Solving Exponential Equations with Logarithms Solve each equation. Round your answers to the nearest ten-thousandth.
1) 3^b = 17 2) 12^r= = 13
3) 9^n = 49 4) 16^v = 67
5) 3^a = 69

Answer:

  10

Step-by-step explanation:

You want the determinant of the matrix ...

  [tex]\left[\begin{array}{cc}1&-2\\3&4\end{array}\right][/tex]

The determinant of a matrix is a sum of products. Each product is the element of a row or column multiplied by its cofactor.

For a 2×2 matrix, this becomes the difference between the product of diagonal terms and the product of off-diagonal terms:

  (1·4) - (3·(-2))

  = 4 +6

  = 10

__

Additional comment

The cofactor of term a[i, j] is (-1)^(i+j)·det(M[i,j]), where M[i,j] is the minor matrix obtained by removing row i and column j from the original. As you can see, this definition of a determinant is recursive.

The above describes the determinant of a 2×2 matrix. The determinant of a 3×3 matrix resolves to a sum of 6 3-element products. For larger matrices, the computation burden can be reduced by taking advantage of rows or columns containing zeros.

If the sides of a right triangle are x, 3x+3, and 4x-3, the value of x is 7

In a right triangle, the Pythagorean theorem states that the sum of the squares of the two shorter sides equals the square of the longest side (the hypotenuse). Using this theorem, we can set up an equation as follows

x^2 + (3x+3)^2 = (4x-3)^2

Expanding the squares, we get

x^2 + 9x^2 + 18x + 9 = 16x^2 - 24x + 9

Simplifying and rearranging terms, we get

6x^2 - 42x = 0

Factorizing out x, we get

x(6x - 42) = 0

Therefore, x = 0 or x = 7. However, since the sides of the triangle cannot be zero, we must choose x = 7 as the solution.

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I have solved the question in general as the given question is incomplete:

The complete question is:

If the sides of a right triangle are x, 3x+3, and 4x-3, then what is the value of x?

The radius of the cylinder is approximately 9.02 mm.

What is cylinder?

A cylinder is a three-dimensional shape that consists of two parallel and congruent circular bases that are connected by a curved lateral surface. It can be thought of as a stack of circular discs or coins of equal size, with a constant distance between them. The axis that passes through the center of the circular bases is called the axis of the cylinder.

The formula for the volume of a cylinder is:

[tex]$$ V = \pi r^2 h $$[/tex]

where [tex]$V$[/tex] is the volume, [tex]$r$[/tex] is the radius, and [tex]$h$[/tex] is the height.

We can rearrange this formula to solve for the radius:

[tex]$$ r = \sqrt{\frac{V}{\pi h}} $$[/tex]

Substituting the given values, we get:

[tex]$$ r = \sqrt{\frac{4700}{\pi \cdot 31}} $$[/tex]

Simplifying this expression, we get:

[tex]$$ r \approx 9.02\ \text{mm} $$[/tex]

Therefore, the radius of the cylinder is approximately 9.02 mm.

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Answer:

The GCF of 16 and 40 is 8. To calculate the greatest common factor of 16 and 40, we need to factor each number (factors of 16 = 1, 2, 4, 8, 16; factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40) and choose the greatest factor that exactly divides both 16 and 40, i.e., 8.

Answer:

The GCF is 8 ; factored is 8(5+2)

Step-by-step explanation:

The gcf of 40 and 16 is 8.

Since 8*5 = 40 and 8*2 = 16:

40+16 = 8(5+2).

The relationship between the height of these objects and their shadow length is approximately proportional because increase in height leads to increase in their length.

Direct proportional relationship is defined as the type of relationship whereby an increase in one variable leads to the increase in the other variable.

From the diagram given above, all the objects are of various heights and the length here f their shadow increased with increase in the heights of the objects.

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Answer:

(4, 1 )

Step-by-step explanation:

3x - 5y = 7 → (1)

2x + 5y = 13 → (2)

the 'best' or 'simplest' way to solve this system is the elimination method

since the signs of 5y are opposites then adding will eliminate the y term

add (1) and (2) term by term to eliminate y

(3x + 2x) + (- 5y + 5y) = 7 + 13 , that is

5x + 0 = 20

5x = 20 ( divide both sides by 5 )

x = 4

substitute x = 4 into either of the 2 equations and solve for y

substituting into (2)

2(4) + 5y = 13

8 + 5y = 13 ( subtract 8 from both sides )

5y = 5 ( divide both sides by 5 )

y = 1

solution is (4, 1 )

Answer:

Step-by-step explanation:

in the middle is 0 in the first one is -5 and last one is 5

Answer: 7[tex]x^{2}[/tex]+13x-7

Step-by-step explanation: (3x + 7)(3x - 1) = 9[tex]x^{2} \\[/tex] + 18x - 7. (2x + 5)x = 2[tex]x^{2}[/tex]+5x. 9[tex]x^{2} \\[/tex] + 18x - 7 - (2[tex]x^{2}[/tex]+5x) = 7[tex]x^{2}[/tex]+13x-7

Answer:

3 2/3.

Probably the easiest question I've answered all day!

Answer:

Find the heigh if a=550cm and r=7cm

C) 92.65%

To find the percent decrease in customers, we first need to find the difference in the number of customers between the two given hours.

Number of customers on Saturday from 6:00 p.m. to 7:00 p.m. = 68

Number of customers on Wednesday from 5:00 p.m. to 6:00 p.m. = 5

Difference in number of customers = 68 - 5 = 63

Next, we need to find the percent decrease in customers, which is given by:

(percent decrease) = (difference / original) x 100%

where "original" refers to the number of customers on Saturday from 6:00 p.m. to 7:00 p.m.

Plugging in the values, we get:

(percent decrease) = (63 / 68) x 100% ≈ 92.65%

Therefore, the percent decrease in customers from the given hour on Saturday to the given hour on Wednesday is approximately 92.65%.

Answer: ~1256.64

Step-by-step explanation:

The equation for the surface area of a sphere is 4πr². So you would substitute 10 in for r which would then be 4π10². That will get you 1256.63706 but rounded it would be approximately 1256.64.

Answer: [tex]\text{400}\pi \ \text{square inches}[/tex]

Step-by-step explanation:

Given: The radius of the sphere = 10 in.

The formula to calculate the surface area of the sphere is given by:

[tex]\text{Surface Area}=4\pi r^2[/tex], where r is the radius of the sphere.

For r = 10 in. we have

[tex]\text{Surface Area of the sphere}=4\pi (10)^2[/tex]

[tex]\implies \text{Surface Area of sphere}=4\pi (100)[/tex]

[tex]\implies \text{Surface Area of the sphere}=\text{400}\pi \ \text{square inches}[/tex]

The two-column proofs of the segments are shown below

The proof is as follows

Statement                                                      Reason

ABCD and BCDE are parallelogram            Given

AG = BC                                                         Opposite sides of

                                                                       parallelogram            

ED = BC                                                         Opposite sides of

                                                                       parallelogram            

AG ≅ ED                                                         Substitution property (proved)

Proving that KLMN is a parallelogram

The proof is as follows

Statement                                                Reason

KL || NM  and ∠L ≅ ∠N                            Given

KN ≅ LM                                                   CPCTC

KL ≅ NM                                                   CPCTC

We've proved that the opposites sides are equal and parallel

So, KLMN is a parallelogram

Proving that STUV is a parallelogram

The proof is as follows

Statement                                                Reason

ST || VU  and W is midpoint of SU          Given

SW = UW                                                   Definition of midpoint

VW = TW                                                   Definition of midpoint

VU = ST                                                     CPCTC

VS = UT                                                     CPCTC

We've proved that the opposites sides are equal and parallel

So, STUV is a parallelogram

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Answer:

-3x^3y^2 - 4x^2y + 8y.

Step-by-step explanation:

When we distribute the negative sign to the terms inside the parentheses, we get:

(2x^3y^2 − 8x^2y + 4y) − (5x^3y^2 − 4x^2y − 4y) = 2x^3y^2 − 8x^2y + 4y - 5x^3y^2 + 4x^2y + 4y

Now, we can combine like terms. The terms with x^3y^2 are 2x^3y^2 and -5x^3y^2, which combine to give -3x^3y^2. The terms with x^2y are -8x^2y and 4x^2y, which combine to give -4x^2y. The terms with y are 4y and 4y, which combine to give 8y. Therefore, we have:

(2x^3y^2 − 8x^2y + 4y) − (5x^3y^2 − 4x^2y − 4y) = -3x^3y^2 - 4x^2y + 8y

Therefore, the simplified expression is -3x^3y^2 - 4x^2y + 8y.

The expression can be simplified by subtracting the coefficients of matching terms from the first and second parentheses. The simplified expression is -3x3y2 - 4x2y + 8y.

To simplify the expression (2x3y2 − 8x2y + 4y) − (5x3y2 − 4x2y − 4y), you must subtract each term in the second parentheses from the corresponding term in the first. In this case, it works as follows:

So, the simplified expression becomes −3x3y2 − 4x2y + 8y.

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There are some traits that rectangles, squares, and rhombuses have in common. These are also some differences between them. Rectangles, squares, and rhombuses all have 4 sides. If a shape is a rectangle the diagonal sides will be congruent. If a shape is a square all the sides will be congruent, the shape will also have a right angle. The diaganles of a rombus are perpendicular.

Answer: yes it is rational because it’s not within a square root symbol

Step-by-step explanation:

Answer:

exact form-512/7

decimal form-73.1

Mixed number form-73 1/7

Step-by-step explanation:

Answer:

41

Step-by-step explanation:

A = LW + (1/2)bh

A = 7 m × 5 m + (1/2)(3 m)(4 m)

A = 35 m² + 6 m²

A = 41 m²

The probability of spending more than 4 days in recovery is 0.8554

To calculate the probability of spending more than 4 days in recovery, we need to use the standard normal distribution since the patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.6 days and a standard deviation of 1.5 days.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

If X is a normally distributed random variable with mean μ and standard deviation σ, then Z = (X-μ) / σ is a standard normal random variable.

So, P(X > 4) = P(Z > (4-5.6) / 1.5) = P(Z > -1.0667)

Using a standard normal distribution table, we find that P(Z > -1.0667) = 0.8554

Therefore, the probability of spending more than 4 days in recovery is 0.8554, rounded to four decimal places.

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Answer:

I did the first image. I'm sorry but I don't remember fully how to do the other two, and I don't want to give you the wrong answer. GL

Step-by-step explanation:

Okay, for the first image if the perimeter is 48 you can divide that by 3 getting you 16. M=16.

Answer:

Now: Carolyn 18 yo, Alfred 12 yo

Step-by-step explanation:

Use X represent Alfred's age now, then Carolyn's age is (X+6)

6 years ago, their ages should be current age minus 6. So Alfred's age was (X-6), Carolyn's age was ((X+6)-6).

As 6 years ago, Carolyn's age was twice as Alfred's. So

((X+6)-6)=(X-6)×3

X=12 which is Alfred' current age

X+6=18 which is Carolyn' current age

The proportion of adults who have little faith in their automobiles, or the chance of success (p), is 44%.

The probability of an event occurring is referred to as probability. We frequently have to make predictions about the future in real life. We may or may not be aware of the outcome of an event.

At the point when this happens, we broadcast that there is plausible that the occasion will happen.

In conclusion, probability can be put to great use in business as well as in this rapidly expanding field of artificial intelligence. By simply dividing the total number of outcomes by the favorable number of possibilities, the probability of an event can be calculated using the probability formula.

Given that "p" stands for probability—that is, the fact that 45 percent of adults in the United States have little faith in their automobiles—p = 0.45, the likelihood of failure is q = 1-p = 1 - 0.45.

The chance of success (p), or the percentage of adults who have little faith in their automobiles, is therefore 44%, with q =0.55.

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Spanish playing cards have four suits: coins, cups, swords, and clubs, with each suit representing a different aspect of life, and consisting of ten numbered cards and three court cards.

Spanish playing cards traditionally have four suits: coins, cups, swords, and clubs. These suits are similar to the suits found in Italian and Latin American playing cards, but different from the suits found in French and German playing cards. The coins suit represents wealth and commerce, the cups suit represents love and pleasure, the swords suit represents warfare and justice, and the clubs suit represents agriculture and prosperity. Each suit typically contains ten numbered cards and three court cards: the knight, the queen, and the king.

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According to the given information, the correct answer is Bus 18, with a median of 13.

What is median?

The median is a measure of central tendency in statistics. It is the value that separates the dataset into two equal halves, with half the values being greater than the median and half being less than the median.

To compare the travel times of the two buses and determine which typically has the faster travel time, we need to look at the measures of central tendency.

The two common measures of central tendency are mean and median.

For Bus 18, the median is 13, which means that half of the students take less than 13 minutes to travel to school and half take more than 13 minutes. The mean can be calculated by finding the sum of all travel times and dividing by the number of students, but this information is not provided in the given line plot.

For Bus 47, the median is 16, which means that half of the students take less than 16 minutes to travel to school and half take more than 16 minutes. The mean can be estimated by calculating the average of all the travel times, which is:

(4+6+10+10+12+12+14+14+18+18+22+22+28+28+28)/15 = 16

Therefore, the mean travel time for Bus 47 is also 16.

Comparing the median travel times, we can see that Bus 47 has a higher median of 16 compared to Bus 18's median of 13. This suggests that the travel times for Bus 47 are typically longer than those for Bus 18, and therefore Bus 18 typically has the faster travel time.

Therefore, the correct answer is Bus 18, with a median of 13.

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For the given congruence triangle:

Congruent triangles are triangles that are precisely the same size and shape. The word congruent has the sign. When the three sides and three angles of one triangle match the same dimensions as the three sides plus three angles of another triangle, two triangles are said to be congruent.

For the given data:

Statement                                  Reasons

U is the midpoint of ST             Given

SV  ≅ TW                                   Given

VU ≅ WU                                  Given

SU = UT                                    U is the midpoint of ST

ΔSUV ≅ ΔTUW                         By Side Side Side (SSS) congruence

∠SVU  ≅  ∠TWU                       BY CPCT

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Answer:

a. 6x^2

b. 70x^2

c. -360s^2

d. -60x^2

Step-by-step explanation:

Toby sold 16 more games than Molly.

A ratio is a comparison of two or more quantities of the same kind. It is often expressed as a fraction or with a colon (:). For example, the ratio of the number of boys to the number of girls in a class of 30 students could be 2:3, which means that for every two boys, there are three girls.

Let's start by finding the total number of parts in the ratio: 5 + 3 + 1 = 9.

Next, we can use the ratio to find out how many parts of the total each person sold:

Katie: 5/9 of 72 games = 40 games

Toby: 3/9 of 72 games = 24 games

Molly: 1/9 of 72 games = 8 games

To find out how many more games Toby sold than Molly, we can subtract Molly's sales from Toby's sales: 24 - 8 = 16.

Therefore, Toby sold 16 more games than Molly.

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Answer:
Toby sold 16 more games than Molly.

Step-by-step explanation:

Let's start by finding the total number of parts in the ratio: 5 + 3 + 1 = 9.

Next, we can use the ratio to find out how many parts of the total each person sold:

Katie: 5/9 of 72 games = 40 games

Toby: 3/9 of 72 games = 24 games

Molly: 1/9 of 72 games = 8 games

Then we subtract Toby's number of games by Molly's. (24-8)

Therefore, Toby sold 16 more games than Molly.

The x-intercepts (zeros), end behaviors, degree and multiplicity of the function f(x) = (x - 2)²·(x + 1)·(x - 4), can be used to graph the polynomial in factored form. Please find attached the graph of the polynomial created with MS Excel.

A zero is an input variable value that makes the value of the output to be zero. A root is a value that satisfies the equation.

The steps to graph the polynomial in factored form are;

1. Find the x-intercepts; The x-intercepts are obtained by setting each  factor as being equivalent to zero and solve for x. The values obtained are the x-intercepts or the zeros of the function.

The specified function, f(x) = (x - 2)²·(x + 1)·(x - 4), the zeros are; x = 2, (with a multiplicity of 2), x = -1, and x = 4.

2. Determine the end behavior of the function: The degree and leading coefficient of a polynomial determines the end behavior of the polynomial.
The degree of a function is the highest exponent of the function. The highest exponent of x in the function f(x) is 4, therefore, the degree of f(x) is 4.

The leading coefficient is the coefficient of the term with the highest exponent of x. The leading coefficient of f(x) is therefore, the coefficient of the term with the highest exponent of x, which is 1

3. Determine the multiplicity of the zeros: The number of times a factor appears in the factored form determines the multiplicity of the zero of the factor.

The zero of x = 2 in the function has a multiplicity of 2 indicating that the graph touch and bounces upwards on the x-axis at x = 2.

4. Determine the behaviour at the x-intercepts: The multiplicity of the zero determines the behavior of the graph at the x-intercept, such that an odd multiplicity indicates that the graph crosses the x-axis and an even multiplicity indicates that the graph touches the x-axis and bounces upwards or downwards at the point.

The information in the steps 1 – 4 can then be used to sketch the graph of the polynomial.

Please find attached the graph of the polynomial created with MS Excel,  using the function, f(x) = (x - 2)²·(x + 1)·(x - 4)

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Let's first define some events:

A: The person has COVID-19.

B: The test result is positive.

Using the information provided, we can calculate the probability of a correct diagnosis using Bayes' theorem:

P(A|B) = P(B|A) * P(A) / [P(B|A) * P(A) + P(B|not A) * P(not A)]

where P(B|A) is the probability of a positive test result given that the person has COVID-19, P(A) is the prevalence of COVID-19 in the population (0.083 or 8.3%), P(B|not A) is the probability of a positive test result given that the person does not have COVID-19 (false positive rate, 0.05 or 5%), and P(not A) is the complement of P(A) (0.917 or 91.7%).

Plugging in the values, we get:

P(A|B) = (0.98 * 0.083) / [(0.98 * 0.083) + (0.05 * 0.917)]

= 0.6037 or 60.37%

Therefore, the probability that the diagnosis is correct is 60.37%, rounded to the nearest hundredth percent.

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Answer:

  (c)  AE = x·cos(α)/sin(θ)

  (d)  BD = x·cos(α)cos(α-θ)/cos(θ)

Step-by-step explanation:

In the given diagram, you want expressions for (c) AE and (d) BD in terms of x, α, and θ.

The law of sines can be used in ∆ACE to give the measure of AE.

[tex]\dfrac{AE}{\sin(C)}=\dfrac{CE}{\sin(\theta)}\\\\\boxed{AE=\dfrac{x\,\cos(\alpha)}{\sin(\theta)}}\qquad\text{where $\sin(C)=\cos(\alpha)$ and $CE=x$}[/tex]

The length BD can be found in terms of AD and θ:

  BD = AD·tan(θ)

The length AD can be found in terms of AE and (α-θ):

  AD = AE·cos(α-θ)

Using the value of AE from part (c), we have ...

  BD = AE·cos(α-θ)·tan(θ) = x·cos(α)cos(α-θ)·tan(θ)/sin(θ)

  BD = x·cos(α)cos(α-θ)/cos(θ) . . . . . . . using tan/sin = 1/cos

__

Additional comment

We assume angle EAC is θ throughout, as it is in part (a).

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Option B :The solution for x in the given circle is x = 5 + 2√6 or x = 5 - 2√6 is near to value 5.

In the given figure, we have two tangents and a secant that intersects the circle at points A and B.

By the tangent-secant theorem, we know that the product of the lengths of the secant segment (AX) and the exterior segment (XB) is equal to the product of the lengths of the entire secant (AB) and the interior segment (CX).

So, we have:

AX * XB = AB * CX

Substituting the given values, we get:

(3 + x) * (7 - x) = 5 * 5

Expanding the left side and simplifying, we get:

[tex]21 - x^2 = 25 - 10x[/tex]

Bringing all the terms to one side and simplifying, we get:

[tex]x^2 - 10x + 4 = 0[/tex]

Using the quadratic formula, we get:

x = (10 ± √([tex]10^2 - 414[/tex])) / (2*1)

x = (10 ± √96) / 2

x = 5 ± 2√6

Therefore, the solutions for x are:

x = 5 + 2√6 or x = 5 - 2√6

So, the answer is (B) 5.

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