the library labels each book with five digit code to identify the books in their computer system each code is 3 digits(0-9)followed by two letters how many codes are possible

Given:

P(x) is a 2nd degree polynomial.

\(P(1)=0,\ P(2)=3,\ P(-3)=0\)

To find:

The polynomial P(x).

Solution:

If P(x) is a polynomial and P(c)=0, then c is a zero of the polynomial and (x-c) is a factor of polynomial P(x).

We have, \(P(1)=0,\ P(-3)=0\). It means 1 and -3 are two zeros of the polynomial P(x) and (x-1) and (x+3) are two factors of the polynomial P(x).

So, the required polynomial is defined as:

\(P(x)=a(x-1)(x+3)\)          ...(i)

Where, a is a constant.

We have, \(P(2)=3\). So, substituting \(x=2,\ P(x)=3\) in (i), we get

\(3=a(2-1)(2+3)\)

\(3=a(1)(5)\)

\(3=5a\)

\(\dfrac{3}{5}=a\)

Putting \(a=\dfrac{3}{5}\) in (i), we get

\(P(x)=\dfrac{3}{5}(x-1)(x+3)\)

Therefore, the required polynomial is \(P(x)=\dfrac{3}{5}(x-1)(x+3)\).

Answer:

a

Step-by-step explanation:

57-13=44. If Sydney is 11, then Thomas is 46. 11x3=33, and 33+13=46. To check our answer, we will add 46+11. 46+11=57.

Declan's reasoning does make sense. This is because 3/4 and 1/2 have the same denominator, and 3/4 is a larger fraction than 1/2.

Therefore, it is reasonable to assume that 3/4 is greater than 6/12 because 6/12 simplifies to 1/2. Simplifying fractions means dividing the numerator and denominator by the same number, in this case, 6 is divisible by 2, so we can reduce the fraction to 1/2.

So, Declan is correct in his reasoning that 3/4 is greater than 6/12. It is important to understand the relationship between fractions and their denominators to make such comparisons accurately.

3/4 = 9/12

1/2 = 6/12

6/12 = 6/12

Since 9/12 (which is equivalent to 3/4) is greater than 6/12 (which is equivalent to 1/2), we can say that 3/4 is indeed greater than 6/12.

Therefore, Declan's reasoning is correct.

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Answer:

x intercepts are (-2, 0).

Step-by-step explanation:

X intercepts: The x intercepts are the points where the graph of the equation crosses the x-axis. In order to find the x intercepts, we can set the equation equal to 0 and solve for x.

x^2 + 4x + 4 = 0

(x + 2)^2 = 0

x + 2 = 0

x = -2

Therefore, the x intercepts are (-2, 0).

\(\hrulefill\)

Alternate method: Another method to find the x intercepts is to use the quadratic formula. The quadratic formula is:

\(x = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}\)

Comparing x^2 + 4x + 4 = 0 with ax^2=bx+c=0,

In this case, we get a = 1, b = 4, and c = 4.

Substituting these values into the quadratic formula, we get:

\(x = \frac{-4 \± \sqrt{4^2 - 4 * 1 * 4} }{2 * 1}\\x =\frac{-4}{2}\\x=-2\)

Therefore, the x intercepts are still (-2, 0).

\(\hrulefill\)

Vertex: The vertex is the point of the parabola that is highest or lowest.

In order to find the vertex, we can use the formula:

\((h, k) = (\frac{-b}{2a}, \frac{-(b^2-4ac)}{4a})\)

Comparing x^2 + 4x + 4 = 0 with ax^2=bx+c=0,

In this case, we get a = 1, b = 4, and c = 4.

Substituting these values into the formula, we get:

\((h, k) = (\frac{-4}{2*1}, \frac{-(4^2-4*1*4)}{4*1})\\(h, k) =(-2,0)\)

Therefore, the vertex is (-2, 0).

Minimum or maximum: The vertex is a minimum because the coefficient of x^2 is positive. This means that the parabola opens upwards, and the vertex is the lowest point of the parabola.

\(\hrulefill\)

Graphing: In order to graph the equation, we can first plot the x intercepts.

Graph: Attachment

Then, we can draw a parabola that passes through the x intercepts and the vertex. We can use symmetry to help us graph the parabola.

The graph can be drawn by inputting values of x into the function f(x) = x² + 4x + 4,

The parabola is symmetric about the vertical line that passes through the vertex. This means that if we reflect the parabola over this line, we will get the same parabola.

As you can see, the parabola opens upwards and has a minimum vertex at (-2, 0).

Answer:

x-intercept:  x = -2

Alternative method:  Quadratic formula

Vertex:  (-2, 0)

Minimum as the leading coefficient is positive.

Step-by-step explanation:

The x-intercepts are the points at which the curve crosses the x-axis, so the x-values when y = 0.

As the quadratic equation is already equal to zero, to find the x-intercepts, factor and solve for x.

\(x^2+4x+4=0\)

\(x^2+2x+2x+4=0\)

\(x(x+2)+2(x+2)=0\)

\((x+2)(x+2)=0\)

\((x+2)^2=0\)

\((x+2)=0\)

\(x=-2\)

Therefore, the x-intercept of the given quadratic equation is x = -2.

\(\hrulefill\)

An alternative method to use to find the x-intercept(s) is the quadratic formula.

\(\boxed{\begin{minipage}{4 cm}\underline{Quadratic Formula}\\\\$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$\\\\\\when $ax^2+bx+c=0$ \\\end{minipage}}\)

\(\hrulefill\)

The vertex of a quadratic function is the turning point of the parabola. It is the point where the graph reaches its minimum or maximum value

As there is only one x-intercept of the given quadratic equation, and the equation can be rewritten in the form (x + 2)² = 0, we can identify that the vertex of the parabola is the x-intercept, (-2, 0).

Since the leading coefficient of the equation is positive, the parabola opens upwards. Therefore, the vertex represents a minimum.

\(\hrulefill\)

The graph of a quadratic equation is a parabola.

A parabola is symmetric with respect to its axis of symmetry, which is a vertical line passing through its vertex. As the vertex of the given quadratic equation is (-2, 0), the axis of symmetry is x = -2.

To graph the equation using the information already calculated, first plot the vertex at (-2, 0) and draw the axis of symmetry through the vertex.

Find other points on the graph by inputting values of x into the function f(x)  = x² + 4x + 4, then reflect those points across the axis of symmetry.

By connecting these points and considering the symmetry, we can sketch the graph of the equation.

Graph D shows the relationship between the number of side dishes and the total cost of the special.

Since we know that the function crosses x axis at some point, then for some polynomial functions, we have those as roots of the polynomial.

In Graph D, the y coordinate, or we can say that the number of boxes is 25 for the x coordinate, or the time (in days).

As we follow the graph, we can say that there is 21 boxes for 3 days, 19 boxes for 4 days, and 13 boxes for 7 days.

Therefore, Graph D shows the relationship given in the table.

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Answer:

8.06 feet

Step-by-step explanation:

use the Pythagorean theorem

9^2 = 4^2 + x^2

81 =16 +x^2

x^2 = 65

x = 8.06 feet

Answer:

Step-by-step explanation:

The specific name of the quadrilateral is kite

The measures of the missing angles are 75 degrees each

To determine the measure of the angle, we need to know the different properties of a kite.

The properties of a kite includes;

Since the adjacent angles are equal, we have that;

105+ x = 180

collect the like terms, we have;

x = 180 - 105

x = 75 degrees

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The quadrilateral is a kite and the missing angles are 105° and 45°

A kite is a quadrilateral that has 2 pairs of equal-length sides and these sides are adjacent to each other.

A kite has a pair of equal angle and the diagonals of a kite meets at 90°.

In a kite , there are two pairs of congruent or equal sides.

This means that the missing angles are

The opposite angle to 105 is also 105°

The fourth angle is calculated as;

105+105+105+x = 360

= 315 +x = 360

x = 360- 315

x = 45°

Therefore the quadrilateral is a kite and the missing angles are 105° and 45°

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The center of the face of a cube is the midpoint of the face and lies along the line that runs between the opposite vertices of the face.

This is the point of intersection of the two diagonals in a face. In other words, it is the point that is equidistant from all four corners of the face.

To visualize this, imagine dividing the face into four equal triangles by drawing lines from each corner to the center. The center of the face is at the intersection of these lines.

In three-dimensional space, the center of the face of a cube is also the center of the cube itself, as all faces of a cube are congruent (the same shape and size) and meet at the center of the cube.

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Answer:

......s.s.s..s.s.s.s.s.s.ss..s.s

9514 1404 393

Answer:

  (1, 7)

Step-by-step explanation:

Fill in x=1 and do the arithmetic.

  h(1) = 4(1) +3 = 7

The coordinate pair is ...

  (x, h(x)) = (1, h(1)) = (1, 7)

Step-by-step explanation:

to select 3 out of 13.

no repetition, and the sequence of the selected items does not matter.

that means combinations :

C(13, 3) = 13! / (3! × (13 - 3)!) = 13! /(3! × 10!) =

= 13×12×11 / (3×2) = 13×2×11 = 286

there can be 286 different random samples of size 3.

The answer is 286.

The number of different simple random samples of size 3 that can be selected from a population consisting of 13 items is 286.Explanation:Simple random sampling is a statistical technique used in research. In this sampling method, a subset of a population is selected randomly, with each member of the population having an equal chance of being selected. There are different types of random sampling methods, such as systematic sampling, stratified sampling, cluster sampling, etc.The number of different simple random samples of size k that can be selected from a population of size N is given by the formula N!/k!(N-k)!, where ! denotes factorial. Here, the population consists of 13 items, and we need to find the number of different simple random samples of size 3 that can be selected from this population.So, the required number of different simple random samples of size 3 that can be selected from this population is 13!/3!(13-3)! = 286.

Therefore, the number of different simple random samples of size 3 that can be selected from a population consisting of 13 items is 286.

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Answer:

140J

Step-by-step explanation:

work done = Mass x acceleration x distance

mass = 7kg

acceleration = 10m/s^2

distance= 2m

work = 7 x 10 x 2

= 140j

Answer:

The answer is 140J or 140Nm

Step-by-step explanation:

Work done=mgh

W=7×10×2

W=140J or 140Nm

The measurement that is closest to the value of x, in centimeters, is given as follows:

14.3.

In this problem, we have a right triangle, in which the legs, which are the sides between the angle of 90º, are of 11.9 cm and 7.9 cm.

Then the hypotenuse x, which is the segment connecting both legs, is obtained using the Pythagorean Theorem.

The Pythagorean Theorem states that the measure of the hypotenuse squared is equals to the sum of the squares of the measures of each side.

Then the length of x is calculated as follows:

x² = 11.9² + 7.9²

x = square root of (11.9² + 7.9²)

x = 14.28.

Closest to 14.3, rounding to the nearest tenth, meaning that the third option is correct.  

We suppose that 11.9 cm and 7.9 cm are the measures of the legs.

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The correct answer to the exponential expression which is as given in the task content is; 8³.

Based on the given task content, Gina's error from her workings is dividing the bases of the exponent.

Gina's working:

8^12 / 8^9

= 0^12 / 0^9

The Correct workings:

8^12 / 8^9

= 8^(12-9)

= 8³

So therefore, the correct answer to the exponential expression which is given in the above expression is 8³.

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The direction of propagation is given by the sign of v.The equation of a travelling wave is given by y(x,t) = 0.2e^(-x - 3t) sin(x + 3t). To prove that this wave satisfies the one-dimensional wave equation, we need to show that:

∂²y/∂x² = 1/v² * ∂²y/∂t² where v is the wave velocity. Let us compute the first and second derivatives of y with respect to x and

t:∂y/∂x = 0.2e^(-x - 3t) cos(x + 3t) - 0.2e^(-x - 3t) sin(x + 3t)

= 0.2e^(-x - 3t) cos(x + 3t - π/4)∂²y/∂x²

= -0.2e^(-x - 3t) sin(x + 3t - π/4)∂y/∂t

= -0.2e^(-x - 3t) (sin(x + 3t) + 3cos(x + 3t))∂²y/∂t²

= -0.2e^(-x - 3t) (cos(x + 3t) + 9sin(x + 3t))

Comparing the two sides of the wave equation, we have:

∂²y/∂x² = (1/v²) ∂²y/∂t²-(1/0.2) sin(x+3t-π/4)

= (1/v²) [-0.2(cos(x+3t)+9sin(x+3t))] -(1/0.2) sin(x+3t-π/4)

On simplification, we get: 9cos(x + 3t) + 17sin(x + 3t) = 0

This equation is satisfied for all x and t only if the coefficients of cos(x + 3t) and sin(x + 3t) are equal to zero, that is: 9/v² = 17/v²

Solving for v, we get:v = ±√(9/17)

The positive sign corresponds to the wave propagating in the positive x-direction, and the negative sign corresponds to the wave propagating in the negative x-direction. Therefore, the direction of propagation is given by the sign of v.

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Answer:

3 miles

Step-by-step explanation:

1/8*24=3

Answer:

Step-by-step explanation:

He ran 24 times around the track1

Circumfrence of track=1/8 Mile

Total distance he covered=1/8*24

Distance = 3 miles

a. This is a combination because the order in which the students are chosen does not matter, only the group of four students is selected.
b. There are 4845 different ways that 4 students can be chosen from a group of 20.

a. This is a combination because the order of the chosen students does not matter. In a permutation, the order matters, whereas in a combination, it does not.

b. To find the number of different ways 4 students can be chosen from a group of 20, use the combination formula:

C(n, k) = n! / (k!(n-k)!)

Where n = the total number of students (20), k = the number of students to be chosen (4), and ! denotes factorial.
The number of different ways 4 students can be chosen from a group of 20 can be calculated using the combination formula:

nCr = n! / r!(n-r)!

Where n is the total number of students (20) and r is the number of students being chosen (4).

So,

20C4 = 20! / 4!(20-4)!
C(20, 4) = 20! / (4!(20-4)!)
C(20, 4) = 20! / (4! * 16!)
C(20, 4) = 2,432,902,008,176,640,000 / (24 * 20,922,789,888,000)
C(20, 4) = 4845

There are 4,845 different ways to choose 4 students from a group of 20.

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Answer:

7^12

Step-by-step explanation:

When dividing two exponents by the same base, it is the same as subtracting the power (14-2).

Answer:

63.16%

Step-by-step explanation:

\(\frac{12}{19}\)  x 100% = 63.16%

So, 12 is 63.16% of the 19

The derivative \(\( y' \)\) of the implicit function \(\( y^2 - xy = -2 \)\) is 0, indicating a constant slope with no change in relation to \(\( x \)\).

To find \(\( y' \)\)for the implicit function \(\( y^2 - xy = -2 \)\), we can differentiate both sides of the equation with respect to \(\( x \)\) using the chain rule. Let's go step by step:

Differentiating \(\( y^2 \)\) with respect to \(\( x \)\) using the chain rule:

\(\[\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}\]\)

Differentiating \(\( xy \)\) with respect to \(\( x \)\) using the product rule:

\(\[\frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + y \cdot \frac{dx}{dx} = x \cdot \frac{dy}{dx} + y\]\)

Differentiating the constant term (-2) with respect to \(\( x \)\) gives us zero since it's a constant.

So, the differentiation of the entire equation is:

\(\[2y \cdot \frac{dy}{dx} - (x \cdot \frac{dy}{dx} + y) = 0\]\)

Now, let's rearrange the terms:

\(\[(2y - y) \cdot \frac{dy}{dx} - x \cdot \frac{dy}{dx} = 0\]\)

Simplifying further:

\(\[y \cdot \frac{dy}{dx}\) \(- x \cdot \frac{dy}{dx} = 0\]\)

Factoring out:

\(\[(\frac{dy}{dx})(y - x) = 0 \]\)

Finally, solving:

\(\[\frac{dy}{dx} = \frac{0}{y - x} = 0\]\)

Therefore, the derivative \(\( y' \)\) of the given implicit function is 0.

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We already know that the solution is (9, -3/5), let's show that

Let's plug our solution into the equation

The second equation is true!

Answer:

Slope = -1

Step-by-step explanation:

Answer:

Slope= -1

Step-by-step explanation:

The formula for slope is (m being slope) m=\(y2-y1/x2-x1\)

You are given the points (-3,4) and (2,-1)

-1 is going to be y2 and 4 is going to be y1

2 is x2 and -3 is x1

Plug these into the formula:

m= -1 - 4/2-(-3) = -5/5=-1

Slope=-1

we can only see a portion of the graph of this function in the standard viewing window of -10 to 10.

What is Graph ?

In mathematics, a graph is a visual representation of a set of objects, such as numbers or data points, and the relationships between them. Graphs can be used to represent a variety of mathematical concepts, including functions, sets of data, and geometric shapes.

The function f(x) = -|x| - 12 is a piecewise function. When x is greater than or equal to zero, the function simplifies to f(x) = -x - 12, and when x is less than zero, it simplifies to f(x) = x - 12.

If we graph both of these pieces separately, we get two straight lines that intersect at the point (0, -12), where the absolute value of x changes sign. The slope of both lines is -1, which means they are both downward sloping with a negative slope.

When we graph this function in the standard viewing window of -10 to 10, we can see the part of the graph where x is greater than or equal to zero, which is the left side of the graph, but the part where x is less than zero, which is the right side of the graph, is outside the viewing window.

Therefore, we can only see a portion of the graph of this function in the standard viewing window of -10 to 10.

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Answer:

16

Step-by-step explanation:

if the ribbon is 32/3 yards long and you need 2/3 yard long segments the denominators cancel out giving you 32 and 2 32/2 = 16

Answer:

16 pieces of ribbon

Step-by-step explanation  

if you divide the length of the ribbon by the length of the pieces she needs to cut it into, you get 16.

The points on the curve where the tangent is horizontal: (1, 0) & (-1, 0),  and where the tangent is vertical: (0, 1) & (0, -1)

To find the points on the curve x=cos(theta) where the tangent is horizontal or vertical, we need to find the derivative of x with respect to theta.

Taking the derivative of x with respect to theta, we get:

dx/dtheta = -sin(theta)

When the tangent is horizontal, the derivative is equal to zero. So we need to solve the equation -sin(theta) = 0 for theta.

This equation is true when theta = n*pi, where n is an integer. So the points on the curve x=cos(theta) where the tangent is horizontal are:

(1, 0) for n=2k, where k is an integer

(-1, 0) for n=2k+1, where k is an integer

When the tangent is vertical, the derivative is undefined. This happens when cos(theta) = 0, which is true when theta = (2k+1)*pi/2, where k is an integer.

So the points on the curve x=cos(theta) where the tangent is vertical are:

(0, 1) for n=2k+1, where k is an integer

(0, -1) for n=2k, where k is an integer

Therefore, the points on the curve where the tangent is horizontal are (1, 0) and (-1, 0), and the points on the curve where the tangent is vertical are (0, 1) and (0, -1).

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The probability of drawing any one of the letters from the given list is 1/16. The probability of drawing either an "F" or "P" tile is 2/16 or 1/8. The probability of drawing a vowel is 4/16 or 1/4. The probability of not drawing a "Q" tile is 15/16.

In the given list, there are a total of 16 tiles with different letters. Since there is an equal chance of drawing any one of the tiles, the probability of drawing any specific letter is 1 out of 16, which can be written as 1/16.

To determine the probability of drawing either an "F" or "P" tile, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes. Since there are two tiles with either "F" or "P" out of the 16 total tiles, the probability is 2 out of 16, which simplifies to 1/8.

To calculate the probability of drawing a vowel, we need to count the number of vowel tiles and divide it by the total number of tiles. In this case, there are four vowel tiles (A, I, O, U) out of the 16 total tiles. Therefore, the probability of drawing a vowel is 4 out of 16, which can be simplified to 1/4.

Finally, to find the probability of not drawing a "Q" tile, we subtract the probability of drawing a "Q" from 1. Since there is only one "Q" tile out of the 16 total tiles, the probability of drawing it is 1 out of 16. Hence, the probability of not drawing a "Q" tile is 15 out of 16, which simplifies to 15/16.

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Answer:

a) C(x)=50x+450

b) R(x) = 140x

c) Company will have to clean 5 offices to break even

d) Company will make money, $1,800

Step-by-step explanation:

a) C(x) = 50x+450

b) R(x)=140x

c) to break even, C(x) = R(x)

140x=50x+450

 90x=450

      x=5

Company will have to clean 5 offices to break even

d) R(x)= 140*25 = 3,500

    C (x)=50*25+450 = 1,700

    total profit will be 3500-1700 = 1,800

    so company will make money

The required requirement of the density curve is should be equal to 1 and the ordinate should not be zero.

Given that,
Two requirements of a density curve are to be determined.

A density curve is defined as a curve on a graph that represents the distribution of values in a dataset.

Here,

The two requirements of a density curve should be,

a. The area under the density curve should be equal to 1.

b. The ordinate of the curve should not be zero i.e. they should always have the vertical height.

Thus, the required requirement of the density curve is mentioned above.

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