what are the possible rational roots of f(r)=2r^3-2r^2+r+8

Answer: x = (-1 + sqrt(85)) / 6, or x = (-1 - sqrt(85)) / 6

Step-by-step explanation:

Let's simplify and solve the equation step by step:

Distribute the -5 and -4 on the left side and combine like terms:

-5x + 10 + 4x = 3x^2 - x - 4x + 8 + x + 2

Simplifying the left side and combining like terms on the right side, we get:

-x + 10 = 3x^2 + 3

Subtract 10 from both sides to isolate the variable on one side:

-x = 3x^2 - 7

Add x to both sides to get the quadratic equation in standard form:

3x^2 + x - 7 = 0

Use the quadratic formula to solve for x:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 3, b = 1, and c = -7. Substituting these values, we get:

x = (-1 ± sqrt(1^2 - 4(3)(-7))) / 2(3)

Simplifying under the square root, we get:

x = (-1 ± sqrt(85)) / 6

Therefore, the solutions to the equation are:

x = (-1 + sqrt(85)) / 6, or x = (-1 - sqrt(85)) / 6

These are approximate values since sqrt(85) is irrational.

if lucky got 20 questions correct then lucky had missed 4 tests.

If Lucy got 4 out of 5 questions correct, then the proportion of questions she got correct is:

4/5 = 0.8

We can use this proportion to find the number of questions she got correct in the larger set of 20 questions:

0.8 x 20 = 16

So, Lucy got 16 questions correct out of 20. To find the number of questions she missed, we can subtract the number she got correct from the total number of questions:

20 - 16 = 4

Therefore, Lucy missed 4 questions on the test.

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Answer:

jiris age is 32

Step-by-step explanation:

39(palma's age)+32(jiris age)=71

Answer:

u = 4 , v = 2[tex]\sqrt{3}[/tex]

Step-by-step explanation:

using the tangent and sine ratios in the right triangle and the exact values

tan30° = [tex]\frac{1}{\sqrt{3} }[/tex] , sin30° = [tex]\frac{1}{2}[/tex] , then

sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{2}{u}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )

u = 2 × 2 = 4

and

tan30° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{2}{v}[/tex] = [tex]\frac{1}{\sqrt{3} }[/tex] ( cross- multiply )

v = 2[tex]\sqrt{3}[/tex]

The possible lengths of the stage (in feet) can be described by the inequality:

L ≥ A/10

where A is the minimum required area of the stage and L is the length of the stage.

Let's denote the length of the rectangular stage as 'L' in feet.

The area of a rectangle is given by the formula A = L × W, where A is the area, L is the length, and W is the width.

We are given that the width of the stage is 'W' feet and the area of the stage must be at least 'A' square feet. So we can write the inequality:

A ≤ L × W

Substituting the given values, we get:

A ≤ L × 10

Dividing both sides by 10, we get:

A/10 ≤ L

Therefore, the possible lengths of the stage (in feet) can be described by the inequality:

L ≥ A/10

where A is the minimum required area of the stage and L is the length of the stage.

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When the circle's radius is 8 cm and it is shrinking at a rate of 0.4 cm per second, the rate that the area is shrinking is -6.4 cm²/s.

Circle area formula: r² = r. The radius squared is multiplied to find the circumference of the circle. When a circle's radius is specified, its area is equal to r². When the diameter "d" is known, the circle's surface area is equal to d²/4.

90 degrees is equal to a quarter turn. A complete turn is 360 degrees, while a half turn is 180. Moreover, the rotation can be expressed in radians or fraction of pi. Under this system, a complete circle turn is equal to 2 radians, a half circle turn is equal to 1, and so on.

Step 1: Variables involved in the question:

The variables involved in the question are:

The radius of the circle r = 8 cm (constant value)

The rate at which the radius is decreasing (dr/dt) = -0.4 cm/s (negative because it's decreasing)

The rate at which the area of the circle is changing dA/dt = ?

Step 2: Chain rule:

We can use the chain rule to link the three rates:

dA/dt = dA/dr x dr/dt

Step 3:

From the given information, we know that (dr/dt) = -0.4 cm/s.

Step 4: Equation for dA/dr

The formula for the area of a circle is  A = πr², where r is the radius of the circle. We can differentiate both sides of the equation with respect to r to get the following formula for dA/dr:

dA/dr = 2πr

Step 5:

Substituting the given value of r = 8 cm in the equation for dA/dr, we get:

dA/dr = 2π(8) = 16π cm

Step 6: dA/dt:

Now we can use the chain rule to find dA/dt by substituting the values we have found for dA/dr and dr/dt:

dA/dt = dA/dr x dr/dt

dA/dt = 16π cm x -0.4 cm/s

dA/dt = -6.4π cm²/s

Therefore, the rate at which the area of the circle is decreasing when its radius is 8cm and decreasing at -0.4 cm/s is  -6.4π cm²/s.

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Answer:

125.6637061

Step-by-step explanation:

Answer:

40π inches, or approximately 125.66 inches

Step-by-step explanation:

The formula to find the circumference of a circle is C = 2πr, where r is the radius of the circle and π is a mathematical constant approximately equal to 3.14.

In this case, the radius of the tire is 20 inches. So, the circumference of the tire is:

C = 2πr

C = 2π(20)

C = 40π

Therefore, the circumference of the tire is 40π inches, or approximately 125.66 inches if you round to two decimal places.

If there were 56 people who were all 48 years old, then the total number of years that the people had lived would be: 56 people x 48 years/person = 2,688 years

This calculation multiplies the number of individuals (56 people) by their age (48 years per person) to determine the total number of years of life experience represented by the group. In essence, it is calculating the cumulative age of everyone in the group. The result, 2,688 years, reflects the sum total of all the years that the 56 people have lived in total. Therefore, the total number of years that the people had lived is 2,688.

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The correct answer is x = 9.

Making use of the logarithm table, Compute the characteristic that the provided integer's whole number component dictates. Using the significant digits of the given number, find the mantissa. Add a decimal point after combining the characteristic and mantissa.

The student's solution has a division by zero error. The wrong step is when ln(x2) = ln(3x) - 0. It would have been better to first simplify the addition of ln(9) - 2ln(3) as follows:

The formula is ln(9) - 2ln(3) = ln(9) - ln(32) = ln(9/32) = ln(1/3).

When we substitute this number into the first equation, we get:

ln(1/3) - ln(3x) = 2ln(x)

ln(3x/1/3) = 2ln(x)

2ln(x) = ln (9x)

If we multiply both sides by their exponential, we get:

E = 2ln(x) + eln (9x)

x^2 = 9x

x^2 - 9x = 0

x(x - 9) = 0

As a result, the solutions are x = 0 and x = 9, but since ln(0) is undefined, we must determine whether x = 0 is a valid solution. So x = 9 is a workable answer.

Hence, the right response is x = 9.

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Mean value of the function = E(X) = ∞/πOption (b) ∞/π is the correct option.

Given a density function of a random variable X as f(x)={ 2 π ​ , 0, ​ 0. The question is to find the mean value of the function.As we know,The mean value of the function = E(X) = ∫xf(x)dx, where x is the random variable and f(x) is the density function,∫ denotes integral from negative infinity to infinityOn substituting the given values,∫xf(x)dx= ∫x (2/π)dx= (2/π) ∫xdx= (2/π)(x^2/2)+C ……(1)Where C is the constant of integration.But given the density function is zero for all negative x values and f(0) = 2/π, so the integral should be calculated from 0 to infinity instead of negative infinity to infinity.On substituting the values,∫0∞ x (2/π)dx= (2/π) ∫0∞xdx= (2/π) (x^2/2) [0,∞]= ∞/πTherefore, mean value of the function = E(X) = ∞/πOption (b) ∞/π is the correct option.

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Answer:

Step-by-step explanation:

The two 'slashes' on the 2 edges indicate that it's an isosceles triangle ( the base angles are equal)

Total angles in a triangle = 180.

Thus,

3x + 4x+2+4x+2= 180.

11x+4=180

11x=176

Therefore, x= 16°

Thus,

3x = 3(16) = 48°

The (4x+2)'s =

4(16)+2

=66° each.

Hope this helps! :)

The function represents the height of the plant in centimetres after n days, so n cannot be negative.

Part A:

To find a reasonable domain to plot the growth function, we need to consider the practical limitations of the growth of the plant. Also, since the function represents the growth of a particular species of plant, there may be an upper limit to how many days the plant can grow.

Assuming that the plant is not a perennial plant and has a limited lifespan, we can choose a reasonable domain for the function as [0, t], where t is the expected lifespan of the plant in days.

Since we do not have information about the expected lifespan of the plant, we can choose a reasonable value such as [tex]t = 365[/tex]  (assuming it is an annual plant). So the domain for the function can be  [tex][0, 365][/tex]  .

Part B:

The y-intercept of the graph of the function f(n) represents the height of the plant when it was planted or started growing, that is, at n = 0. To find the y-intercept, we can substitute n = 0 in the equation:

[tex]f(0) = 15(1.02)^0 = 15[/tex]

Therefore, the y-intercept of the graph of the function f(n) is [tex]15[/tex] cm.

Part C:

The average rate of change of the function f(n) from n = 1 to n = 4 can be calculated using the formula:

average rate of change [tex]= [f(4) - f(1)] / (4 - 1)[/tex]

Substituting the values in the equation, we get:

average rate of change [tex]= [15(1.02)^4 - 15(1.02)^1] / 3[/tex]

average rate of change [tex]≈ 1.42 cm/day[/tex]

Therefore, The average rate of change of the function f(n) from n = 1 to n = 4 represents the average daily growth rate of the plant during this period.

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The value of h of the parallelogram is = 4.2

A parallelogram is a special kind of quadrilateral made up of parallel lines. A parallelogram can have any angle between its adjacent sides, but for it to be a parallelogram, its opposite sides must also be parallel.

If the opposing sides of a quadrilateral are parallel and congruent, the shape will be a parallelogram. Hence, if both sets of opposite sides are parallel and equal, a quadrilateral is said to be a parallelogram.

A parallelogram's area is determined as follows:

Area = base × height

In the given parallelogram, we can note that the base is 8.4 inches, and the corresponding height is 5 inches.

This means that:

Area of parallelogram = 8.4 × 5 = 42 in²

This same area can be calculated using the other base (6 in) and its corresponding height (h)

This means that:

Area of parallelogram = 10 × h

42 = 10 × h

h = 42/10

h = 4.2 inches.

Hence, value of h in the parallelogram is equals to 4.2.

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Answer:

sin(2x) = 2sin(x)cos(x)

= 2(-3/5)(-2/5)

= 12/25

cos^2(x) = (2/√13)^2 = 4/13

sin^2(x) = (-3/√13)^2 = 9/13

cos(2x) = cos^2(x) - sin^2(x) = (4/13) - (9/13) = -5/13

= cos(2x) = -5/13

tan(2x) = 3/4

Explanation:

Given tan(x) = -3/2 and x terminates in quadrant II.

We know that tan(x) = sin(x) / cos(x)

Using this, we can find sin(x) and cos(x):

tan(x) = sin(x) / cos(x) = -3/2

cos(x) = -2/√13

sin(x) = 3/√13

Using double angle formulas, we can find sin(2x), cos(2x), and tan(2x):

sin(2x) = 2sin(x)cos(x) = 2(3/√13)(-2/√13) = -12/13

cos(2x) = cos²(x) - sin²(x) = (-2/√13)² - (3/√13)² = 4/13 - 9/13 = -5/13

tan(2x) = (2tan(x)) / (1 - tan²(x)) = (2(-3/2)) / (1 - (-3/2)²) = 3/4

We know that tan(x) = -3/2 and x is in quadrant II. Therefore, we can use the Pythagorean theorem to find the opposite side (y) and the hypotenuse (r) of a right triangle with an angle of x in quadrant II.

Let r = 2, so y = -3 and x = arctan(-3/2) ≈ -56.31°.

Using the double angle formulas:

sin(2x) = 2sin(x)cos(x) = 2(-3/2)(√5/2) = -3√5/2

cos(2x) = cos²(x) - sin²(x) = (√5/2)² - (-3/2)² = (5-9)/4 = -1/2

tan(2x) = 2tan(x)/(1-tan²(x)) = 2(-3/2)/(1-(-3/2)²) = 3/4

So, the answers are: sin(2x) = -3√5/2, cos(2x) = -1/2, and tan(2x) = 3/4.

Hope this helps you! Sorry if it's wrong. If you need more help, ask me! :]

Answer:$8,890.83

Step-by-step explanation:

The first term of the sequence is 30 (36 divided by 5 and then subtract 2).

A term by term rule is used for a sequence in which the next term is obtained from the previous term. Example: Arithmetic sequence. In an arithmetic sequence, each term (other than the first term) is obtained by adding or subtracting a constant value from the preceding term.

To work out the term to term rule, give the starting number of the sequence and then describe the pattern of the numbers. The first number is 3. The term to term rule is 'add 4'. Once the first term and term to term rule are known, all the terms in the sequence can be found.

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Answer:

15/13 OR 1.154

Step-by-step explanation:

2 1/2=5/2

2 1/6=13/6

5/2 / 13/6

=5/2 x 6/13

=30/26

=15/13

=1.15384615385

≈ 1.154

The value of angle 1 , angle 2 and angle 3 are 60°, 30° and 60° respectively

A polygon is called regular if it has equal sides and angles. Thus, a regular triangle is an equilateral triangle, and a regular quadrilateral is a square.

An hexagon is a six sided polygon. This means that;

The sum of angle in an hexagon = (6-2)180 = 180× 4 = 720

Therefore each angle in the hexagon = 720/6 = 120°

The angles are equally bisected

therefore, angle 3 = 120/2 = 60°

angle 1 = 180-(60+60)

= 180-120 = 60°

angle 2 = 180-(90+60)

= 180-150

= 30°

therefore the value of angle 1 , angle 2 and angle 3 are 60°, 30° and 60° respectively

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The polynomials P(x) can be formulized as P(x) =  (1/5)(x-1)²(x)(x+3).

The given problem states that a degree 4 polynomial, P(x), has the following properties:

The degree of the polynomial is 4, so the polynomial can be written as;

P(x) = a(x-1)²(x-0)(x+3)

To find the value of 'a', substitute the given point (5,128) into the equation, P(x);

P(5) = a(5-1)²(5-0)(5+3) = 128

P(5) = a(4)²(5)(8) = 128

Simplifying, we get;

128 = 640a,

a = 1/5

Thus, the formula for P(x) is; P(x) = (1/5)(x-1)²(x)(x+3)

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According to given conditions, the biggest number is 46.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as 1 : 3 (for every one boy there are 3 girls).

Let the two numbers in the first ratio be 3x and 5x, where x is some constant of proportionality.

According to the problem, if we subtract 9 from each number, the new ratio is 12:23. So, we have:

(3x - 9) : (5x - 9) = 12 : 23

We can cross-multiply to get:

23(3x - 9) = 12(5x - 9)

Simplifying this equation, we get:

69x - 207 = 60x - 108

9x = 99

x = 11

So, the two numbers in the first ratio are 3x = 33 and 5x = 55.

To find the biggest number, we need to determine which of these numbers is larger after subtracting 9.

33 - 9 = 24

55 - 9 = 46

Therefore, according to given conditions, the biggest number is 46.

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The length of PR is 6 units. To find the length of PR, we need to use the fact that the triangles are similar.

Since the triangles are similar, their corresponding sides are proportional. That is,

AB/DE = BC/EF = AC/DF

We know that AB = 8, BC = 6, AC = 10, DE = 6, EF = 4, and DF = 8. Therefore,

AB/DE = 8/6 = 4/3

BC/EF = 6/4 = 3/2

AC/DF = 10/8 = 5/4

Since we are looking for the length of PR, which corresponds to BC in the smaller triangle, we can use the ratio of BC/EF from above. We have:

BC/EF = 3/2 = PR/4

Solving for PR, we get:

PR = (3/2) * 4 = 6

Therefore, the length of PR is 6 units.

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The triangles are similar. PR = ___ units

In response to the given question, we can state that Rounding this to the  Pythagorean theorem nearest inch gives an answer of [tex]$\boxed{\textbf{(D) }45}$[/tex]

The Pythagorean Theorem is the foundational Euclidean geometry relationship among a right triangle's three sides. The area of a cube with the velocity vector side is the sum of something like the areas of squares that have the other two sides, according to this rule. The rectangle that spans the slope of a triangular shape across the sharp angle equals the total of the rectangles that span its sides, as defined by the Pythagorean Theorem. In general algebraic notation, it is written as a2 + b2 = c2.

The height of each triangle can be found using the Pythagorean theorem:

[tex]$15^2 - \left(\frac{20}{2}\right)^2 = 225 - 100 = 125$, so the height of each triangle is $\sqrt{125} = 5\sqrt{5}$.[/tex]

The overall height of the kite is the sum of the heights of the two triangles, which [tex]is $2 \cdot 5\sqrt{5} = 10\sqrt{5}$.[/tex]

Rounding this to the nearest inch gives an answer of [tex]$\boxed{\textbf{(D) }45}$[/tex]

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Answer: C 27

Step-by-step explanation:

there would be 24 sheep, which means there is 36 horses.

if ratio from horses to goats is 4-3

then there is only 27 goats.

Some of the features of the quadratic function y = 2(x - 2)² + 3 are

The vertex form of a quadratic equation is

y = a(x - h)² + k

Where

Using the above features, we have the following key features

Quadratic function 1

y = 2(x - 2)² + 3

Quadratic function 2

y = 3(x + 2)² - 2

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Wei-Ling's fifteenth-generation ancestor is estimated to have around 1,220,703 descendants.

Assuming that each descendant has an average of 2.5 children, we can use the formula:

where n is the number of generations.

In this case, n = 15 (fifteenth-generation), so we can plug in the values and get:

number of descendants = [tex](2.5)^{15}[/tex]= 1,220,703.125

Therefore, Wei-Ling's fifteenth-generation ancestor is estimated to have around 1,220,703 descendants. However, it is important to note that this is just an estimation and may not be entirely accurate due to various factors such as reproductive rates and other variables.

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The number of 11-inch softballs ordered is 70 and the number of 12-inch softballs ordered  is 50

a + b = 120 equation 1

2.50a + 3.50b = 350 equation 2

Where:

The elimination method would be used to determine the number of each size of softball ordered.

Multiply equation 1 by 2.5

2.5a + 2.5b = 300 equation 3

Subtract equation 3 from equation 2

b = 50

Substitute for b in equation 1

a + 50 = 120

a = 120 - 50

a = 70

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To streamline your betting odds and increase your chances of making a profit from your initial 100 scrap, you can use a combination of different betting strategies.

Here are a few possible strategies:

1) Spread your bets: Instead of betting all of your scrap on one number, you can spread your bets across multiple numbers to increase your chances of winning.

For example, you could bet 25 scrap on 1 (yellow), 25 scrap on 3 (green), 25 scrap on 5 (blue), and 25 scrap on 10 (pink). This way, you have a 48% chance of winning on 1 (yellow), a 24% chance of winning on 3 (green), a 16% chance of winning on 5 (blue), and an 8% chance of winning on 10 (pink).

2) Bet on the most likely outcomes: Another strategy is to bet on the numbers with the highest probabilities of winning. For example, you could bet 50 scrap on 1 (yellow) and 50 scrap on 3 (green), which have the highest probabilities of winning at 48% and 24%, respectively.

3) Use a combination of the above strategies: You could also combine the above strategies to increase your chances of winning. For example, you could bet 25 scrap on 1 (yellow), 25 scrap on 3 (green), 25 scrap on 5 (blue), and 25 scrap on 10 (pink), and then also bet an additional 50 scrap on 1 (yellow) and 50 scrap on 3 (green).

Overall, the key to streamlining your betting odds is to spread your bets across multiple numbers and focus on the numbers with the highest probabilities of winning. By using a combination of different betting strategies, you can increase your chances of making a profit from your initial 100 scrap.

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Using logarithmic identities and quadratic equation, the value of lg a * lg c is 4/3

Using logarithmic identities, we can rewrite the given equations as:

logₐa + log_b c = 4

log_b b + logₐ c = 3

Simplify the first equation using the fact that logₐa = 1/log_aₐ = 1, and simplify the second equation using the fact that log_b b = 1:

1 + log_b c = 4

1 + logₐ c = 3

Solve for log_b c and logₐ c:

log_b c = 3

logₐ c = 2

Now we can use the fact that logₐ c = logₐ (a · c)/a = logₐ a + logₐ c/a to write:

logₐ a + logₐ c/a = 2

Substitute logₐ c = 2 into this equation to get:

logₐ a + 2/a = 2

Multiply both sides by a to get:

logₐ a · a + 2 = 2a

Rearrange this equation as a quadratic equation:

logₐ a · a - 2a + 2 = 0

This quadratic has a maximum value when the coefficient of logₐ a is zero, i.e., when:

a = 2^(2/3)

Substituting this value of a into the equation logₐ c = 2, we get:

log₂ c = log₂ a^2 = 2/3 · log₂ 2^2 = 4/3

Therefore, logₐ c = log₂ c/log₂ a = (4/3) / (2/3) = 2

Finally, we can find the maximum value of logₐ a · logₐ c by multiplying logₐ a and logₐ c:

logₐ a · logₐ c = log₂ a/log₂ a · log₂ c/log₂ a = log₂ c = 4/3

Therefore, the maximum value of logₐ a · logₐ c is 4/3.

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