Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 45 degrees at midnight and the high and low temperature during the day are 50 and 40 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer:

a) 4 calories per minute

b) 0.25

Step-by-step explanation:

a) If you look at the line it intercepts the x and y axis at the origin (0,0). therefore if you take any point on the line you will see that the calories per minute are constant:

Look at point (40,10)

Calories per minute = x/y = 40/10 = 4

Look at point (80,20)

Calories per minute = x/y = 80/20 = 4

b) you can use any two points on the line. Lets use point 1 as (20,5) and point 2 as (60,15).

The slope of a straight line is defined as:

slope = (y2-y1)/(x2-x1) = (15-5)/(60-20) = 0.25

Sample size--

It is the collections of all the possible outcomes of an event.

(A)

It is given that:

Shona spins a spinner with three equal-sized spaces—red, green, and yellow and then rolls a six-sided die numbered from 1 to 6.

This means that the possible outcomes are given as follows:

(Red,1)          (Green,1)           (Yellow,1)

(Red,2)         (Green,2)          (Yellow,2)

(Red,3)         (Green,3)           (Yellow,3)

(Red,4)         (Green,4)           (Yellow,4)

(Red,5)         (Green,5)           (Yellow,5)

(Red,6)         (Green,6)           (Yellow,6)

This means that the total number of outcomes are: 18

Hence, the sample size for this compound event is:  18

(B)

If the spinner has four colored spaces.

Let the fourth color be: Blue

Then the possible outcomes are given by:

(Red,1)          (Green,1)           (Yellow,1)           (Blue,1)

(Red,2)         (Green,2)          (Yellow,2)          (Blue,2)

(Red,3)         (Green,3)           (Yellow,3)          (Blue,3)

(Red,4)         (Green,4)           (Yellow,4)          (Blue,4)  

(Red,5)         (Green,5)           (Yellow,5)          (Blue,5)

(Red,6)         (Green,6)           (Yellow,6)          (Blue,6)

Hence, the total number of outcomes are:  24

The sample size of this compound event would be 24.

Answer:

a- 18

b- 24

Step-by-step explanation:

Answer:

Δ ABC was dilated by a scale factor of 1/2, reflected across the x-axis

and moved through the translation (4 , 1)

Step-by-step explanation:

* Lets explain how to solve the problem

- The similar triangles have equal ratios between their

  corresponding side

- So lets find from the graph the corresponding sides and calculate the

  ratio, which is the scale factor of the dilation

- In Δ ABC :

∵ The length of the vertical line is y2 - y1

- Let C is (x1 , y1) and B is (x2 , y2)

∵ B = (-2 , 0) and C = (-2 , -4)

∴ CB = 0 - -4 = 4

- The corresponding side to BC is FE

∵ The length of the vertical line is y2 - y1

- Let F is (x1 , y1) , E is (x2 , y2)

∵ E = (3 , 3) and F = (3 , 1)

∵ FE = 3 - 1 = 2

∵ Δ ABC similar to Δ DEF

∵ FE/BC = 2/4 = 1/2

∴ The scale factor of dilation is 1/2

* Δ ABC was dilated by a scale factor of 1/2

- From the graph Δ ABC in the third quadrant in which y-coordinates

 of any point are negative and Δ DFE in the first quadrant in which

 y-coordinates of any point are positive

∵ The reflection of point (x , y) across the x-axis give image (x , -y)

* Δ ABC is reflected after dilation across the x-axis

- Lets find the images of the vertices of Δ ABC after dilation and

 reflection  and compare it with the vertices of Δ DFE to find the

 translation

∵ A = (-4 , -2) , B = (-2 , 0) , C (-2 , -4)

∵ Their images after dilation are A' = (-2 , -1) , B' = (-1 , 0) , C' = (-1 , -2)

∴ Their image after reflection are A" = (-2 , 1) , B" = (-1 , 0) , C" = (-1 , 2)

∵ The vertices of Δ DFE are D = (2 , 2) , F = (3 , 1) , E = (3 , 3)

- Lets find the difference between the x-coordinates and the

 y- coordinates of the corresponding vertices

∵ 2 - -2 = 4 and 2 - 1 = 1

∴ The x-coordinates add by 4 and the y-coordinates add by 1

∴ Their moved 4 units to the right and 1 unit up

* The Δ ABC after dilation and reflection moved through the

  translation (4 , 1)

Answer:ABC was dilated by a scale factor of 1/2, reflected across the x-axisand moved through the translation (4 , 1)

Step-by-step explanation:

Answer:

See explanation

Step-by-step explanation:

We can express

[tex] \tan( \theta) = \frac{ \sin \theta}{ \cos \theta } [/tex]

in so many ways using trigonometric identities.

Let us rewrite to obtain:

[tex]\tan( \theta) = \frac{1}{ \cos \theta } \times \sin \theta[/tex]

This implies that

[tex]\tan( \theta) = \sec \theta \sin \theta[/tex]

When we multiply the right side by

[tex] \frac{ \cos \theta}{ \cos \theta} [/tex]

we get:

[tex]\tan( \theta) = \frac{ \sin \theta \cos \theta }{ \cos ^{2} \theta } [/tex]

[tex]\tan( \theta) = \frac{ \sin 2\theta }{ 2 - 2\sin^{2} \theta } [/tex]

Etc

Answer:

46.80 m

Step-by-step explanation:

Given:

Magnetic field, B = 2.6 kG = 2600 G = 0.26T

Diameter of the plastic tube = 12 cm = 0.12m

Length of the plastic tube = 60 cm

Current, I = 2 A

The formula for the magnetic field (B) at the center of a solenoid is calculated as:

[tex]B=\frac{\mu_oNI}{L}[/tex]

where,

I = current

N = Turns

L = Length

[tex]\mu_o[/tex]= permeability of the free space

on substituting the values in the above equation, we get

[tex]0.26=\frac{4\pi \times10^{-7}\times N\times 2}{0.6}[/tex]

or

N = 62070.42 Turns

also, each turn is a circumference of the plastic tube.

The circumference of the plastic tube, C = 2π×0.12 =  0.7539 m

Thus,

The total length of the wire required, L = (62070.42) × 0.7539 m = 46799.99 ≈ 46800 m  = 46.80 km

Answer:

Perpendicular intersecting lines.

Step-by-step explanation:

A '+' has intersecting perpendicular lines.

Answer:

1. If an individual from Swaziland has tested positive, what is the probability that he carries HIV ?

P=0.8249 or 82.49%

2. If an individual from Swaziland has tested negative, what is the probability that he is HIV free ?

P=0.9988 or 99.88%

Step-by-step explanation:

Make the conditional probability table:

  Individual

              Infected       Not infected

ELISA

Positive              

Negative  

Totals      

The probability of an infected individual with a positive result from the ELISA is obtained from multiplying the probability of being infected (25.9%) with the probability of getting a positive result in the test if is infected (99.7%), the value goes in the first row and column:

P=0.259*0.997=0.2582 or 25.82%

              Individual

              Infected       Not infected Totals

ELISA

Positive    25.82%        

Negative    

Totals      

The probability of a not infected individual with a negative result from the ELISA is obtained from multiplying the probability of not being infected (100%-25.9%=74.1%) with the probability of getting a negative result in the test if isn't infected (92.6%), the value goes in the second row and column:

P=0.741*0.926=0.6862 or 68.62%

              Individual

              Infected       Not infected Totals

ELISA

Positive    25.82%        

Negative                     68.62%

Totals      

In the third row goes the total of the population that is infected (25.9%) and the total of the population free of the HIV (74.1%)

Individual:

              Infected       Not infected Totals

ELISA

Positive    25.82%        

Negative                        68.62%          

Totals       25.9%             74.1%          

Each column must add up to its total, so the probability missing in the first column is 25.9%-25.82%=0.08%, and the ones for the second column is 74.1%-68.62%=5.48%.

             Individual

              Infected       Not infected Totals

ELISA

Positive    25.82%          5.48%            

Negative    0.08             68.62%          

Totals       25.9%             74.1%            

             Individual

The third column is filled with the totals of each row:

              Infected       Not infected Totals

ELISA

Positive    25.82%          5.48%            31.3%

Negative    0.08             68.62%          68.7%

Totals       25.9%             74.1%            100%

The probability A of tested positive is 31.3% and the probability B for tested positive and having the virus is 25.82%, this last has to be divided by the possibility of positive.

P(B/A)=0.2582/0.313=0.8249 or 82.49%

The probability C of tested negative is 68.7% and the probability D for tested negative and not having the virus is 68.62%, this last has to be divided by the possibility of negative.

P(D/C)=0.6862/0.687=0.9988 or 99.88%

Answer:

  x = 10^10

Step-by-step explanation:

You are right to question the question. As posed, it makes no sense.

The idea of an x-intercept is applicable to a relation involving two variables that can be graphed on a coordinate plane.

If you graph this equation on an x-y plane, it will be a vertical line at x = 10^10, so that would be the x-intercept.

_____

I suggest you ask for an explanation from your teacher.

_____

The graph of y=log(x) is something else entirely, as you know. The x-intercept of that graph is x=1.

Answer:

1 -> f

2 -> e

3 -> d

4 ->  a

5 ->  c

6 ->  g

7 ->  b

Step-by-step explanation:

Let consider the options as a,b,c,d,e,f,g

1. Bell curve  ->     f

2. Data set   ->      e

3. Histogram   ->   d

4. Mean       ->      a

5. Normal distribution   ->   c

6. Standard deviation   ->    g

7. Variance      ->     b

The task is to match vocabulary words to their definitions, a common task in an English class. It involves comparing words with definitions and identifying matches. This helps students understand new words.

Given that the task at hand is to match vocabulary words to their correct definitions, this is a task typically found in an English class. This activity involves comparing a list of words with a list of definitions, then identifying which definition corresponds to each word. For example, the word 'gratify' can be matched with the definition 'give (someone) pleasure or satisfaction'. This process helps students better understand and remember the meanings of new words.

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Answer:

A

Step-by-step explanation:

Given

[tex]x^{4}[/tex] - 1 ← a difference of squares which factors in general as

a² - b² = (a - b)(a + b)

here [tex]x^{4}[/tex] = (x²)² ⇒ a = x² and b = 1

[tex]x^{4}[/tex] - 1 = (x² - 1)(x² + 1)

x² - 1 ← is a difference of squares and factors as

x² - 1 = (x - 1)(x + 1), so

(x² - 1)(x² + 1) = (x - 1)(x + 1)(x² + 1), hence

[tex]x^{4}[/tex] - 1 = (x - 1)(x + 1)(x² + 1) → A

Answer:

A.   (x + 1)(x - 1)(x^2 + 1).

Step-by-step explanation:

Using  the difference of 2 squares (a^2 - b^2 = (a + b)(a - b) :

x^4 - 1 = (x^2 - 1)(x^2 + 1).

Now repeating the difference of 2 squares on x^2 - 1:

(x^2 - 1)(x^2 + 1 = (x + 1)(x - 1)(x^2 + 1).

Answer:

Step-by-step explanation:

The base of the exponential factor in a growth formula is the growth factor. Here, that is 1.30. It represents the multiplier of the number of spores each week.

Putting 4 into the formula, we find ...

  f(4) = 345×1.30^4 ≈ 985 . . . . mold spores after 4 weeks

Answer:

george floyd

Step-by-step explanation:

cmon start bouncing

Answer:

Option D is correct.

Step-by-step explanation:

We need to find the value of sin 34°.

We can find the value by putting it in the calculator

sin 34° = 0.5592

So, Option D is correct.

Answer:option d is correct

Step-by-step explanation:

He can conclude that the system of equations has no solution since one equation in it always results with false equivalency.

Hope this helps.

r3t40

He can conclude that the system of equations is inconsistent, i.e. it has no solution.

We know that we get a no solution when we get a expression which is always false.

Here,

Vineet solved a system of equations by substitution.

In his work, he substituted an expression for one of the variables and solved for the value of the other.

This resulted in the equation 7 = 9.

which is a false identity.

since 7 can never be equal to 9.

Hence, the system of equations is inconsistent , it has no solution.

Answer:

3 < x OR 5 > x

Step-by-step explanation:

Divide 3 on both sides; move 9 to the other side of the inequality symbol to get 6x < 30. Then divide both sides by 6.

**NOTE: The ONLY time you reverse the inequality sign is when you are dividing\multiplying by a negative [this does not apply so no need to worry].

I am joyous to assist you anytime.

The solution to both the inequalities will lie in (3 , 5) region, this is represented by line C.

An Inequality is the statement formed when two algebraic expressions are equated using an Inequality operator.

The inequalities are

7x>21 and  6x-9<21

    Dividing 7 on the both sides

     x >3

      Adding 9 on both sides

       6x < 30

     Dividing 6 on both sides

      x < 5

Therefore, the solution to both the inequalities will lie in (3 , 5) region, this is represented by line C.

The complete question is attached with the answer.

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Step-by-step explanation:

[tex]l \times w + l \sqrt{( \frac{w}{2}) ^{2} + {h}^{2} } + w \times \sqrt{( \frac{l}{2} )^{2} + {h}^{2} } [/tex]

L = length base

w = width base

h = height

Answer:

[tex]S=\dfrac{ns^2}{4}\left(\cot{\left(\dfrac{180^{\circ}}{n}\right)}+\sqrt{3}\right)[/tex]

Step-by-step explanation:

A "regular pyramid" is a pyramid whose base is a regular polygon and whose edges are all the same length. Thus each face is an equilateral triangle.

A hexagonal regular pyramid will look like a hexagonal pancake, as the vertical height of it will be zero. A regular pyramid with 7 or more faces cannot exist, because the apexes of the triangular faces cannot meet at a point.

__

For an edge length of "s", the area of each triangular face is (√3)/4×s². There are n of those faces, so the lateral area (LA) will be ...

   LA = ns²(√3)/4

The area of the regular polygon base will be the product of half its perimeter and the length of its apothem (a). The apothem is ...

  a = (s/2)cot(180°/n)

So, the area of the base (BA) is ...

  BA = (1/2)(ns)(s/2)cot(180°/n) = ns²cot(180°/n)/4

The total surface area of the regular pyramid is then ...

  S = BA + LA

  S = (ns²/4)(cot(180°/n) +√3) . . . . for edge length s and n faces (3≤n≤5)

"Alia gets 4^2 fewer than Kelly, who gets 3^3."

So it's about a subtraction.

[tex]\boxed{3^3-4^2}[/tex]

Hope this helps.

r3t40

Answer:

  D. 3^3 - 4^2

Step-by-step explanation:

4^2 fewer than 3^3 is ...

  3^3 - 4^2 . . . . . matches choice D

The question is vague and lacks crucial contextual information required to provide a reliable mathematical proof.

Unfortunately, the question is ambiguous and it lacks the sufficient details to be able to provide a reliable proof. Based on the information provided it appears to be algebraic or geometric. If it's an algebraic equation, such as m+5 = m+6, the proof m+3 = m+4 would not hold since this would imply that 3 = 4 which is not true.

If it's related to geometric figures like angles or sides of a triangle where m represents the measure of an angle or length of a side, we need concrete contextual information to proceed. With more specific details, the required steps to solving your problem could be accurately outlined.

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To prove m3 = m4, we use the given information that m5 = m6 and apply the transitive property of equality.

Given: m5 = m6

We need to prove: m3 = m4

Using the given information, we can see that m5 = m6. This means that the measures of angles 5 and 6 are equal.

By the transitive property of equality, if m5 = m6 and m6 = m3, then m5 = m3.

Similarly, if m5 = m6 and m6 = m4, then m5 = m4.

Therefore, we have proven that m3 = m4.

Answer:

  58°

Step-by-step explanation:

A right triangle can be drawn to model the geometry of the problem. The hypotenuse of the triangle is the length of the string, 100 ft. The side opposite the angle is the height of the kite above the ground, 85 ft.

The mnemonic SOH CAH TOA reminds you of the relationship between sides and angles.

  Sin = Opposite/Hypotenuse

  sin(α) = (85 ft)/(100 ft) = 0.85

The angle whose sine is 0.85 is found using the arcsine (inverse sine) function:

  α = arcsin(0.85) ≈ 58.2°

The angle of elevation is about 58°.

_____

When using your calculator to find the values of inverse trig functions, make sure it is in degrees mode. Otherwise, you're likely to get the answer in radians (≈ 1.01599 radians).

Answer:

C.) 4

Step-by-step explanation:

You can solve this a couple ways but I solved it by looking at the graph. g(x) is 4 units above f(x). Adding four to f(x) would shift it up 4 units. Hope that helped.

Answer:

The correct option is C.

Step-by-step explanation:

The translation is defined as

[tex]g(x)=f(x)+k[/tex]

Where, a is horizontal shift and b is vertical shift.

If k>0, then the graph shifts b units up and if k<0, then the graph shifts b units down.

In the given graph red line represents the the function g(x) and blue line represents the function f(x).

y-intercept of g(x) = 1

y-intercept of f(x) = -3

[tex]k=1-(-3)=1+3=4[/tex]

It means the graph of f(x) shifts 4 unit up to get the graph of g(x). So, the value of k is 4.

Therefore the correct option is C.

Answer:

The distance between the sun and Mercury is 58 343 220 km.

Step-by-step explanation:

1 AU/149,598,000 kilometers = 0.39 AU/x

1x = (149,598,000)(0.39)

x = 58 343 220

For this case we have by definition that an Astronomical Unit (AU) equals 149,598,000 kilometers.

We propose a rule of three to find the distance of Mercury to the sun in kilometers.

1 AU -------------> 149,598,000 km

0.39 AU ---------> x

Where "x" represents the distance in kilometers:

[tex]x = \frac {0.39 * 149,598,000} {1}\\x = 58,343,220[/tex]

Answer:

58,343,220 km

Answer:

5pi/9 radians

Step-by-step explanation:

2 pi radians = 360 deg

pi radians = 180 deg

100 deg * pi rad/(180 deg) = 10/18 pi rad = 5/9 pi rad = 5pi/9 radians

Answer: [tex]ZYX=60\°[/tex]

Step-by-step explanation:

It is important to remember that, by definition:

[tex]Tangent\ chord\ Angle=\frac{1}{2}Intercepted\ Arc[/tex]

 In this case you know that for the circle shown in the figure, the arc XY measures 120 degrees, therefore you can find the measure of the angle ZYX. Then you get that the measure of the this angle is the following:

[tex]ZYX=\frac{1}{2}XY\\\\ZYX=\frac{1}{2}(120\°)\\\\ZYX=60\°[/tex]

Answer:

∠ZYX = 60°

Step-by-step explanation:

The measure of an inscribed angle or tangent- chord angle is one half the measure of its intercepted arc, hence

∠ZYX = 0.5 × 120° = 60°

Answer:  

Outcome     :  A(Country)    B(Rock & roll)    C(blues)

Probability  :     [tex]\dfrac{11}{36}[/tex]                     [tex]\dfrac{17}{36}[/tex]                     [tex]\dfrac{1}{9}[/tex]

Step-by-step explanation:

A probability model is a mathematical display of a random situation S contain various sets .

Let A be the event that they play a country music, B be the event that they play rock & roll and C be the event that they play blues.

Then , n (A) = 11, n(B)=17 and n(C)=8

Let S be the combined set of number of times music played in local restaurant.

Then ,  [tex]n(S)=11+17+8=36[/tex]

Then , [tex]P(A)=\dfrac{n(A)}{n(S)}=\dfrac{11}{36}[/tex]

[tex]P(B)=\dfrac{n(B)}{n(S)}=\dfrac{17}{36}[/tex]

[tex]P(C)=\dfrac{n(C)}{n(S)}=\dfrac{8}{36}=\dfrac{1}{9}[/tex]

Now, the required probability model:-

Outcome     : A(Country)    B(Rock & roll)    C(blues)

Probability  :     [tex]\dfrac{11}{36}[/tex]                     [tex]\dfrac{17}{36}[/tex]                     [tex]\dfrac{1}{9}[/tex]

Answer:

country = 0.31

rock and roll =0.47

Blues = 0.22

Step-by-step explanation: Here we go :O

Let's put the count of each type of music from the sample into a table.

country = 11

Rock and roll= 17

blues = 8

Total = 36

We get the probabilities by dividing the frequencies by the total. (Remember to round to the nearest hundredth.)

11/36 = country

17/36 = rock and roll

8/36 = blues

Divide these

country = 0.31

rock and roll =0.47

Blues = 0.22

Answer:

Around 31 hours

Step-by-step explanation:

So since the formula for cylinder is pi × r squared × depth, you will get 215.51 cubic meters. When it's 0.9 m, it will be 138.54.

The pool has a leak and loses 5 cubic meters every 2 hours.

Subtract 5 from 215.51 until you get 140.51 when you can no longer subtract 5.  

It will be 15 times. Which is 30 hours. Subtract 2.5 from 140.51 and you get around 138.01. Now that was an hour. 30+1= 31 hours

Answer:

C) 24

Step-by-step explanation:

The patter here is that you multiply the previous term by 2 to get the next term. So 3*2 is 6, 6*2 is 12, and 12*2 is 24.

The next term in the sequence: 3, 6, 12, … is 24. This can be obtained by finding the nth term and finding the 4th term.

A collection of items in a particular order and repetitions are allowed.

a, a+d, a+2d, ..., a+(n-1)d, where a is the first term, d is the common difference and a+(n-1)d is the nth term.

a, ar, ar¹, ..., arⁿ⁻¹, where a is the first term, r is the common ratio and arⁿ⁻¹ is the nth term.

Given that, 3, 6, 12, …⇒this is a Geometric Sequence with nth term,

aₙ=3rⁿ⁻¹, n=1,2,3,...

From the given sequence, a₁=3, r=2

Hence the next term in the sequence: 3, 6, 12, … is 24. Thus option C is the correct answer.

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Answer:

  a.  6x^3 - 24x^2 + 6x + 36

Step-by-step explanation:

These are expanded using the distributive property, which is also used for collecting terms.

  (2x - 6)(3x2 - 3x - 6) = 2x(3x^2 - 3x - 6) - 6(3x^2 - 3x - 6)

  = 6x^3 -6x^2 -12x -18x^2 +18x +36

  = 6x^3 +(-6-18)x^2 +(-12 +18)x +36

  = 6x^3 -24x^2 +6x +36 . . . . . . matches selection A

_____

Comment on strategy

For multiple-choice questions, it often works well to find a way to identify a viable answer with the minimum amount of work. Comparing these answer choices, you can see that determining the correct values of the x-term and the constant term will let you pick the right choice.

The constant term is easy, because it is simply the product of the constants (-6)(-6) = 36. This narrows the choices to A and C.

The x-term will be the sum of products of x-terms and constants:

  2x(-6) +(-6)(-3x) = -6(2x-3x) = -6(-x) = 6x

This narrows the choices to A alone.

Answer:

proof my answer is correct

Step-by-step explanation:

The exact value of cosθ in simplified form is -5√61 / 61.

One area of mathematics known as trigonometry examines the relationship between the sides and angles of a right triangle. The relationship between sides and angles is defined for 6 trigonometric functions.

We have,

The terminal side of θ passes through the point (−5,−6).

Now, Hypotenuse

= √(-5)² + (-6)²

= √25+36

=√61

Using Trigonometry

cos θ = Adjacent side/ Hypotenuse

cos θ = -5/ √61

Now, rationalizing

cos θ = -5/√61 x √61/√61

cos θ = -5√61 / 61

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Answer:

Top to Bottom:

Step-by-step explanation:

Subtract 1 from the number in parentheses (the base of the exponential factor). Multiply the result by 100%. This gives you the percentage growth (positive) or decay (negative).

  (1.12 -1)×100% = +12% (growth)

  (0.88 -1)×100% = -12% (decay)

  (0.80 -1)×100% = -20% (decay)

  (1.22 -1)×100% = +22% (growth)

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The sign of the change (+ or -) and the description (growth or decay) convey the same information. It can be confusing to say -12% decay. Rather, the decay is 12%, or the growth is -12%. Above, we tried to indicate that positive is growth and negative is decay. We're not trying to say that the decay is -12%.

The answer x = 7 is false here is why.

[tex]3x+52=13\Longrightarrow3x=-39[/tex]

This further simplifies to solution,

[tex]x=-\dfrac{39}{3}\Longrightarrow\boxed{x=-13}[/tex]

Hope this helps.

r3t40

Final answer:

The two-column proof format shows that the solution to the given equation 3x + 52 = 13 is x = -13, which contradicts the statement x = 7 to be proven. Either the given equation or the proof is incorrect as x = 7 cannot be derived from the equation provided.

Explanation:

Two Column Proof

When given the equation 3x + 52 = 13, we are asked to prove that x = 7. To do this, we can use a methodical approach in a two-column proof format where we list each step of the solution process alongside the reason for each step, as follows:

3x + 52 = 13: Given equation.

3x = 13 - 52: Subtract 52 from both sides.

3x = -39: Simplify.

x = -39 / 3: Divide both sides by 3.

x = -13: Simplify.

This results in x = -13, which contradicts the initial statement that we need to prove (x = 7). Therefore, either there is a mistake in the given equation or the proof is incorrect. Based on the provided information, x = 7 cannot be directly proven from the equation 3x + 52 = 13.

Answer: 1) Each friend had contributed $6.25.

2)The expression to show how many books Aaron read in p months :

[tex]y=9p[/tex]

He reads 54 books in 6 months.

Step-by-step explanation:

1) Given : The total number of friends who contributed equal amounts of money to buy a present. : 4

The total amount they have = $25

Since all the friends contributed the same amount.

Thus , the amount each had contributed :

[tex]=\dfrac{\text{Amount collected}}{\text{Number of friends}}\\\\=\dfrac{25}{4}=6.25[/tex]

Hence, each friend had contributed $6.25.

2) Given : The number of books Aron reads each month = 9 books

Let the number of months be 'p'.

Then the expression to show how many books Aaron read in p months .

[tex]y=9p[/tex]

Then , the  number of books Aaron read if he read for 6 months.

[tex]y=9(6)=54[/tex]

Hence, he reads 54 books in 6 months.