In the problems below, f(x) = log2x and
9(x) = 10910x.
How are the graphs of fand g similar? Check all that apply.

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the second line

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{4}) ~\hfill~ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{4}}} \implies \cfrac{ -1 }{ 2 } \implies - \cfrac{1}{2}[/tex]

so we're really looking for the equation of a line whose slope is -1/2 and it passes through (2 , -2)

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-2)}=\stackrel{m}{- \cfrac{1}{2}}(x-\stackrel{x_1}{2}) \implies y +2 = - \cfrac{1}{2} ( x -2) \\\\\\ y+2=- \cfrac{1}{2}x+1\implies {\Large \begin{array}{llll} y=- \cfrac{1}{2}x-1 \end{array}}[/tex]

Answer:

Line passing through (4, 5) and (6, 4):

[tex]m = \frac{4 - 5}{6 - 4} = - \frac{1}{2} [/tex]

Line passing through (2, -2) and with slope -1/2:

-2 = (-1/2)(2) + b

-2 = -1 + b, so b = -1

y = (-1/2)x - 1

-2y = x + 2

-x - 2y = 2

x + 2y = -2

The correct answer is: OD. (x+7)(x-7)

The scale factor and the value of x for each figure is given as follows:

A) Scale factor of 1/3, x = 7 m.

B) Scale factor 0.4747, x = 4.5 in.

For Figure A, we have that the ratio between the areas is given as follows:

510/4590 = 1/9.

As the area is measured in square units, while the side lengths are measured in units, the scale factor is the square root of 1/9, hence it is given as follows:

1/3.

Then the value of x is obtained as follows:

x = 21 x 1/3

x = 7 m.

For Figure B, we have that the ratio between the areas is given as follows:

16/71 = 0.22535.

The scale factor is then the square root of 0.22535, which is given as follows:

0.4747.

Then the value of x is given as follows:

x = 9.5 x 0.4747

x = 4.5 in.

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Out of the possible outcomes HH, HT, TH, and TT when a coin is tossed twice, the missing outcome is TT. Option B.

To determine the missing outcome when a coin is tossed twice, we need to consider all the possible combinations of heads (H) and tails (T) that can result from two coin tosses.

Given that three of the possible outcomes are HH, HT, and TH, we can deduce that the missing outcome must be TT.

Let's analyze each option to confirm:

(A) TH: This outcome is already mentioned, so it is not the missing outcome.

(B) TT: This is the missing outcome as it has not been mentioned among the given options.

(C) HT: This outcome is already mentioned, so it is not the missing outcome.

(D) TH: This outcome is already mentioned, so it is not the missing outcome.

Therefore, the missing outcome is TT. Option B is correct.

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The probability that a point chosen at random lies on the shaded region is given as follows:

4/7.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The area of the shaded region in this problem is given as follows:

4² = 16.

The total area of the figure is given as follows:

16 + 2 x 1/2 x 3 x 4 = 28.

Hence the probability is given as follows:

16/28 = 4/7.

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Step-by-step explanation:

Slope 1/2    point    5,1  

  in point slope form would be

    (y-1) = 1/2 (x-5)

The measure of angle Q is 70°

A parallelogram is a quadrilateral with two pairs of parallel sides.

Some of the properties of a parallelogram are ;

1. They have two pair of parallel lines

2. The opposite sides are equal

3. The sum of the adjascent sides is 180°

Since we have known that the sum of the adjascent sides of a parallelogram is 180°, then we can say that;

6x+4 + 10x = 180

16x = 180 -4

16x = 176

x = 11

angle Q = 6x +4

Q = 6(11)+4

Q = 70°

Therefore the value of angle Q is 70°

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Answer:

The growth model that represents the population of this city is not linear--it is exponential:

[tex]f(t) = 10000( {1.025}^{t} )[/tex]

[tex]t = 0 \: represents \: 2010[/tex]

The correct choice is option A, "All real numbers greater than or equal to 0," as it encompasses the appropriate range of values for the time variable in the given context.

In the given context, the function [tex]A(t) = 50(2)^t[/tex]represents the number of bacteria in the culture after t hours, where the population doubles every 60 minutes.

To determine the appropriate domain for the function A(t), we need to consider the practical limitations and restrictions of the problem.

Since time is measured in hours and the function represents the population at any given hour, it is reasonable to assume that t must be a non-negative real number.

We cannot have negative time or fractional hours in this scenario, as it wouldn't make sense to evaluate the population of bacteria at those points.

Option A, "All real numbers greater than or equal to 0," is the appropriate domain for the function A(t) in terms of the given context.

It allows us to consider all non-negative real values for t, meaning we can evaluate the function for any non-negative amount of time in hours.

Options B and C, "All integers greater than or equal to 50" and "All integers greater than or equal to 0," respectively, are not suitable domains because they restrict the values of t to integers only, while time can be measured in fractional hours or non-integer values.

Option D, "All real numbers greater than or equal to 50," is not an appropriate domain either, as it excludes values of t less than 50, which contradicts the fact that we can evaluate the function for any non-negative amount of time.

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X = 45 degrees. ------------------

The solution to the equation log2(3x - 7) = 3 is x = 5.

To find the solution of the equation log2(3x - 7) = 3, we can use logarithmic properties to rewrite the equation in exponential form. The logarithmic equation states that log(base 2) of (3x - 7) equals 3. In exponential form, this can be expressed as:

2^3 = 3x - 7

Simplifying the left side of the equation, we have:

8 = 3x - 7

To isolate the variable term, we add 7 to both sides of the equation:

8 + 7 = 3x

15 = 3x

Next, we divide both sides of the equation by 3 to solve for x:

15/3 = x

5 = x

Therefore, the solution to the equation log2(3x - 7) = 3 is x = 5. By substituting x = 5 back into the original logarithmic equation, we can verify the solution:

log2(3(5) - 7) = 3

log2(15 - 7) = 3

log2(8) = 3

Simplifying further:

[tex]2^3 = 8[/tex]

8 = 8

Both sides are equal, confirming that x = 5 is indeed the solution to the given equation.

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Answer:

C.  5x(-x + 2)

Step-by-step explanation:

To factor the expression -5x² + 10x, we need to look for a common factor that can be factored out.

Finding a common factor involves identifying a term or expression that can be factored out from each term of a given expression.

Both terms have the common factor of 5x, so we can factor out 5x:

5x(-x + 2)

Therefore, the factored form of -5x² + 10x is -5x(x - 2).

[tex]\hrulefill[/tex]

Additional notes:

If we expand the expressions in the given answer options, we get:

  A.  −5x(x + 2) = -5x² - 10x

  B.  5(−x² + 10x) = -5x² + 50x

  C.  5x(−x + 2) = -5x² + 10x

  D.  x(5x + 10) = 5x² + 10x

Hence confirming that the correct answer is option C.

To factor [tex]-5x^2+10x[/tex], we can begin by factoring out the greatest common factor, which is [tex]-5x[/tex]:

We can check our answer by distributing [tex]-5x[/tex] to the expression inside the parentheses:

[tex]\therefore[/tex] The answer is [tex]-5x(x-2)[/tex].

[tex]\blue{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]

Among the surveyed students, 50% are boys and 50% are girls. Out of the boys, 30% plan to attend the school play. Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C.

To determine the probability that a student surveyed plans to attend the school play given that the student is a boy, we need to examine the data provided in the table.

From the table, we can see that the probability of a student attending the school play is 70% in total, and the probability of not attending is 30% in total.

Out of the total surveyed students, 50% are boys and 50% are girls. Among the boys, 30% plan to attend the school play, while 20% do not plan to attend.

To calculate the probability that a student plans to attend the school play given that the student is a boy, we divide the number of boys attending the school play by the total number of boys:

Probability = (Boys attending) / (Total boys)

Probability = 30% / 50%

Probability = 0.6 or 60%

Therefore, the probability that a student surveyed plans to attend the school play given that the student is a boy is 60%. Option C is correct.

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Using simultaneous method to solve the system of linear equations, 56 $10 tickets, 1310 $20 tickets, and 1902 $30 tickets were sold.

Let's solve the problem step by step.

Let:

x = number of $10 tickets sold

y = number of $20 tickets sold

z = number of $30 tickets sold

From the given information, we can form the following equations:

Equation 1: x + y + z = 3268 (Total number of tickets sold)

Equation 2: y = x + 259 (259 more $20 tickets than $10 tickets were sold)

Equation 3: 10x + 20y + 30z = 63920 (Total sales from ticket sales)

We can use these three equations to solve for the values of x, y, and z.

First, let's substitute Equation 2 into Equation 1:

x + (x + 259) + z = 3268

2x + 259 + z = 3268

2x + z = 3009 (Equation 4)

Now, let's substitute the value of y from Equation 2 into Equation 3:

10x + 20(x + 259) + 30z = 63920

10x + 20x + 5180 + 30z = 63920

30x + 30z = 58740

x + z = 1958 (Equation 5)

We now have a  (Equations 4 and 5) with two variables (x and z). We can solve this system to find the values of x and z.

Multiplying Equation 4 by 30, and Equation 5 by 2, we get:

60x + 30z = 60270 (Equation 6)

2x + 2z = 3916 (Equation 7)

Now, subtract Equation 7 from Equation 6:

(60x + 30z) - (2x + 2z) = 60270 - 3916

58x + 28z = 56354

Simplifying, we have:

29x + 14z = 28177 (Equation 8)

Now, we can solve Equations 5 and 8 simultaneously:

x + z = 1958 (Equation 5)

29x + 14z = 28177 (Equation 8)

Multiplying Equation 5 by 14, and Equation 8 by 1, we get:

14x + 14z = 27332 (Equation 9)

29x + 14z = 28177 (Equation 8)

Now, subtract Equation 9 from Equation 8:

(29x + 14z) - (14x + 14z) = 28177 - 27332

15x = 845

Divide both sides of the equation by 15:

x = 56

Substituting the value of x into Equation 5, we can find z:

56 + z = 1958

z = 1958 - 56

z = 1902

Now that we have the values of x and z, we can substitute them back into Equation 1 to find y:

56 + y + 1902 = 3268

y + 1958 = 3268

y = 3268 - 1958

y = 1310

Therefore, the solution to the problem is:

x = 56 (number of $10 tickets sold)

y = 1310 (number of $20 tickets sold)

z = 1902 (number of $30 tickets sold)

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The result of the sum in this problem is given as follows:

675

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The sum of the first n terms is given by the equation presented as follows:

[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]

The parameters for this problem are given as follows:

Hence the sum is given as follows:

S = 10/2 x (27 + 108)

S = 675.

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In 12 years, Maxwell will deposit $290 into the savings account each month.

To calculate how much money Maxwell will deposit into the account each month in 12 years, we need to determine the pattern of increasing deposits over time.

Maxwell deposits $125 into the savings account each month this year, which we can consider as Year 1. Starting from Year 2, he plans on increasing the monthly deposit by $15.

In Year 2, the monthly deposit will be $125 + $15 = $140.

In Year 3, the monthly deposit will be $140 + $15 = $155.

This pattern continues, increasing the deposit by $15 each year.

Therefore, in Year 12, the monthly deposit will be $125 + ($15 * 11) = $125 + $165 = $290.

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The time required to get an accrued amount of $6,450.00 with compoundeded interest on a principal of $3,030.00 at an interest rate of 10.2% per year and compounded 4 times per year is 7.5 years.

The formula accrued amount in a compounded interest is expressed as;

[tex]A = P( 1 + \frac{r}{n} )^{(nt)}[/tex]

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given that:

Principal P = $3,030

Accrued amount A = $6,450

Compounded quarterly n = 4

Interest rate r = 10.2%

Time t = ?

First, convert R as a percent to r as a decimal

r = R/100

r = 10.2/100

r = 0.102

Now, plug these values into the above formula and solve for time t.

[tex]A = P( 1 + \frac{r}{n} )^{(nt)}\\\\t = \frac{In(\frac{A}{p}) }{n[In( 1 + \frac{r}{n}) ]} \\\\t = \frac{In(\frac{6450}{3030}) }{n[In( 1 + \frac{0.102}{4}) ]} \\\\t = 7.501 \ years[/tex]

t = 7.5 years.

Therefore, the time taken to get the accrued amount is approximately 7.5 years.

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You need to multiply by 5 instead of 6 because each pair of opposite faces on a cuboid has the same area, so by considering one face from each pair, you ensure that you don't count any face twice.

When calculating the total surface area of a cuboid, you need to understand the concept of face pairs.

A cuboid has six faces, but each face has a pair that is identical in size and shape.

Let's break down the reasoning behind multiplying by 5 instead of 6 in the given scenario.

To find the surface area of a cuboid, you can add up the areas of all its faces.

However, each pair of opposite faces has the same area, so you avoid double-counting by only considering one face from each pair. In this case, you have five pairs of faces:

(1) top and bottom, (2) front and back, (3) left and right, (4) left and back, and (5) right and front.

By multiplying the average area of a pair of faces by 5, you account for all the distinct face pairs.

Essentially, you are considering one face from each pair and then summing their areas.

Since all the pairs have the same area, multiplying the average area by 5 gives you the total surface area.

When dividing 150 by 4 (to find the average area of a pair of faces), you are essentially finding the area of a single face.

Then, by multiplying this average area by 5, you ensure that you account for all five pairs of faces, providing the total surface area of the cuboid.

Thus, multiplying by 5 is necessary to correctly calculate the total surface area of the cuboid by accounting for the face pairs while avoiding double-counting.

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a) The estimated average yearly increase in the percentage of​ first-year college males claiming no religious affiliation is 0.5%.

b) Based on the above average, the percentage of​ first-year college males who will claim no religious affiliation in 2020 is 22.7%

Year     Male

1980     6.6%

1990    10.6%

2000   13.5%

2012    21.8%

Percentage of first-year college males claiming no religious affiliation in 2012 = 21.8%

Percentage of first-year college males claiming no religious affiliation in 1980 = 6.6%

The number of years between 2012 and 1980 = 32 years

The percentage increase from 1980 to 2012 = 15.2% (21.8% - 6.6%)

a. Average yearly increase = 0.475% (15.2% ÷ 32)

= 0.5%

b. The number of years between 2020 and 2012 = 8 years

In 2020, the percentage of first-year college males who will claim no religious affiliation based on the average yearly increase above =

Percentage in 2012 x (1 + Yearly Average)^8

21.8% = 0.218 (21.8 ÷ 100)

0.5% = 0.005 (0.5 ÷ 100)

= 0.218(1.005)⁸.

= 0.2269

= 22.69%

= 22.7%

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The solution to the equation 2(x - 6) + 25 = 49 - 2x is x = 9.

To solve the equation 2(x - 6) + 25 = 49 - 2x for x, we will simplify and isolate the variable x.

Let's start by simplifying both sides of the equation:

2(x - 6) + 25 = 49 - 2x

Expanding the parentheses:

2x - 12 + 25 = 49 - 2x

Combining like terms:

2x + 13 = 49 - 2x

Now, let's isolate the variable x by moving the terms involving x to one side of the equation. We can do this by adding 2x to both sides:

2x + 2x + 13 = 49 - 2x + 2x

Simplifying:

4x + 13 = 49

Next, we'll get rid of the constant term on the left side by subtracting 13 from both sides:

4x + 13 - 13 = 49 - 13

Simplifying:

4x = 36

To solve for x, we'll divide both sides of the equation by 4:

4x/4 = 36/4

Simplifying:

x = 9

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The line of best fit and the correlation coefficient are both useful tools for determining the correlation between two variables on a graph.

The line of best fit is a straight line that represents the trend or average relationship between the variables. It is drawn to minimize the overall distance between the line and the data points. By examining the slope of the line, we can determine whether there is a positive or negative correlation. If the line slopes upwards, it indicates a positive correlation, while a downward slope suggests a negative correlation. Additionally, the steepness of the line indicates the strength of the correlation. A steeper line signifies a stronger correlation.

The correlation coefficient, often denoted as r, is a numerical measure of the strength and direction of the correlation. It ranges from -1 to +1. A positive value of r indicates a positive correlation, while a negative value indicates a negative correlation. The magnitude of r represents the strength of the correlation, with values closer to -1 or +1 suggesting a stronger correlation, and values closer to 0 indicating a weaker correlation.

By analyzing both the line of best fit and the correlation coefficient, we can gain insights into the nature and strength of the correlation between the variables on the graph.

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The length of unknown side x is 58.

The correct answer is option D.

To find the unknown side length, x, in a right triangle with the base measuring 42 and the perpendicular measuring 40, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let x be the hypotenuse. Applying the Pythagorean theorem, we have:

[tex]x^2 = 42^2 + 40^2[/tex]

Simplifying:

[tex]x^2[/tex] = 1764 + 1600

[tex]x^2[/tex]= 3364

Taking the square root of both sides to solve for x:

x = [tex]\sqrt{3364}[/tex]

Simplifying the square root:

x = ([tex]\sqrt{4 * 841)}[/tex]

Since 841 is a perfect square ([tex]29^2[/tex]), we can further simplify:

x = 2 * 29

x = 58

Therefore, the unknown side length, x, is equal to 58.

From the options provided the correct option is D.

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Answer:

Since all four sides are equal in length, quadrilateral ABCD is a rhombus

Step-by-step explanation:

Answer:

[tex]x = \dfrac{9}{2}=4.5[/tex]

Perimeter = 52

Step-by-step explanation:

A tangent is a straight line that touches a circle at only one point.

The given diagram shows a circle with three points of tangency: S, T and U.

According to the Two-Tangent Theorem, if two tangents to a circle meet at one exterior point, the tangent segments are congruent.

The exterior points are P, Q and R. Therefore, the congruent segments are:

[tex]\overline{PS} = \overline{PT} = 9[/tex]

[tex]\overline{QT} = \overline{QU} = 4[/tex]

[tex]\overline{RS} = \overline{RU} = 13[/tex]

To find the value of x, use the equation PS = PT:

[tex]\overline{PS} = \overline{PT}[/tex]

 [tex]2x = 9[/tex]

  [tex]x = \dfrac{9}{2}=4.5[/tex]

To calculate the perimeter of triangle PQR, sum the tangent segments.

[tex]\begin{aligned}\textsf{Perimeter}&=\overline{PS}+\overline{PT}+\overline{QT}+\overline{QU}+\overline{RS}+\overline{RU}\\&=9+9+4+4+13+13\\&=52\end{aligned}[/tex]

Therefore, the perimeter of triangle PQR is 52 units.

Answer:

If AJKL is similar to ARST, the angles of AJKL must be congruent to the

Step-by-step explanation:

The estimate for the percent body fat in 75-year-old men would be 24%.

with aging, body fat increases and muscle mass declines, and this means that that the percent body fat is likely to increase as the age progresses.

Looking at the given vertical components, we see that the values are decreasing as we move from top to bottom and can be inferred as that the percent body fat decreases as the age increases.

The horizontal component for the age are :

15

25

35

45

55

65

75

The age values are evenly spaced. In this case, the difference between each age value is 10.

The decreasing trend of the vertical components and evenly spaced data, we can estimate the percent body fat in 75-year-old men to be closer to the value of 24.

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Step-by-step explanation:

The inscribed angle MPN intercepts twice as many degrees of arc as its measure

 so  MN = 62 degrees

   the lower NP is 180 degrees

      the remainder of the 360 degree circle is MP

               360 - 180 - 62 = MP = 118 degrees

Answer:

[tex]m\overset\frown{MP} =118^{\circ}[/tex]

Step-by-step explanation:

The diagram shows a circle with an inscribed angle NPM and an intercepted arc NM.

To find the measure of arc MP, we first need to find the measure of the intercepted arc NM.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m \angle NPM = \dfrac{1}{2} \overset\frown{NM}[/tex]

         [tex]31^{\circ} = \dfrac{1}{2} \overset\frown{NM}[/tex]

       [tex]\overset\frown{NM}=62^{\circ}[/tex]

The minor arcs in a semicircle sum to 180°. Therefore:

[tex]\overset\frown{MP} + \overset\frown{NM} = 180^{\circ}[/tex]

Substitute the found measure of arc MN into the equation:

[tex]\overset\frown{MP} +62^{\circ} = 180^{\circ}[/tex]

[tex]\overset\frown{MP} +62^{\circ} -62^{\circ}= 180^{\circ}-62^{\circ}[/tex]

[tex]\overset\frown{MP} =118^{\circ}[/tex]

Therefore, the measure of arc MP is 118°.

[tex]\hrulefill[/tex]

Additional information

The approximate market value of the bond is $225.

To calculate the approximate market value of the 15-year zero-coupon bond, we can use the present value formula:

Market Value = Par Value * PV Factor

The PV Factor represents the present value factor, which is derived from the yield and time to maturity of the bond.

Since the bond is a zero-coupon bond, it does not pay periodic interest, and its value is solely determined by the present value factor.

Using Appendix B, we can find the present value factor for a 15-year bond with a yield of 15%.

Let's assume the PV Factor is 0.225.

Market Value = $1,000 * 0.225

= $225

The approximate market value of the bond is $225.

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