Exponential function by = x can be expressed as _______.logby = xlogbx = ylogbx = elogxb = y

If x + 1, x - 5, and x - 2 are in a geometric progression, then there is some constant r for which

x - 5 = r (x + 1)

==>   r = (x - 5) / (x + 1)

and

x - 2 = r (x - 5)

==>   r = (x - 2) / (x - 5)

Then

(x - 5) / (x + 1) = (x - 2) / (x - 5)

Solve for x :

(x - 5)² = (x - 2) (x + 1)

x ² - 10x + 25 = x ² - x - 2

-9x = -27

x = 3

It follows that the ratio between terms is

r = (3 - 5) / (3 + 1) = -2/4 = -1/2

Now, assuming x + 1 = 4 is the first term of the G.P., the n-th term a(n) is given by

a(n) = 4 (-1/2)ⁿ⁻¹

The sum of the first 12 terms - denoted here by S - is then

S = 4 (-1/2)⁰ + 4 (-1/2)¹ + 4 (-1/2)² + … + 4 (-1/2)¹¹

Solve for S :

S = 4 [(-1/2)⁰ + (-1/2)¹ + (-1/2)² + … + (-1/2)¹¹]

(-1/2) S = 4 [(-1/2)¹ + (-1/2)² + (-1/2)³ + … + (-1/2)¹²]

==>   S - (-1/2) S = 4 [(-1/2)⁰ - (-1/2)¹²]

==>   3/2 S = 4 (1 - 1/4096)

==>   S = 8/3 (1 - 1/4096)

==>   S = 1365/512

Step-by-step explanation:

a. The null hypothesis is generally an exact value and the alternative hypothesis is the one we are trying to show. We are trying to show that the population mean is different from $24.57, so the hypotheses are as follows:

Null: The population mean hourly wage in the manufacturing industry is the same as the population mean hourly wage in the goods-producing industries

Alternative: The population mean hourly wage in the manufacturing industry differs from the population mean hourly wage in the goods-producing industries

b.

Because we know the population standard deviation, we can use a z test

Plugging in the values:

Expected population mean: 24.57

Sample average: 23.89

Sample size: 30

Population standard deviation: 2.4, we get 0.1207 as our p-value.

c. Using a = 0.05, our p-value is greater than that, so we can not conclude the alternative hypothesis (The population mean hourly wage in the manufacturing industry differs from the population mean hourly wage in the goods-producing industries)

d. Using a critical value calculator, we can find that for a two-tailed approach, the critical value would be 1.96. This would mean that the z score would have to be greater than 1.96 or less than -1.96 for the results to be significant. The population mean we would plug in here is 24.57, while the raw score would be 23.89 and the standard deviation would be 2.4. The z score we get is -0.283, which is not in the values specified (>1.96 or <-1.95) so we cannot conclude the alternative hypothesis

Answer:

You will have earned $113 in interest.

Step-by-step explanation:

After investing for 4 years at 3% interest, your initial investment of $900 will have grown to $1,013.

To find the amount you need to invest, we can use the formula for simple interest:

\(\displaystyle\sf \text{{Interest}} = \text{{Principal}} \times \text{{Rate}} \times \text{{Time}}\)

Here, the principal is the amount you need to invest, the rate is 3% (or 0.03 as a decimal), and the time is 4 years. We can rearrange the formula to solve for the principal:

\(\displaystyle\sf \text{{Principal}} = \frac{{\text{{Interest}}}}{{\text{{Rate}} \times \text{{Time}}}}\)

Plugging in the given values, we get:

\(\displaystyle\sf \text{{Principal}} = \frac{{900}}{{0.03 \times 4}}\)

Simplifying further:

\(\displaystyle\sf \text{{Principal}} = \frac{{900}}{{0.12}}\)

Calculating the division:

\(\displaystyle\sf \text{{Principal}} = 7500\)

Therefore, you will need to invest $7500 at 3% to have earned $900 in interest in 4 years.

Answer:

The population reached a total of 19880 people in 1967.

Step-by-step explanation:

The population in t years after 1951 is given by:

\(P(t) = 130t + 17800\)

Which year the population reached a population of 19880 people.

This is t years after 1951

t is found when \(P(t) = 19880\)

So

\(P(t) = 130t + 17800\)

\(19880 = 130t + 17800\)

\(130t = 2080\)

\(t = \frac{2080}{130}\)

\(t = 16\)

1951 + 16 = 1967

The population reached a total of 19880 people in 1967.

Answer:

29.4°

Step-by-step explanation:

The equation can be rearranged to give C directly.

 C = arccos((a^2 +b^2 -c^2)/(2ab))

 C = arccos((90^2 +55^2 -50^2)/(2·90·55))

 C = arccos(8625/9900) ≈ 29.4002°

 C ≈ 29.4°

The measure of angle C is approximately 29.46 degrees.

To find the measure of angle C (cos(C)) using the given values a = 55, b = 90, and c = 50, we can use the Cosine Rule formula:

cos(C) = (a² + b² - c²) / (2 * a * b)

Substitute the given values:

cos(C) = (55² + 90² - 50²) / (2 * 55 * 90)

Now, calculate the numerator:

cos(C) = (3025 + 8100 - 2500) / (2 * 55 * 90)

cos(C) = 8625 / 9900

Now, divide to get the final value of cos(C):

cos(C) ≈ 0.8707

To find the measure of angle C itself, we can take the inverse cosine (arccos) of this value:

C ≈ arccos(0.8707)

Using a calculator, you'll find:

C ≈ 29.46 degrees (rounded to two decimal places)

So, the measure of angle C is approximately 29.46 degrees.

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Answer: 21.15 ft²

Step-by-step explanation:

We can use the formula for the area of a triangle:

(b×h)/2

In this case, the base is 9 and the height is 4.7.

So, we substitute the variables with the numbers in this problem.

(9 × 4.7)/2

9 × 4.7 = 42.3

42.3/2 = 21.15

So, our final answer is 21.15 ft²

The sec 132 degrees 15 minutes in degrees is approximately -1.638 degrees.

The secant function, denoted as sec(x), is a trigonometric function that is defined as the reciprocal of the cosine function, cos(x).

In other words,

sec(x) = 1 / cos(x)

The secant function is defined for all values of x except for those where the cosine function is equal to zero, which corresponds to the values x = (2n+1)π/2 where n is an integer. At these points, the secant function is undefined.

To convert sec(132 degrees 15 minutes) to degrees, we first need to convert the angle to decimal degrees.

We know that 1 degree is equal to 60 minutes, so 15 minutes is equal to 15/60 = 0.25 degrees.

Therefore, the angle 132 degrees 15 minutes is equal to 132 + 0.25 = 132.25 degrees.

Now we can find the secant of this angle:

sec(132.25 degrees) = 1/cos(132.25 degrees) ≈ -1.638

So the answer is approximately -1.638 degrees.

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Answer:

20%

Step-by-step explanation:

Total = 45

Absentees = 9

Percentage  = (No. of absentees / total) × 100

                    = 9 / 45 × 100

                    = 0.2 × 100

                    = 20%

Answer:

20% of the children are absent

Step-by-step explanation:

Number of children absent = 9

Total children in the class = 45

Percent of absent children / total children is defined as the quotient 9/45 multiplied by 100. That is;

\(\frac{9}{45} \,*\,100 = 0.2 \,*\,100=20\,%\)

SO the percentage is 20%

Answer:seahawks | 20

sharks | 14

marlins. | 12

dolphins. | 9

penguins | 5

Step-by-step explanation:

Answer:

Here ,

the value of b = 6...

Answer:

the leaf would be 9 and 2

Step-by-step explanation:

Stem | leaf

  3      9 & 2

Answer:

1

Step-by-step explanation:

Square roots are always the oposite

The square root of negative -1 is positive 1.

Answer:

1 chocolate bar=$2

Step-by-step explanation: Please mark brainliest

$8/4=$2

which means that every chocolate bar is $2.

\( the \: slope \: is \: \frac{15}{4} \: or \: 3.75 \\ \\\)

\(using \: the \: slope \: fourmula \: = \\ \\ m = \frac{y {}^{2} }{x {}^{2} } - \frac{y {}^{1} }{x {}^{1} } \\ \\ plugging \: in \: our \: points \: = \\ \\ m = \frac{5}{ - 1} - \frac{ - 10}{ - 5} \\ \\ = \frac{5}{ - 1} + \frac{10}{5} \\ \\ \)

\( = \frac{15}{4} \\ \\ = 3.75\)

Answer:

\(42a^{2}\)

Step-by-step explanation:

Split them into sections!

First, \(3a\) x \(5a\) = \(15a^{2}\)

Next, \((3a+3a+3a)(3a)\) = \(9a\) x \(3a\) = \(27a^{2}\)

Add \(15a^{2}\) and  \(27a^{2}\) to get \(42a^{2}\)

Hope I helped!

Answer:

C

Step-by-step explanation:

Did the test and got it right.

The function g(x) = 2x + 3 if  f(x) = 2(x + 1)² and f(g(x)) = 2(2x + 4)² option (C) is correct.

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

The question is incomplete.

The complete question is in the picture, please refer to the attached picture.

Suppose f(x) = 2(x + 1)² and f(g(x)) = 2(2x + 4)². What is g(x)?

A.g(x) = x + 3

B.g(x) = x + 4

C.g(x) = 2x + 3

D.g(x) = 2x + 4

We have:

f(x) = 2(x + 1)²

Let take g(x) = 2x + 3

Find f(g(x))

f(2x + 3) = 2[(2x+3) + 1]²

= 2(2x + 4)² = f(g(x))

Thus, the function g(x) = 2x + 3 if  f(x) = 2(x + 1)² and f(g(x)) = 2(2x + 4)² option (C) is correct.

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Answer:

15% of eighty is equal to twelve

The proof to show that ΔTSV ≅ ΔRSV, given that V is the midpoint of TR.

Two right angled triangles are said to be congruent if they have the same corresponding properties when compared.

A midpoint is said to divide a given line into two equal parts. So that the measure of length of the two parts are exactly the same. It may be determined by bisecting the line.

The following is the proof that ΔTSV ≅ ΔRSV;

<RSV ≅ <TSV                                   (definition of an angle bisector)

<RVS ≅ <TVS                                   (definition of right angle)

RS ≅ TV                                            (sides of congruent triangles)

RT = RV + VT                                    (addition property of a line bisector)

RV ≅ VT                                            (definition of a midpoint)

Therefore it can be deduced that;

ΔTSV ≅ ΔRSV                              (Side-Angle-Angle congruency theorem)

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The transformation is not isometric because the side lengths did not remain the same and hence option C is the correct answer.

An isometric transformation is one that keeps the angles and distances between the original and changed shapes the same. There are numerous techniques that can be used to alter any image in a plane.

The two figures are isometric only if they are congruent. In the given figure the angles remain the same however the lengths of the side are transformed as the figure is dilated.

Hence, the transformation is not isometric because the side lengths did not remain the same and hence option C is the correct answer.

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The equation that describes the line that passes through (2, 1) and (0, -5) is: B. y = 3x - 5.

To find the equation of a line, find the value of the slope (m) and the y-intercept (b), and plug in the values into y = mx + b.

Given the points:

(2, 1) and (0, -5)

Slope (m) of the line = change in y / change in x = (-5 - 1) / (0 - 2)

Slope (m) = -6/-2

Slope (m) = 3

The y-intercept (b) = -5 (this is because it is the value of y when x = 0).

To write the equation, substitute m = 3 and b = -5 into y = mx + b:

y = 3x - 5

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The values x = -5 and x = 3 make the second expression undefined. The correct answers are:

x = -5

x = 3

x= -3

To determine the values for which the given expressions are undefined, we need to find the values that make the denominators equal to zero.

First expression: \(\frac{3x}{(x^2 - 9)}\)

For this expression, the denominator is (x^2 - 9). It will be undefined when the denominator equals zero:

x^2 - 9 = 0

Factoring the equation, we have:

(x - 3)(x + 3) = 0

Setting each factor equal to zero, we get:

x - 3 = 0 --> x = 3

x + 3 = 0 --> x = -3

So, the values x = 3 and x = -3 make the first expression undefined.

Second expression: \(\frac{(x + 4)}{(x^2 + 2x - 15)}\)

For this expression, the denominator is (x^2 + 2x - 15). It will be undefined when the denominator equals zero:

x^2 + 2x - 15 = 0

Factoring the equation, we have:

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

Setting each factor equal to zero, we get:

x + 5 = 0 --> x = -5

x - 3 = 0 --> x = 3

So, The second expression is ambiguous because x = -5 and x = 3.

Consequently, the right responses are x = -5, x = 3 and x= -3.

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Answer:

A. \(\frac{1}{5} log_3(x) + log_3(y)\)

Step-by-step explanation:

Log rules tell us that multiplication inside a log can be expressed as two summands with the same base.
In other words:

\(\log_c(a*b) =\log_c(a)+\log_c(b)\)

In this case:

\(log_3(\sqrt[5]{x} * y) = log_3(\sqrt[5]{x}) + log_3(y)\)

Using log exponent rules(power can be moved to the front of a log), we get our final answer:

\(\frac{1}{5} log_3(x) + log_3(y)\)

Answer:

Answers below

Step-by-step explanation:

Marion: 42 inches

Elijah: 38 inches

Lorie: 57 inches

Convert each to inches and solve.

Hope this helps!

A histogram would be the best graphical representation to display the data in this case.

What is Histogram?

A histogram is a graphical representation of the distribution of a set of continuous data. It is commonly used in statistics to represent the frequency distribution of a set of data.

A histogram would be the best graphical representation to display the data in this case.

A histogram is a graph that shows the distribution of a continuous variable by dividing it into intervals (bins) and counting the number of observations that fall into each bin. In this case, the bins would correspond to the four categories of items purchased (Health and Medicine, Beauty, Household, and Grocery), and the height of each bar would represent the number of purchases in each category.

A box plot, on the other hand, is used to display the distribution of a continuous variable by showing the median, quartiles, and outliers of the data. However, since the data in this case is not continuous, a box plot would not be the most appropriate choice.

A line plot and stem-and-leaf plot are typically used to display the distribution of a single variable, rather than the relationship between two variables, as in this case.

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Answer:

y = 16.6

Step-by-step explanation:

y - 4.6 = 12

Add 4.6 to each side

y - 4.6+4.6 = 12+4.6

y = 16.6

Answer:Supplementary

Step-by-step explanation:

The two angles add up to 180

Answer:

supplementary since 136+44= 180 and any angles that sum up to 180 are supplementary

The constant differences between the consecutive terms are 2 (a); 2 (b), -3 (c), 7 (d), 1(e), and 6(f).

In math, the constant difference can be defined as the number that defines the pattern of a sequence of numbers. This means that number that should be added or subtracted to continue with the sequence.

Due to this, to determine the constant difference it is important to observe the pattern and find out the number that should be added. For example, if the sequence is 2, 4, 6, 8, there is a difference of 2 between each of the numbers and this is the constant difference.

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