transformations of figures in a plane. Images are the new (modified) forms, whereas preimages are the old, unchanged ones.

What in geometry is a pre-image?

The pre-image of a figure in a transformation process is that figure's original look.

Let x represent a 3 picture. Preimages of 3 are 13 and 15, respectively.

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The  probability of the card being an ace, given that it is a non-face card is option A: 8% .

To find the probability of an event occurring, we can use the formula:

Probability = number of ways the event can occur / total number of outcomes.

In this case, we are trying to see the probability of drawing an ace from a standard deck of 52 cards, given that the card is a non-face card. There are 4 aces in a standard deck of 52 cards, so the number of ways the event (drawing an ace) can occur is 4. There are 52 total cards in the deck, and 48 of them are non-face cards. So, the total number of outcomes is 48.

Therefore, the probability of the card being an ace, given that it is a non-face card, is 4/48.

4/48 = 0.083

When converted to percentage, it is  approximately 8.3%.

Therefore,  This means that the probability of the card being an ace, given that it is a non-face card, is approximately 8%, which is seen as the closest to the answer choice "8%".

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The number of small candles (x) and large candles (y) that were

sold 16 large candles and 12 small candles were sold.

number of small candles (x)

large candles (y)

4x+6y=144  ....equation 1.

x+y=28  ....equation 2.

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A quadratic inequality of the form

y>a[tex]x^{2}[/tex]+bx+c

(or substitute <,≥ or ≤ for > ) represents a region of the plane bounded by a parabola

A quadratic inequality of the form

y>a[tex]x^{2}[/tex]+bx+c

(or substitute <,≥ or ≤ for > ) represents a region of the plane bounded by a parabola .

To graph a quadratic inequality, start by graphing the parabola. Then fill in the region either above or below it, depending on the inequality.

If the inequality symbol is ≤ or ≥ , then the region includes the parabola, so it should be graphed with a solid line.

Otherwise, if the inequality symbol is < or > , the parabola should be drawn with a dotted line to indicate that the region does not include its boundary.

Example:

Graph the quadratic inequality.

y≤[tex]x^{2}[/tex]−x−12

The related equation is:

y=[tex]x^{2}[/tex]−x−12

First we notice that a , the coefficient of the [tex]x^{2}[/tex] term, is equal to 1 . Since a is positive, the parabola points upward.

The right side can be factored as:

y=(x+3)(x−4)

So the parabola has x -intercepts at −3 and 4 . The vertex must lie midway between these, so the x -coordinate of the vertex is 0.5 .

Plugging in this x -value, we get:

y=(0.5+3)(0.5−4)

y=(3.5)(−3.5)

y=−12.25

So, the vertex is at (0.5,−12.25) .

We now have enough information to graph the parabola. Remember to graph it with a solid line, since the inequality is "less than or equal to".

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The rectangle's length and width can be replaced by variables. and can make use of the area. The area of the garden under consideration can be expressed as x^2-11x+28 square feet.

What mathematical expressions can be created using the above description?

Variables can be used to represent the unknowable amounts. Follow the description and mathematically convert each item one by one. For instance, if you are instructed to multiply an item by 4, you can do so by adding 4. You can multiply anything by two, for instance, if it is doubled, and so on to translate a description into a mathematical expression.

How can you describe the size of the garden in words?

Let the garden's length be y feet.

Since the garden's length is one foot longer than its breadth, which is specified as being (x-7) feet, y=3+ (x-7)

y=3+(x-7)

y=x-4 feet

Since the length and breadth of the rectangle under consideration are multiplied to determine the area of the rectangle, the following results are obtained:

A= length * width

A= (x-4)(x-7)

A= x^2-7x-4x+28

A= x^2-11x+28

Consequently, one phrase for the size of the garden under consideration is x^2-11x+28 square feet.

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The limiting speed of particle can be obtained as 12 cm/sec.

A linear equation for the given case can be written by assuming any variable as the unknown quantity. Then, as per the given data the required operations are done and it is equated to some value.

The mass of particle is given as 6 grams.

Suppose the limiting speed be x cm/s.

As per the given question, the following equation can be formed as below,

6 × 980 = 490x

⇒ x = (6 × 980)/490

⇒ x = 12

Hence, the value of the limiting speed is 12 cm/sec.

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The ideal way to find the intercepts of an equation with the "x" and "y" axes is to evaluate both values when the image is 0, as described below:

First let's define equation

An equation is the equality that has mathematical expressions on both sides of it, they can be functions, polynomials or others, there are no limits for them.

  Equations can be graphed, just like a function.

Now that we know what an equation is and its structure, let's explain it with an example.

Let the equation

2y = 4x + 2, determine its intersections with the axes "x" and "y".

First, we clear:

y = (4x + 2)/2

y = 2x + 1

0 = 2x + 1

2x = -1

x = -1/2

y = 2(0) + 1

y = 1

As we can see, to find the intercepts, we only have to evaluate the equation when y = 0 and when x = 0.

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Answer:

Soft drink: $3

Burger: $4

Step-by-step explanation:

We have equations

4x + 6y = 34

4x + 2y = 22

Multiply the second equation by -1

4x + 6y = 34

-4x + -2y = -22

4y = 12

y = $3

Now put 3 back in to solve for burger

4x + 6(3) = 34

4x +18 =34

4x = 16

x = $4

Answer:

[tex]f^{-1}[/tex](x) = [tex]\sqrt{x+11}[/tex] + 4, x=-11

Step-by-step explanation:

y = x^2 - 8x + 5, x<=4

x = y^2 - 8y + 5, y<=4  ==> switch the x and y variables to find the inverse

x = y^2 - 8y + 5  ==> solve for y

x + 16 = y^2 - 8y + 16 + 5  ==> add 16 to get a perfect polynomial square

x + 16 = (y - 8/2)^2 + 5  ==> simplify

x + 16 = (y - 4)^2 + 5

x + 16 - 5 = (y - 4)^2 + 5 - 5  ==> isolate y by subtracting 5 on both sides

x + 11 = (y - 4)^2  ==> simplify

[tex]\sqrt{x+11}[/tex] = [tex]\sqrt{(y - 4)^2}[/tex]  ==> take the square root of both sides to remove the

                                        square

[tex]\sqrt{x+11}[/tex] = y - 4

y = [tex]\sqrt{x+11}[/tex] + 4  ==> add 4 on both sides to isolate y

[tex]f^{-1}[/tex](x) = [tex]\sqrt{x+11}[/tex] + 4  ==> substitute [tex]f^{-1}[/tex](x) for y

[tex]f^{-1}[/tex](x) = [tex]\sqrt{x+11}[/tex] + 4, [tex]f^{-1}[/tex](x)<=4  ==> add in the domain restriction

4 = [tex]\sqrt{x+11}[/tex] + 4  ==> plugin the domain restriction for [tex]f^{-1}[/tex](x)

0 = [tex]\sqrt{x+11}[/tex] ==> subtract 4 on both sides

x + 11 = 0  ==> the square root of 0 is 0

x = -11  ==> subtract 11 on both sides

Hence, answer is: [tex]f^{-1}[/tex](x) = [tex]\sqrt{x+11}[/tex] + 4, x=-11

The AAA (angle-angle-angle) similarity theorem can be used to express it as follows: two triangles have equal corresponding angles if and only if their corresponding sides are proportionate.

The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.

Proof:

Consider two triangles ABC and DEF, such that ∠A = ∠D, ∠B = ∠E and ∠C = ∠F as shown in the figure.

Now, cut DP= AB and DQ= AC and join PQ. Hence, we can say that a triangle ABC is congruent to the triangle DPQ.

(i.e) ∆ ABC ≅ ∆ DPQ, which gives ∠B = ∠P = ∠E and also, the line PQ is parallel to EF.

Therefore, by using the basic proportionality theorem, we can write

DP/PE = DQ/QF.

(i.e) AB/DE = AC/DF. …(1)

Similarly,

AB/DE = BC/EF …(2)

From (1) and (2), we can write:

AB/DE = BC/EF = AC/DF.

Therefore, the two triangles ABC and DEF are similar.

Hence, AAA(Angle-angle-angle) is proved.

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