Consider the quadratic polynomial ax 2 bx c. suppose the coefficients are given by rolling a fair die three times. what is the probability that the roots of the polynomial are equal?

775757874875875345634634563456

205×3/100=6.15

Total students

205-6.15=198.85

we do not use decimal numbers

198.89=199 students

Answer:

y = (9/4)x + 2, given that I've assume the correct meaning of "-5x4."

Step-by-step explanation:

A parallel line has the same slope as the reference line,

The reference line is 9x-4y = -5x4, as written.  I'm not sure the 4 belongs, but I'll assume it is written correctly.  I'm not sure if the term -5x4 is meant to mean -5x^4, -20x or -20.  I'll assume -20:

9x-4y = -20

Put the equation into standard slope-intercept form:  Y = mx + b, where m is the slope and b is the y-intercept (the value of y when  x = 0).

-4y = -9x - 20

y = (9/4)x + 5

This line has a slope of (9/4).  The parallel line will have the same slope.

y = (9/4)x + b

We can find a value of b that would shift this line to include (4,7) by using this point in the equation:

y = (9/4)x + b

7 = (9/4)(4) + b

7  = 9 + b

b = 2

The equation of a line parallel to the reference line is y = (9/4)x + 2

Answer:

x = 9.4m

Hope this helps...Have a good day!!

Answer:

C, (x-4)(x+5)

Step-by-step explanation:

when u simplify the equetion (x-4)(x+5)

we get x2 + z +20

Answer:

\((x-4)(x+5)=x^2+x-20\)

Step-by-step explanation:

\((x*x)+(x*5)+(-4*x)+(-4*5)=x^2+5x+-4x+-20\)

\(x^2+5x+-4x+-20=x^2+(5-4)x-20\)

\(x^2+(5-4)x-20=x^2+x-20\)

Answer:

x=2 , y=3

Step-by-step explanation:

Answer:

12.92

Step-by-step explanation:

4.14 * 104 = 430.56

3% of 430.56 = 430.56 * 0.03 = 12.92

The man does 14,400 ft-lbs of work against gravity while climbing to the top of the helical staircase. A 160-lb man carrying a 20-lb can of paint climbs a helical staircase around a silo with a radius of 20 ft. The silo is 40 ft high, and the man makes two complete revolutions.

To calculate the work done by the man against gravity, we first need to determine the total vertical distance he climbs.

The height gained in one revolution can be found using the Pythagorean theorem. The man moves along the circumference of the circle with radius 20 ft, so the horizontal distance in one revolution is 2 * π * 20 = 40π ft. Thus, the helical path forms a right-angled triangle, with the height gained as one side, 40π ft as the other side, and the helical path's length as the hypotenuse. If the man makes two complete revolutions, the total horizontal distance traveled is 80π ft.

Let h be the height gained in one revolution. Then, h² + (40π)² = (80π)². Solving for h, we find that h = 40 ft. Since there are two revolutions, the total height gained is 80 ft.

The man's total weight (including the paint can) is 160 + 20 = 180 lbs. Work done against gravity is the product of force, distance, and the cosine of the angle between the force and displacement vectors. In this case, the angle is 0° since the force and displacement are in the same direction (vertically). So, the work done is:

Work = (180 lbs) * (80 ft) * cos(0°) = 180 * 80 * 1 = 14,400 ft-lbs.

Therefore, the man does 14,400 ft-lbs of work against gravity while climbing to the top of the helical staircase.

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Answer:

\(\displaystyle y' = - \csc (x) \big[ \cot^2 (x) + \csc^2 (x) \big]\)

General Formulas and Concepts:

Algebra I

Terms/Coefficients

Functions

Calculus

Differentiation

Derivative Rule [Product Rule]:                                                                             \(\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)\)

Step-by-step explanation:

Step 1: Define

Identify.

\(\displaystyle y = \csc (x) \cot (x)\)

Step 2: Differentiate

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

The standard form of the equation of this ellipse is \(\frac{(x\;+\;5)^2}{5^2} +\frac{(y\;-\;6)^2}{8^2}=1\)

In Mathematics, the standard form of the equation of an ellipse can be represented by the following mathematical equation:

\(\frac{(x\;-\;h)^2}{a^2} +\frac{(y\;-\;k)^2}{b^2}=1\)

Where;

From the information provided above, we have the following parameters about the equation of this ellipse:

Vertices = (-5, -2) and (-5, 14)

Point = (0, 6)

Since the vertices are located at (-5, -2) and (-5, 14), we would determine the coordinates of the center (h, k) by using the midpoint formula as follows:

Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint = [(-5 - 5)/2, (-2 + 14)/2]

Midpoint or center (h, k) = (-5, 6).

Since the point (0, 6) lies on the ellipse, we would determine the value of a and b as follows:

\(\frac{(0\;-\;h)^2}{a^2} +\frac{(6\;-\;k)^2}{b^2}=1\)

b² = (k - 14)²             b² = (k + 2)²

b² = (6 - 14)²             b² = (6 + 2)²

b² = (-8)²                   b² = 8²

b = √64 = 8.             b = √64 = 8.

a = x - h

a = 0 - (-5)

a = 5

Therefore, the required equation of this ellipse is given by;

\(\frac{(x\;-\;h)^2}{a^2} +\frac{(y\;-\;k)^2}{b^2}=1\\\\\frac{(x\;+\;5)^2}{5^2} +\frac{(y\;-\;6)^2}{8^2}=1\)

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The difference in volume between the two pyramids is 12,371,740 cubic feet.

To find the volume of a square pyramid, you use the formula :

V = (1/3)Bh, where B is the area of the base and h is the height.

Using the dimensions given in the table, we can calculate the volumes of the two pyramids:

- Volume of Bent Pyramid = (1/3)(619 feet * 619 feet)(332 feet) = 26,384,053.33 cubic feet
- Volume of Red Pyramid = (1/3)(722 feet * 722 feet)(341 feet) = 38,755,793.33 cubic feet

To find the difference in volume between the two pyramids, we subtract the smaller volume from the larger volume:

38,755,793.33 - 26,384,053.33 = 12,371,740 cubic feet.

Rounding to the nearest cubic foot, the difference in volume between the two pyramids is 12,371,740 cubic feet.

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The percentage change in quantity demanded, rate of change of -19 means that for every one dollar increase in price, the demand for the portable USB battery charger decreases by 19 units.

a. The demand of a product with respect to price is known as price elasticity of demand.

The rate of change of demand with respect to price can be found by differentiating the demand function with respect to price.

So, we differentiate D(p) with respect to p,

we get;

D'(p) = -2p+5

Therefore, the rate of change of demand with respect to price is -2p + 5.

b. When the price of the portable USB battery charger is $12, the demand is given by D(12) = -12²+5(12)+1

= -143 units.

The rate of change of demand when the price is $12 can be found by substituting p = 12 into D'(p) = -2p + 5,

we get;

D(p) = -p² + 5p + 1

Taking the derivative with respect to p:

D'(p) = -2p + 5

D'(12) = -2(12) + 5= -19.

Interpretation:The demand for a portable USB battery charger is inelastic at the price of $12, since the absolute value of the rate of change of demand is less than 1.

This means that the percentage change in quantity demanded is less than the percentage change in price.

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Answer:

1. A

2. I

3. R

Step-by-step explanation:

The opposite of a number is just adding or removing a minus sign.

For -6, we just remove the - and get 6.

For 0, this is a special case. It always stays as 0, because -0 does not exist.

For 18, we just put a minus sign and get -18.

Answer:

20

Step-by-step explanation:

In a given rock sample, the amount of silicon-32 isotope gets divided in half about every 170 years. The half-life can be used to determine the amount of silicon-32 that has decayed from the time the closure temperature was reached.

There are 23 known isotopes of silicon (14Si), with masses ranging from 22 to 44. The stable isotopes of silica include 28Si (which has a 92.23% abundance), 29Si (4.67%), and 30Si (3.1%). The radioisotope 32Si, which is created when cosmic rays collide with argon, has the longest half-life. With a decay energy of 0.21 MeV, it has been estimated to have a half-life of roughly 150 years. It decays through beta emission first to 32P, which has a 14.28-day half-life, and then to 32S. At 157.3 minutes, 31Si has the second-longest half-life after 32Si. Half-lives for the others are all under 7 seconds.

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The surface area needed to be paint is 3,008 sq. ft. The cans requires for the paint are 15.

A cuboid is a 3 solid with six (rectangular) faces, eight vertices, and twelve edges.

Total Surface Area (TSA) = 2(lb + bh + hl) sq.units

where, i is the length, b is the breadth and h is the height.

The given values are;

length l = 12 ft

breadth b = 10 ft

height h = 16 ft

Put the values in the formula;

Total Surface Area (TSA) = 2(12×10 + 10×16 + 16×12)

                                         = 2×(120 + 160 +192)

                                          = 2×(472)

Total Surface Area (TSA) = 1504

Now, reduce the area of the floor as it is not to be painted;

Painted area = Total Surface Area (TSA) - area of floor

                     = 1504 -12×10

Painted area = 3,008 sq. ft

The area covered by 1 paint can is 200 sq. ft.

Can required = total painted area/painted area by one can

Can required = 3,008/200

                      = 14.44

Can required = 15 (as half can is not considered)

Therefore, the area painted is 3,008 sq. ft with the total number of cans as 15.        

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...........................................................H.............................................I..............:)

h....o....w:)

a......r........e:)

y........o..........u:)

The quotient of the expression is 7x² + 8x +  1

From the question, we have the following expression that can be used in solving the quotient

(7x^3-13x^2-23x-5)/(x-3)

Express properly

So, we have

(7x³ - 13x² - 23x - 5)/(x-3)

Set the divisor to 0

x - 3 = 0

So, we have

x = 3

Using a synthetic setup, we have

3 | 7 - 13 - 23 - 5

The synthetic division is then carried out as follows

3 | 7 - 13 - 23 - 5

        21   24    3

    7   8     1    -2

So, we have

Quotient = 7   8   1

Remainder = 2

Introduce the variable

Quotient = 7x² + 8x +  1

Remainder = 2

Hence, the result is 7x² + 8x +  1 remainder 2

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Answer:

above beat me to it he is right

The identity can be verified by converting the left-hand side into sines and cosines and simplifying at each step, resulting in the simplified form: 8 cos(x) + 8 sin(x) tan(x) = 8 sec(x).

Starting with the left-hand side (LHS) of the given equation:

8 cos(x) + 8 sin(x) tan(x)

Using the identity tan(x) = sin(x)/cos(x), we can substitute this into the equation:

8 cos(x) + 8 sin(x) * sin(x)/cos(x)

Next, simplifying the expression by multiplying sin(x) with sin(x)/cos(x):

8 cos(x) + 8 sin²(x)/cos(x)

Using the identity sin²(x) + cos²(x) = 1, we can rewrite sin²(x) as 1 - cos²(x):

8 cos(x) + 8 (1 - ²(x))/cos(x)

Expanding the numerator:

8 cos(x) + 8/cos(x) - 8 cos(x)

Combining like terms:

8/cos(x)

Finally, using the identity sec(x) = 1/cos(x), we can rewrite 8/cos(x) as 8 sec(x):

8 sec(x)

Thus, we have verified that the LHS is equal to the RHS, and the given identity is valid. Overall, the left-hand side (LHS) of the equation is simplified to 8 sec(x), which is equal to the right-hand side (RHS), confirming the identity.

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The probability that we will roll exactly one number exactly three times is 125/7776.

Given a fair die is rolled six times.

A fair die has 6 faces, hence, the sample space will be {1, 2, 3, 4, 5, 6}

So, on rolling a die thrice, the number of elements in sample space is

6⁶ = 46,656

Let us assume that the die in the first roll shows 3, hence the second die and third die can show any of the other 5 numbers.

First, the probability of rolling a single value (in this case 3) on a fair 6-sided die would be one out of six = 1/6

The probability of NOT rolling that number would be five out of six.

Let P be the probability of getting 3 only once

P= 1/ 6 × 1/6 × 1/6 × 5/6 × 5/6 × 5/6 = 125/46656

The number 3 can be showed in any of the six dies at a time

Hence required probability ,

= 125/46656 + 125/46656 + 125/46656 + 125/46656 + 125/46656 + 125/46656  

= 750/46656

P = 125/7776

Therefore, the probability that exactly one 3 is rolled is 125 / 7776.

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The average speed of Renee over the last 2 hours of her trip is 9 miles/hour.

Let the average speed of Renee in 2 hours in between 6th and 8th hour be x miles / hours.

So the distance covered by Renee in 6 hours is = 6x

So the distance covered by Renee in 8 hours is = 8x

So the distance covered by Renee in this 2 hours in between 6th and 8th hours is = 8x - 6x

So according to the information the distance covered by Renee in this same 2 hours = 74 - 56 = 18 miles.

The equation best situated to the situation is,

8x - 6x = 18

2x = 18

x = 18/2

x = 9

Hence the average speed of Renee over the last 2 hours of her trip is 9 miles/hour.

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The annual appreciation of the coin from 1968 to 2019 was 6.7%. Hence the coin appreciated at a rate of 6.7% per year from 1968 to 2019.

In 2019, a coin was sold at auction for $4,120,500. The coin had a face value of $20 when it was first issued in 1786. The coin had been previously sold for $395,000 in 1968. At what price did the coin appreciate annually from 1968 to 2019? Solution:We can start this problem by using the compound interest formula which is given as follows;A = P (1+r/n)^(nt)Where:A is the future value of the investment;P is the principal investment;r is the annual interest rate;n is the number of times that interest is compounded per year;t is the number of years the money is invested.To solve the problem we will consider the following;We will use the initial price of the coin which was sold in 1968 which is $395,000 as the principal investment, PWe will use the price of the coin sold at auction in 2019 which is $4,120,500 as the future value of the investment, AWe will use the number of years between 1968 and 2019 which is 51 years as t.The interest rate is not given, but we can solve for it by substituting the known values into the formula as shown below;A = P (1+r/n)^(nt)=> 4,120,500 = 395,000 (1+r/1)^(1 x 51)=> (1+r) = (4,120,500/395,000)^(1/51)=> (1+r) = 1.067=> r = 1.067 - 1=> r = 0.067 = 6.7%Therefore the annual appreciation of the coin from 1968 to 2019 was 6.7%. Hence the coin appreciated at a rate of 6.7% per year from 1968 to 2019.



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False,False,True,False,True respectively.

The statement is false. Simultaneous increase or decrease in variables does not necessarily imply a causal effect. Causal effect requires establishing a relationship where changes in one variable directly affect the other.

The statement is false. While econometric models can be derived from formal economic models, they can also be based on empirical data or theoretical considerations. Valid conclusions can be drawn as long as the econometric model is correctly specified and meets the necessary statistical assumptions.

The statement is true. In a time series, the ordering of observations is crucial as it reflects the temporal dimension of the data. The sequence of events can reveal patterns, trends, and seasonality, which are essential for understanding and analyzing time-dependent phenomena.

The statement is false. Statistical significance is determined by comparing the p-value (probability of obtaining the observed result by chance) with the significance level (often denoted as alpha). If the p-value is smaller than the significance level, the result is considered statistically significant, indicating evidence against the null hypothesis.

The statement is true. Consistency refers to the property of an estimator to approach the true parameter value as the sample size increases. In other words, if an estimator is consistent, its expected value will equal the true parameter value for all possible values of the true parameter as the sample size grows indefinitely. Consistency is a desirable property of estimators as it ensures accuracy and reliability in estimating population parameters.

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Answer: x-3y=6

Step-by-step explanation:

so first subtract x to get -x+3y = -6 because standford is ax+by = c

and then multiply by -1 since x has to be positive

x-3y = 6 That is your answer

The sorted array after applying the PARTITION algorithm using the first element (5) as the pivot.

To illustrate the operation of the PARTITION algorithm on the given array {5, 1, 2, 7, 9, 3, 7, 8, 4}, we'll follow the steps of the algorithm and show the array at each step.

1: Choose the first element, 5, as the pivot.

{5, 1, 2, 7, 9, 3, 7, 8, 4}

2: Reorder the array so that all elements smaller than the pivot (5) come before it, and all elements larger than the pivot come after it. This is done by swapping elements.

{4, 1, 2, 3, 5, 9, 7, 8, 7}

3: The pivot is now in its final sorted position. Divide the array into two subarrays, one to the left of the pivot (elements smaller than the pivot) and one to the right of the pivot (elements larger than the pivot).

Left subarray: {4, 1, 2, 3}

Right subarray: {9, 7, 8, 7}

4: Recursively apply the PARTITION algorithm to the left and right subarrays.

For the left subarray:

Choose the first element, 4, as the pivot.

{4, 1, 2, 3}

Reorder the array:

{3, 1, 2, 4}

The left subarray is now sorted.

For the right subarray:

Choose the first element, 9, as the pivot.

{9, 7, 8, 7}

Reorder the array:

{7, 7, 8, 9}

The right subarray is now sorted.

5: The entire array is now sorted.

{3, 1, 2, 4, 5, 7, 7, 8, 9}

This is the final sorted array after applying the PARTITION algorithm using the first element (5) as the pivot.

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Answer:  the value of f′′′(4) is 3/256.

Step-by-step explanation:

Given the third-degree Taylor polynomial:

f(x) = (x−4)³/512 − (x−4)²/64 + (x−4)⁴/2

To find the value of f′′′(4), we need to differentiate the polynomial three times and evaluate it at x = 4.

First derivative:

f'(x) = 3(x−4)²/512 − 2(x−4)/64 + 4(x−4)³/2

Second derivative:

f''(x) = 6(x−4)/512 − 2/64 + 12(x−4)²/2

Third derivative:

f'''(x) = 6/512 + 24(x−4)/2

Now, substitute x = 4 into f'''(x):

f'''(4) = 6/512 + 24(4−4)/2

= 6/512 + 0

= 6/512

= 3/256

Therefore, the value of f′′′(4) is 3/256.

the number is really "x", which oddly enough is the 100%, but we also know that 15% of that is 107.6.

\(\begin{array}{ccll} Amount&\%\\ \cline{1-2} x & 100\\ 107.6& 75 \end{array} \implies \cfrac{x}{107.6}~~=~~\cfrac{100}{75} \implies\cfrac{x}{107.6} ~~=~~ \cfrac{4}{3} \\\\\\ 3x=430.4\implies x=\cfrac{430.4}{3}\implies x=\cfrac{2152}{15}\implies x=143\frac{7}{15}\implies x=143.4\overline{66}\)

Answer:

the fourth one

Step-by-step explanation:

y-intercept is when x equals 0