multiply (−3x+4)(2x−1)

Log3(14)

= log(14) / log(3) (Change of base formula)

= 4 / 1.965 (log(3) = 1.965)

= 2.041 (rounded to 3 decimal places)

2.041

Answer:

Step-by-step explanation:

To emulate the logarithm using the change of base formula, we need to use a base that is more convenient to evaluate. Let's use base 10:

log3(14) = log10(14) / log10(3)

Using a calculator, we can evaluate the numerator and denominator:

log10(14) ≈ 1.146

log10(3) ≈ 0.477

Dividing the numerator by the denominator gives:

log3(14) ≈ 2.402

Rounding to three decimal places, the final result is:

log3(14) ≈ 2.402

Answer:

8.5 yrs

Step-by-step explanation:

parent: 41 yrs

eldest son: x

Firstly, we will need to subtract 24 from 41 to get 2x the eldest son's age.

By doing that, we will have an equation that looks like this:

41-24=2x

This equation would basically get us through the whole problem.

Simplify the equation:

17=2x

x=8.5

The eldest son's age is 8.5.

Let's check our work!

8.5 x 2 = 17

17+24=41 (parent's age.)

Hope this helps :)

The complete equation for the cosine rule is c² = 3² + 10² - 2(3)(10) × cos(34).

The cosine rule, also known as the law of cosines, is a formula used to find the lengths of sides or measures of angles in a triangle. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The cosine rule is typically used when you have a triangle with at least one known side length and its opposite angle, and you want to find either another side length or another angle in the triangle.

The cosine rule is given by the following formula:

c² = a² + b² - 2ab × cos(C)

where:

For the given side c;

c² = 3² + 10² - 2(3)(10) × cos(34)

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Check the picture below, so the angle looks more or less like so.

now, let's recall that the value pair in a terminal point is the adjacent and opposite sides

[tex]\cot(\theta )=\cfrac{\stackrel{adjacent}{0}}{\underset{opposite}{5}}\implies \cot(\theta )=0[/tex]

The surface area of the rectangular prism is 502 mm²

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional state.

The surface area of the prism = 2(8 mm * 13 mm) + 2(8 mm * 7 mm) + 2(7 mm * 13 mm) = 502 mm²

The surface area is 502 mm²

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* Mary's solar panels cost $2,000.

* The panels will reduce her monthly electricity bill by $40.

* She can earn 12% interest compounded monthly.

* So her monthly savings is $40

* And her 12% monthly interest rate is 12% / 12 = 1% per month.

* So each month her balance grows by 1% of the current balance.

* Let's think through this step-by-step:

* Initial balance = $2,000 (from paying for the solar panels)

* Month 1:

** Savings = $40 (from lower electric bill)

** Interest = 1% of $2,000 = $20

** Balance after Month 1 = $2,000 + $40 + $20 = $2,060

* Month 2:

** Savings = $40

** Interest = 1% of $2,060 = $20

** Balance after Month 2 = $2,060 + $40 + $20 = $2,120

* Month 3: (continue the calculations for Months 3 through 24)

** Savings = $40

** Interest = 1% of $2,121 = $21

** Balance after Month 3 = $2,121 + $40 + $21 = $2,182

* After 24 months, the balance is $3,149 (calculated step-by-step as shown above)

* The initial investment was $2,000

* So it took about 24 months to recoup her investment.

Does this help explain the steps? Let me know if you have any other questions!

Answer:

Valid

Step-by-step explanation:

The volume of the batter in the pan can be found by multiplying the length, width, and height of the right rectangular prism-shaped pan. In this case, the length is 9 inches, the width is 5 inches, and the height is 3 inches.

Using the formula for the volume of a rectangular prism, V = lwh, we can substitute the given values to find:

V = (9 in.)(5 in.)(3 in.)

V = 135 cubic inches

Therefore, the volume of the batter in the pan is 135 cubic inches. This argument is valid because it follows the basic formula for calculating the volume of a rectangular prism and uses the specific measurements provided for the length, width, and height of the pan.

The mean and the median of the following gasoline prices that are listed above would be =3.29 and 4.85 respectively.

The formula that is used to calculate the mean of a data set is given as follows;

mean = sum of data set/number of data set

= 3.14+3.21+3.28+3.53/4

= 13.16/4 = 3.29

The median = 3.21+3.28/2 = 4.85

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[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$879\\ r=rate\to 19.5\%\to \frac{19.5}{100}\dotfill &0.195\\ t=years\dotfill &1.5 \end{cases} \\\\\\ A = 879[1+(0.195)(1.5)] \implies A=879(1.2925)\implies A \approx 1136.11[/tex]

The functions which have a y-intercept which is greater than the y-intercept of the function g(x) = |x + 3| + 4 are :

h(x) = -5 |x| + 10

k(x) = 1/4 (x - 4)² + 4

m(x) = 1/4 |x - 8| + 6

Given function is,

g(x) = |x + 3| + 4

y intercept of a function is the value of the function when the input value is 0.

g(0) = |0 + 3| + 4 = 7

f(x) = -2 (x - 8)²

f(0) = -2 (0 - 8)² = -128

h(x) = -5 |x| + 10

h(0) = 10

j(x) = -4(x + 2)² + 8

j(0) = -4 (0 + 2)² + 8 = -8

k(x) = 1/4 (x - 4)² + 4

k(0) = 1/4 (0 - 4)² + 4 = 8

m(x) = 1/4 |x - 8| + 6

m(0) = 1/4 |0 - 8| + 6 = 8

Hence the functions h(x) = -5 |x| + 10, k(x) = 1/4 (x - 4)² + 4 and m(x) = 1/4 |x - 8| + 6 have the y intercept greater than that of g(x) = |x + 3| + 4.

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The slope of the line that is parallel to the line that contains side WV is 1.

What about slope of line?

The slope of a line is a measure of its steepness or inclination, defined as the ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis) between any two points on the line.

In other words, the slope of a line represents the rate of change of the dependent variable (y) with respect to the independent variable (x). It is commonly denoted by the letter "m" and is calculated as:

m = (y₂ - y₁) / (x₂ - x₁) where (x₁, y₁) and (x₂, y₂) are any two points on the line.

The slope can be positive, negative, zero, or undefined. A positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls as we move from left to right. A slope of zero indicates that the line is horizontal, and an undefined slope indicates that the line is vertical.

According to the given information:

W(a+3, b-3) and V(a+5, b-1)

So, the slope WV,

[tex]\frac{b-3-(b-1)}{a+3 - (a+5)} \\\\\frac{-2}{-2} = 1[/tex]

So, the slope of the given condition is 1.

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An elephant weighs 110 pounds more on Day 60 than it does on Day 5

Here, function w = 2d +200 represents the weight w of an elephant on a given day during his first year.

From above function , the weight of an elephant on Day 60 would be,

w₆₀ = 2(60) + 200

w₆₀ = 120 + 200

w₆₀ = 320 pounds

and the weight of an elephant on Day 5 would be,

w₅ = 2(5) + 200

w₅ = 10 + 200

w₅ = 210

The difference between these weights is:

w = w₆₀ - w₅

w = 320 - 210

w = 110 pounds

Therefore, an elephant weighs 110 pounds more on Day 60 than on Day 5

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The points that is in the solution set of y < x² - 2x - 8 is option A: (-2, -1)

When you Check the inequality by (-2,-1) then Put x=-2 and y=-1 in the given inequality.

1 < (-2)² - 2(-2) - 8

-1 < 4 + 4 - 8

-1 < 0

Hence:

(-2,-1) can be in the solution

( 0,-2) Cannot

(4,0) will not.

Therefore, The points that is in the solution set of y < x² - 2x - 8 is option A: (-2, -1)

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See full question below

Which of the following points is in the solution set of y < x2 - 2x - 8?

(-2, -1)(0, -2)(4, 0)

Answer:

∠ BAC = 66°

Step-by-step explanation:

the inscribed angle BAC is half the angle at the centre, subtended on the same arc BC , then

∠ BAC = [tex]\frac{1}{2}[/tex] × 132° = 66°

So the area of the triangle is 216 square inches.

To find the area of a triangle, you can use the formula A = 1/2 * b * h, where A is the area, b is the length of the base, and h is the height of the triangle. In this case, we have a height of 12 inches and a base of 36 inches.

To find the length of the missing side of the triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. So, we have:

c² = a² + b²

c² = 12² + 34²

c² = 144 + 1156

c² = 1300

c = √(1300)

c = 36.06 (rounded to two decimal places)

Now that we know all three sides of the triangle, we can plug them into the formula for the area:

A = 1/2 * b * h

A = 1/2 * 36 * 12

A = 216

So the area of the triangle is 216 square inches.

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The solution to the system of equations is x = 6 and  y = -5, which is the same as we obtained using the elimination method.

A system of equations is a collection of one or more equations that are considered together. The system can consist of linear or nonlinear equations and may have one or more variables. The solution to a system of equations is the set of values that satisfy all of the equations in the system simultaneously. The given system of equations is:

2x + y = 7 ---(1)

x + y = 1 ---(2)

To solve this system, we can use the method of elimination or substitution.

Method 1: Elimination

In this method, we eliminate one of the variables by adding or subtracting the two equations. To do this, we need to multiply one or both equations by a suitable constant so that the coefficients of one of the variables become equal in magnitude but opposite in sign.

Let's multiply equation (2) by -2, so that the coefficient of y in both equations becomes equal in magnitude but opposite in sign:

-2(x + y) = -2(1) --

Multiplying equation

(2) by -2-2x - 2y = -2

Now we can add the two equations (1) and (-2x - 2y = -2) to eliminate y:

2x + y = 7(-2x - 2y = -2)0x - y = 5

We now have a new equation in which y is isolated.

To solve for y, we can multiply both sides by -1:

-1(-y) = -1(5)y = -5

Now that we know y = -5, we can substitute this value into equation (2) to find x:x + y = 1x + (-5) = 1x = 6

Therefore, the solution to the system of equations is (x,y) = (6,-5).

Method 2: Substitution

In this method, we solve one of the equations for one variable in terms of the other variable and substitute this expression into the other equation to get an equation with only one variable.

From equation (2), we can solve for y in terms of x:y = 1 - x

We can then substitute this expression for y into equation (1):2x + y = 72x + (1 - x) = 7 --Substituting y = 1 - xx + 1 = 7x = 6

Now that we know x = 6, we can substitute this value into equation (2) to find y:x + y = 16 + y = 1 --Substituting x = 6y = -5

Therefore, the solution to the system of equations is (x,y) = (6,-5), which is the same as we obtained using the elimination method.

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Expanding the left-hand side of the equation, we get:

(x+6)2 = (x+6)(x+6) = x(x+6) + 6(x+6) = x^2 + 6x + 6x + 36 = x^2 + 12x + 36

So now we have the equation:

x^2 + 12x + 36 = 8

Subtracting 8 from both sides, we get:

x^2 + 12x + 28 = 0

We can factor this quadratic equation as:

(x+2)(x+14) = 0

This gives us two possible solutions:

x+2 = 0, so x = -2

x+14 = 0, so x = -14

Therefore, the solutions to the equation (x+6)2 = 8 are x = -2 and x = -14.

Step-by-step explanation:

Here is one way :

(x+6)^2 = 8        Take the square root of both sides

x+6 =   +- sqrt 8

x = -6 +- sqrt8     = -6 + 2 sqrt 2   or   -6 - 2 sqrt 2  =   -  3.17   or - 8.83

Triangles RLG and PCN are similar. The similarity is by SSS criteria and the scale factor is 2/3.

Two triangles that are similar in shape but not necessarily in size are called similar triangles. To put it another way, the two triangles' angles are similar and their corresponding sides are proportional.

If two triangles are comparable, then all pairs of related sides have the same length-to-length ratio. The scale factor of similar triangles is the name given to this ratio. The scale factor between two triangles is 2:1, for instance, if one side of a triangle is twice as long as its counterpart side in another triangle.

In the given figure we have triangle RLG and triangle PCN.

Here, the ratio of the sides of the triangle are given as:

RG / PN = 32 / 48 = 2/3

Also we have:

LG / PC = 18/27 = 2/3

LR / CN = 30 / 45 = 2/3

RG / PN = LG / PC = LR / CN = 2/3

Hence, triangles RLG and PCN are similar. The similarity is by SSS criteria and the scale factor is 2/3.

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The complete question is:

On the basis of this information, the ratio of married male non-members to the married female non-members is (b) 3 : 1

Total married male employees = 80% × 50% = 40%

Total male Trade Union members = 91.67%

Total male non-Trade Union members = 100% - 91.67% = 8.33%

Married male Trade Union members = 40% × 91.67% = 36.67%

Married male non-Trade Union members = 40% - 36.67% = 3.33%

Now, let's find the number of married female non-members:

The ratio based on the information is 3 to 1.

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Answer:

1.5=slope

Step-by-step explanation:

To do this, we need to solve the slope.

We will use rise/run

0-3=-3

0-2=-2

-3/-2=1.5

Answer:

Karl has 2.67 pounds of flour left.

Step-by-step explanation:

First, let us dissect the given.

Second, let us identify the correct solution for this problem.

A recursive definition for the set S can be given as follows:

What is recursion?

Recursion is a programming technique where a function calls itself to solve a problem.

Base case: (1, n) is in S for all positive integers n, since 1 is a factor of all positive integers.

Recursive case: If (a, b) is in S, then (a', b) is in S for all positive integers a' that are factors of a, and (a, b') is in S for all positive integers b' that are multiples of b.

In other words, the set S contains all pairs (a,b) where a is a positive integer that divides b, and b can be obtained by multiplying any such a with another positive integer. The base case includes all pairs where a=1 and b is any positive integer.

The recursive case states that if (a,b) is in S, then all pairs where a' is a factor of a and b is a positive integer such that b=a'b are also in S, as well as all pairs where b' is a multiple of b and a is a positive integer that divides b'.

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The maximum height attained by the arrow is 2128 feet

What is kinematic equation for vertical motion?

The kinematic equation for vertical motion is [tex]h = h_0 + v_0 \times (\frac{1}{2} )t - gt^2[/tex]

where:

h = height attained by the arrow (maximum height)

[tex]h_0 = [/tex]initial height (80 feet)

[tex]v_0 = [/tex]initial velocity (256 feet per second)

g = acceleration due to gravity (-32 feet per second squared)

t = time taken to reach maximum height

At the maximum height, the velocity of the arrow is zero. So we can use the fact that the final velocity is zero to find the time taken to reach the maximum height,

[tex]0 = v_0 - g \times t \\ t = \frac{ v_0}{g}[/tex]

Substituting this value of t into the first equation, we get:

[tex]h = h_0 + v_0 \times ( \frac{v_0}{g}) - ( \frac{1}{2} )g( \frac{v_0}{g})^2 \\ h = 80 + (256) \frac{256}{(-32)} - ( \frac{1}{2} )(-32)\frac{256^2}{(-32)^2} \\ h = 80 + 4096 - ( \frac{1}{2} ) \frac{(256)^2}{(32)} \\ h = 80 + 4096 - 2048 \\ h = 2128 \: feet[/tex]

Therefore, the maximum height attained by the arrow is 2128 feet.

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The angles of the triangle are:

A =  95.7°, B = 33.6°, C = 50.7°.

What are the angles of the triangle?

The area created between two of a triangle's side lengths is known as the angle. Both internal and external angles are present in a triangle. In a triangle, there are three interior angles. When the sides of a triangle are stretched to infinity, exterior angles are created.

Here, we have

Given: a = 9, b= 5, c = 7

We have to find all angles in degrees.

Using law of cosines,  a² = b²+c² - 2bc cos(A),

9² = 5²+7² - 2(5)(7) cos(A)

81 = 25 + 49 - 70 cos(A)

7 = -70 cos(A)

cos(A) = -7/70

cos(A) = -1/10

A = cos⁻¹(-1/10)

A = 95.7°

Now, using the law of sines,  

[sin (A)]/(a) =  [sin (C)]/(c)

[sin (95.7°)]/(9) =  [sin (C)]/(7)

0.9955/9 = sin (C)/(7)

0.7742 = sinC

C = sin⁻¹(0.7742)

C = 50.7°

The sum of all the angles of a triangle= 180°

A + B + C = 180°

95.7° + B + 50.7° =  180°

B = 33.6°

Hence, the angles of the triangle are:

A =  95.7°, B = 33.6°, C = 50.7°.

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With a p-value of 0.022, there is convincing evidence of a difference in the proportion of men and women who intended to vote for Trump in the city. Type I error could have been made. No, there is not convincing evidence to support the claim that a greater proportion of women than men intended to vote for Trump. Increasing the significance level would increase power, but would also increase the likelihood of making a Type I error.

We can use a two-sample z-test to test for the difference in proportions between men and women who intended to vote for Trump.

Let p1 be the proportion of men who intended to vote for Trump and p2 be the proportion of women who intended to vote for Trump. Then the null and alternative hypotheses are

H0: p₁ = p₂

Ha: p₁ ≠ p₂

We can calculate the pooled sample proportion

p = (x₁ +  x₂) / (n₁ +  n₂)

= (120 + 132) / (250 + 240)

= 0.508

where x₁ = 120, x₂ = 132, n₁ = 250, and n₂ = 240.

We can calculate the test statistic

z = (p₁ - p₂) / √(p * (1 - p) * (1/n₁ + 1/n₂))

= (0.48 - 0.55) / √(0.508 * 0.492 * (1/250 + 1/240))

= -2.29

Using a standard normal distribution table, the p-value for a two-tailed test with a test statistic of -2.29 is 0.022. Since the p-value is less than 0.05, we reject the null hypothesis and conclude that there is convincing evidence that there is a difference in the proportion of all men and the proportion of all women in this city who intended to vote for Trump at the 0.05 significance level.

The mistake we could have made is a type I error, which is rejecting the null hypothesis when it is actually true. In this case, it would mean concluding that there is a difference in proportions between men and women who intended to vote for Trump when there is actually no difference.

The confidence interval for the difference in proportions is (-0.158, 0.018), which includes 0. Since 0 is in the interval, we cannot reject the null hypothesis that there is no difference in proportions between men and women who intended to vote for Trump.

Therefore, based on the interval, there is not convincing evidence to support the claim that a greater proportion of women than men intended to vote for Trump.

One way to increase the power of the test is to decrease the significance level (i.e., increase the alpha level). This would allow us to reject the null hypothesis more easily and increase the chance of detecting a true difference in proportions.

However, the drawback of making this change is that it increases the chance of making a type I error, which means rejecting the null hypothesis when it is actually true.

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The current yield of a 4.95% coupon at a price of $102.53 is [tex]4.82%[/tex]%.

The current yield of bond means return on an investment based on its current market price.

To calculate the current yield, we will use (annual interest payment/current market price * 100).

Annual interest payment = $1,000 face value x 4.95% coupon

Annual interest payment = $49.50

Current market price = 102.53% of the face value

Current market price = $1,025.30

Current yield = $49.50 / $1,025.30 x 100

Current yield = 4.82%

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(a) G'(a) = 2x - 3

(passing through (0, 3)) y1(x) = -3

(passing through (4, 7)) y2(x) = 5

b) The illustration of the graph is defined below.

In calculus, finding the derivative of a function is an important tool to understand the behavior of a curve at a specific point. One application of this concept is determining the equation of the tangent line to a curve at a given point. In this problem, we will use the derivative of a quadratic function to find the equations of tangent lines to the curve y = x² − 3x + 3 at the points (0, 3) and (4, 7).

To begin, we need to find the derivative of G(x) = x² − 3x + 3. Using the power rule, we have:

G'(x) = 2x - 3

Next, we can use this derivative to find the slope of the tangent line to the curve y = x² − 3x + 3 at any given point (a, G(a)). At the point (0, 3), we have a = 0, so the slope of the tangent line is:

G'(0) = 2(0) - 3 = -3

Using the point-slope equation of a line, we can find the equation of the tangent line passing through (0, 3). The equation of the tangent line is:

y - 3 = -3(x - 0)

Simplifying, we get:

y = -3x + 3

Similarly, at the point (4, 7), we have a = 4, so the slope of the tangent line is:

G'(4) = 2(4) - 3 = 5

Using the point-slope equation again, we can find the equation of the tangent line passing through (4, 7). The equation of the tangent line is:

y - 7 = 5(x - 4)

Simplifying, we get:

y = 5x - 13

To graph these tangent lines on the same screen as the curve y = x² − 3x + 3, we can plot the curve and the two tangent lines using a graphing calculator or software. The graph should show the curve as a parabola and the tangent lines as straight lines intersecting the curve at the points (0, 3) and (4, 7), respectively.

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By subtraction between two points P₁(x, y) = (- 5, 3) and P₂(x, y) = (- 2, - 7), the vector V is 3 i - 10 j.

According to linear algebra, a vector can be formed by subtracting the initial point (P₁) from the final point (P₂). The equation is introduced below:

V = P₂(x, y) - P₁(x, y)

V = (- 2, - 7) - (- 5, 3)

V = (3, - 10)

V = 3 i - 10 j

The vector V = 3 i - 10 j is the result of subtracting the two points P₁(x, y) = (- 5, 3) and P₂(x, y) = (- 2, - 7).

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Answer:

Class Rank | Frequency

--- | ---

Fr | 7

So | 3

Jr | 4

Se | 6