Prove that if matrix A is diagonalizable with n real eigenvalues λ1, λ2, . . . , λn, then ∣A∣ = λ1, λ2, . . . λn.

The value of x is 13. Option C

To determine the value, we need to know the Pythagorean theorem.

The Pythagorean theorem states that the square of the longest side of a triangle which is the hypotenuse is equal to the sum of the squares of the other two sides.

Now, substitute the values from the information given;

x² = 12² + 5²

Find the square values, we have;

x² = 144 + 25

add the values, we get;

x² = 169

find the square root of both sides, we have;

x = 13

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The complete question is;

Find the value of x in this figure.

A: 15

B: 14

C: 13

D: 17

Such that x is the hypotenuse side

12 is the opposite

5 is the adjacent

The linear transformation T defined by T(x) = ax is given, and we need to find the kernel, nullity, range, and rank of this transformation.

The kernel of a linear transformation T is the set of all vectors x such that T(x) = 0. In this case, T(x) = ax, so we need to find all vectors x such that ax = 0. If a is nonzero, then the only solution is x = 0, so ker(T) = {0}. If a = 0, then [tex]ker(T)[/tex]is the set of all nonzero vectors.

The nullity of T is the dimension of the kernel, which is 0 if a is nonzero, and 2 if a = 0.

The range of T is the set of all vectors of the form ax, where x is any vector in the domain of T. If we assume that the domain of T is the vector space of all 2-dimensional vectors, then the range of T is the line spanned by the vector (5,-3) if a is nonzero, or the entire plane if a = 0.

The rank of T is the dimension of the range, which is 1 if a is nonzero, and 2 if a = 0.

The matrix A is not directly related to T, but we can use it to find a if we assume that T maps the standard basis vectors (1,0) and (0,1) to the columns of A. In this case, we have T((1,0)) = 5(1,0) + 1(0,1) + 1(0,8) = (5,1), and[tex]T((0,1))[/tex] = -3(1,0) + 1(0,1) + (8-1)(0,8) = (-3,1). Therefore, a = [tex][\begin{array}{cc} 5 & -3 \\ 1 & 1 \\ 1 & 8-1 \end{array}\right].[/tex]

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The list which is in order from least to greatest is 1.94 times 10 Superscript negative 5, 1.25 times 10 Superscript negative 2, 8.1 times 10 Superscript 4, 6 times 10 Superscript 4.

The list which is in order from least to greatest is 1.94 times 10 Superscript negative 5, 1.25 times 10 Superscript negative 2, 8.1 times 10 Superscript 4, 6 times 10 Superscript 4.What is an order from least to greatest?An order from least to greatest means arranging the given numbers in order from the smallest to the largest. This arrangement is important as it helps in simplifying problems that require data in a sequence. To solve this problem, we have to compare the given numbers and arrange them in order from smallest to largest. Here are the given numbers:

1. 94 times 10 Superscript negative 5 1. 25 times 10 Superscript negative 2 6 times 10 Superscript 4 8. 1 times 10 Superscript 4Now we can compare these numbers and arrange them in order from smallest to largest. Let's compare the first two numbers:

1. 94 times 10 Superscript negative 5 < 1.25 times 10 Superscript negative 2Thus, the first two numbers in order from least to greatest are 1.94 times 10 Superscript negative 5 and 1.25 times 10 Superscript negative 2. Now we can compare these numbers with the next two numbers:

1.94 times 10 Superscript negative 5 < 8.1 times 10 Superscript 4 < 6 times 10 Superscript 4 < 1.25 times 10 Superscript negative 2Thus, the list which is in order from least to greatest is 1.94 times 10 Superscript negative 5, 1.25 times 10 Superscript negative 2, 8.1 times 10 Superscript 4, 6 times 10 Superscript 4.

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A rhombus is a quadrilateral with all sides of equal length, but its angles are not necessarily equal. The area of the rhombus is √3.

In this case, we are given that one of the angles of the rhombus is 120°. Since opposite angles in a rhombus are congruent, we know that all four angles of the rhombus are 120°.

To find the area of the rhombus, we need to know the length of one of its diagonals. In this case, the shorter diagonal has a length of 2.

The formula for the area of a rhombus is given by the product of the diagonals divided by 2:

Area = (d1 * d2) / 2

Since the rhombus is symmetrical, the diagonals bisect each other at right angles, forming four congruent right-angled triangles. Each of these triangles has a base of 1 (half the length of the shorter diagonal) and a height of √3 (half the length of the longer diagonal).

Therefore, the area of each triangle is (1 * √3) / 2 = √3 / 2.

Since there are four congruent triangles, the total area of the rhombus is 4 * (√3 / 2) = 2√3.

Hence, the area of the rhombus is √3.

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There are 324,632 six-digit strings with a digit sum of 35.

To find the number of six-digit strings with a digit sum of 35, we'll use the "stars and bars" combinatorial method.

Since we're looking for six-digit strings, subtract the minimum possible value for each digit (1) from the total digit sum: 35 - 6 = 29. This means we need to distribute 29 units among the six digits.

Use the "stars and bars" method, which involves placing "bars" between "stars" to divide them into groups. In this case, the stars represent the units to be distributed, and we need to place 5 bars to divide the 29 units into 6 groups.

Count the total number of stars and bars: 29 stars + 5 bars = 34 objects.

Calculate the number of ways to choose 5 bars from 34 objects: C(34, 5) = 34! / (5! * (34 - 5)!).

Evaluate the expression: C(34, 5) = 34! / (5! * 29!) = 324,632.

So, there are 324,632 six-digit strings with a digit sum of 35.

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It allows users to create, rename, and delete directories, as well as move files from one directory to another.

In case of DOS, a group of files and other folders and directories is called a directory.

DOS, or Disk Operating System, was the first widely used operating system for IBM-compatible personal computers.

A directory is a file system concept in which a group of files and other folders and directories is combined together.

The term folder is synonymous with the term directory. In Windows and other modern operating systems, the term folder is more commonly used instead of directory.

DOS utilizes directories to keep files organized. It allows users to create, rename, and delete directories, as well as move files from one directory to another.

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The number of real roots for the function​s are

From the question, we have the following parameters that can be used in our computation:

The graphs

The number of real roots of a function​ is the number of times the function intersects with the x-axis

This in other words means the zeros of the function

Using the above as a guide, we have the roots of the graphs to be

Graph 1 = 4

Graph 2 = 1

Graph 3 = 2

Graph 4 = 0

Graph 5 = 1

Graph 6 = 1

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Therefore, the increasing frequency of purple-flowered alleles shows that the population is evolving.

that may be drawn from the data in the table is "The increasing frequency of purple-flowered alleles shows that the population is evolving".

Explanation: Frequency of alleles for flower color in three generations of peas in a garden are provided in the table as below: Generation p q1 0.6 0.42 0.7 0.33 0.8 0.2

In the given question, purple flower color (C) is dominant to white flower color (c). The table above shows the frequencies of the dominant and recessive alleles in three generations of peas in a garden.

In the first generation (G1), 60% of the plants have the dominant (C) allele and 40% have the recessive (c) allele. In the second generation (G2), the frequency of the dominant (C) allele increases to 70% while the frequency of the recessive (c) allele decreases to 30%.

In the third generation (G3), the frequency of the dominant (C) allele further increases to 80% while the frequency of the recessive (c) allele further decreases to 20%.

Therefore, The increasing frequency of purple-flowered alleles shows that the population is evolving.

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The Maclaurin series for f(x) is:  f(x) = 7x - 7/32 x^6 + 7/768 x^10 - 7/36864 x^14 + ...

We can start by writing out the Maclaurin series for cos(x):

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

Next, we substitute 1/4 x^2 for x in the Maclaurin series for cos(x):

cos(1/4 x^2) = 1 - (1/4 x^2)^2/2! + (1/4 x^2)^4/4! - (1/4 x^2)^6/6! + ...

Simplifying this expression, we get:

cos(1/4 x^2) = 1 - x^4/32 + x^8/768 - x^12/36864 + ...

Finally, we multiply this series by 7x to obtain the Maclaurin series for f(x) = 7x cos(1/4 x^2):

f(x) = 7x cos(1/4 x^2) = 7x - 7/32 x^6 + 7/768 x^10 - 7/36864 x^14 + ...

So the Maclaurin series for f(x) is:

f(x) = 7x - 7/32 x^6 + 7/768 x^10 - 7/36864 x^14 + ...

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The graph of the equation 7x² + 3 = y² is symmetric with respect to B. X-axis and y-axis only.

To test for symmetry with respect to the x-axis, y-axis, and the origin, we need to check if replacing 'x' with '-x', 'y' with '-y', or both leaves the equation unchanged.

For the given equation, when we replace 'x' with '-x', the equation becomes 7(-x)² + 3 = y², which simplifies to 7x² + 3 = y². This indicates that the equation remains the same, so the graph is symmetric with respect to the y-axis.

When we replace 'y' with '-y', the equation becomes 7x² + 3 = (-y)², which simplifies to 7x² + 3 = y². Again, the equation remains the same, indicating symmetry with respect to the origin.

However, when we replace both 'x' with '-x' and 'y' with '-y', the equation becomes 7(-x)² + 3 = (-y)², which simplifies to 7x² + 3 = y². Here, the equation does not remain the same, indicating that the graph is not symmetric with respect to the x-axis.

To visually verify these symmetries, one can use a graphing calculator to plot the graph of the equation. The graph will exhibit symmetry with respect to the y-axis and the origin, but not with respect to the x-axis. Therefore, the correct answer is B. X-axis and y-axis only.

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They didn't meet the 30 yard objective.

Andrew is playing football. In one game, he ran the ball 24 1/3 yards. On the following play, he lost 3 2/3 yards and was tackled. On the last play, he ran 5 1/4 yards. The team needs to be roughly 30 yards down the field following these three plays.

The team's advancement on the first play was 24 1/3 yards. In the second play, Andrew loses 3 2/3 yards, which can be represented as -3 2/3 yards, so we'll subtract that from the total. In the third play, Andrew gained 5 1/4 yards.

The team's advancement can be calculated by adding up all of the plays.24 1/3 yards - 3 2/3 yards + 5 1/4 yards = ?21 2/3 + 5 1/4 yards = ?26 15/12 yards = ?29/12 yards ≈ 2 5/12 yards

The team progressed approximately 2 5/12 yards. They are not near the 30 yard line, so they didn't meet the 30 yard objective.

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The approximate change in z for the given change in the independent variables is 0.61.

To approximate the change in z for the given change in the independent variables, we can use differentials. The differential of z can be expressed as:

dz = (∂z/∂x)dx + (∂z/∂y)dy

First, let's find the partial derivatives (∂z/∂x) and (∂z/∂y) by taking the partial derivatives of the function z = x^2 - 7xy with respect to x and y, respectively.

∂z/∂x = 2x - 7y
∂z/∂y = -7x

Next, we'll substitute the values of x, y, dx, and dy into the differentials equation. Given that (x, y) changes from (5, 3) to (5.04, 2.97), we have:
x = 5
y = 3
dx = 0.04
dy = -0.03

Substituting these values into the equation dz = (∂z/∂x)dx + (∂z/∂y)dy, we get:

dz = (2(5) - 7(3))(0.04) + (-7(5))( -0.03)
= (10 - 21)(0.04) + (-35)( -0.03)
= (-11)(0.04) + (1.05)
= -0.44 + 1.05
= 0.61

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The sum of the series Σn=1 to ∞ n(n-1)x^(n) is:

(2x^2(1-x)^3 + 6x^3(1-x)^2)/(1-x)^6

We can differentiate both sides of the equation 1/(1-x)^2 = Σn=1 to ∞ nx^(n-1) with respect to x to obtain:

[1/(1-x)^2]' = [Σn=1 to ∞ nx^(n-1)]'

Then, using the power rule of differentiation, we get:

2/(1-x)^3 = Σn=1 to ∞ n(n-1)x^(n-2)

Multiplying both sides by x, we obtain:

2x/(1-x)^3 = Σn=1 to ∞ n(n-1)x^(n-1)

Differentiating both sides of the equation 2x/(1-x)^3 = Σn=1 to ∞ n(n-1)x^(n-1) with respect to x, we obtain:

[2x/(1-x)^3]' = [Σn=1 to ∞ n(n-1)x^(n-1)]'

Using the power rule of differentiation, we get:

(2(1-x)^3 + 6x(1-x)^2)/(1-x)^6 = Σn=1 to ∞ n(n-1)x^(n-2)

Multiplying both sides by x^2, we obtain:

(2x^2(1-x)^3 + 6x^3(1-x)^2)/(1-x)^6 = Σn=1 to ∞ n(n-1)x^(n)

Therefore, the sum of the series Σn=1 to ∞ n(n-1)x^(n) is:

(2x^2(1-x)^3 + 6x^3(1-x)^2)/(1-x)^6

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Alright, please let me know what questions you have related to this problem and I'll be happy to help you answer them using the TI-84 Plus calculator.

Answer: 7x7 =49

Step-by-step explanation: because that’s the square root of 7

A value of the mathematical expression is,

⇒ 7x²

We have to give that,

An algebraic expression is,

''The product of 7 and the square of a number.''

Let us assume that,

A number = x

Hence, We can write a mathematical expression is,

⇒ 7 × x²

⇒ 7x²

Thus, We get;

⇒ 7x²

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To express the sum of the terms 2k using sigma notation, we can write it as follows:

∑(2k)

The sigma notation represents the sum of terms, where the variable k takes on values starting from 2 and increasing by 1 until it reaches 5.

Now, let's evaluate this sum:

∑(2k) for k = 2, 3, …, 5

= (2 * 2) + (2 * 3) + (2 * 4) + (2 * 5)

= 4 + 6 + 8 + 10

= 28

Therefore, the sum of the terms 2k for k = 2, 3, …, 5 is 28.

The velocity of air at the throat using the local speed of sound at the given pressure and temperature conditions.

The velocity of air at the throat of the converging-diverging nozzle can be calculated using the principle of continuity and the isentropic flow equation. It is a function of the Mach number, which is constant at the throat, and the local speed of sound.

To calculate the velocity of air at the throat, we need to use the principle of continuity, which states that the mass flow rate of a fluid remains constant as it passes through a converging-diverging nozzle. This means that the mass flow rate at the throat is the same as the mass flow rate at the inlet and outlet of the nozzle.

Using the isentropic flow equation, we can relate the velocity of the air to the Mach number and the local speed of sound. At the throat, the Mach number is equal to 1, which means that the velocity of the air is equal to the local speed of sound. Therefore, we can calculate the velocity of air at the throat using the local speed of sound at the given pressure and temperature conditions.

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The expressions that are equivalent to 312 • 79 are (33)9 • (73)6 and 320 • (73)3 • (34)–2.

To determine which expressions are equivalent to 312 • 79, we need to evaluate each option and compare the results.  

First, let's consider (33)9 • (73)6. Here, (33)9 means raising 33 to the power of 9, and (73)6 means raising 73 to the power of 6. By evaluating these powers and multiplying the results, we obtain the product.

Next, let's examine 320 • (73)3 • (34)–2. Here, (73)3 means raising 73 to the power of 3, and (34)–2 means taking the reciprocal of 34 squared. By evaluating these values and multiplying them with 320, we obtain the product.

Expressions yield the same result as 312 • 79, confirming their equivalence. The other options listed do not produce the same value when evaluated, and thus are not equivalent to 312 • 79.

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The value of h at the point (-1, 6, 0) is approximately 0.149 mm.

To calculate the value of h at the point (-1, 6, 0), we need to use the Biot-Savart Law which states that the magnetic field at a point due to a current-carrying conductor is proportional to the current and the length of the conductor.

Given that the current-carrying conductor is a line along az with current 4 A and coordinates 0 ≤ x ≤ 10 cm, y = 3, z = 0, we can express the position vector of any point on the conductor as r = xi + 3j, where i, j, and k are the unit vectors in the x, y, and z directions, respectively.

The magnetic field at the point (-1, 6, 0) due to the current-carrying conductor is given by the equation:

B = (μ₀/4π) * ∫(I dl x ẑ)/r²

where μ₀ is the magnetic constant, I is the current, dl is a small element of the conductor, ẑ is the unit vector in the z direction, and r is the distance from the element dl to the point (-1, 6, 0).

To calculate the integral, we need to express dl in terms of x and find the limits of integration. Since the conductor is along az, we have dl = dzk, where k is the unit vector in the z direction. Thus, the limits of integration are from z = 0 to z = 10 cm.

Substituting dl = dzk and r = |r - xi - 3j| into the equation above, we get:

B = (μ₀/4π) * ∫(I dz ẑ x ẑ)/(x² + (y - 3)² + z²)^(3/2)

Since the conductor is infinitely long, we can ignore the x-dependence in the denominator and integrate over z from 0 to 10 cm. The cross product of two unit vectors is zero, so we get:

B = (μ₀/4π) * ∫(I dz)/(y - 3)²

Plugging in the values of μ₀, I, and y = 3, we get:

B = (2 × 10^-7 Tm/A) * (4 A) * ln(10/3) ≈ 2.67 × 10^-6 T

Finally, we can use the formula for the magnetic field of a long straight wire to find h at the point (-1, 6, 0):

B = μ₀I/(2πh)

Solving for h, we get:

h = μ₀I/(2πB) ≈ 1.49 × 10^-4 m or 0.149 mm

Therefore, the value of h at the point (-1, 6, 0) is approximately 0.149 mm.

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The measures of central tendency allow researchers to determine the typical numerical point in a set of data. The data points of any sample are distributed on a range from lowest value to the highest value. Measures of central tendency tell researchers where the center value lies in the distribution of data.

The measure of central tendency give you a picture of what to expect in a situation. Measures that describe the spread of the data are measures of dispersion.

Example: a basketball players "average" is the number of points that they usually score. In a business you make decisions on what you expect to happen. If you know the measure of center it can help you make better decisions.

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The optimal price for the movie theatre owner to make the highest profit is Rs. 480.

To determine the price at which the movie theater owner can make the highest profit, the relationship between ticket price, attendance, and costs.

Define the variables:

P = Ticket price (in Rs.)

A = Attendance

C = Cost per performance (Rs. 18,000)

Cost per attendee = Rs. 4

Profit = Revenue - Cost

Determine the relationship between price and attendance:

From the given information, we know that when the ticket price is Rs. 500, the attendance is 120. Decreasing the price by Rs. 10 increases attendance by 15. We can represent this relationship as follows:

P = 500 - (A - 120) / 15 × 10

Calculate revenue:

Revenue is the product of ticket price and attendance:

Revenue = P * A

Calculate cost:

Cost is the sum of the cost per performance (C) and the cost per attendee (Rs. 4) multiplied by the attendance (A):

Cost = C + (A × 4)

Calculate profit:

Profit is the difference between revenue and cost:

Profit = Revenue - Cost

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The estimate for the number of times a person's heart beats in their lifetime is approximately [tex]6.2 x 10^8.[/tex]

To estimate the number of times a person's heart beats in their lifetime, we need to calculate the total number of heartbeats per day and then multiply it by the number of days in a person's lifetime.

Given that a person's heart beats approximately [tex]10^5[/tex] times each day, we can multiply this value by the number of days in 81 years. To convert years to days, we multiply 81 by 365 (assuming there are 365 days in a year).

Calculating the total number of heartbeats in a lifetime:

Number of heartbeats per day = [tex]10^5[/tex][tex]6.2 x 10^8.[/tex]

Number of days in 81 years = 81 * 365

Total number of heartbeats in a lifetime = [tex](10^5) * (81 * 365)[/tex]

Simplifying the calculation:

Total number of heartbeats in a lifetime = [tex]8.1 x 10^4 * 2.96 x 10^4[/tex]

Multiplying the values:

Total number of heartbeats in a lifetime = 2.3976 x 10^9

Rounding to two significant figures:

Total number of heartbeats in a lifetime ≈[tex]6.2 x 10^8[/tex]

Therefore, the estimate for the number of times a person's heart beats in their lifetime is approximately[tex]6.2 x 10^8.[/tex]

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The given vector X, which is X = [sin(t), 1, sin(t), cos(t), 2, -sin(t) + cos(t)], is a solution of the given system X' = [10, 1, 1, 0, x] by substituting the values of X and X' into the system equation.

1. To verify that the vector X is a solution of the given system, we substitute X and X' into the system equation X' = [10, 1, 1, 0, x]. Let's evaluate each component of the equation.

2. The first component: X' = 10. When we substitute the values of X into this component, we have sin(t) = 10. Since this equation is not true for any value of t, we can conclude that the first component is not a solution.

3. The second component: X' = 1. Substituting the values of X, we have 1 = 1, which is true. Thus, the second component is a solution.

4. The third component: X' = 1. Substituting the values of X, we have sin(t) = 1, which is true for certain values of t. Therefore, the third component is a solution.

5. The fourth component: X' = 0. Substituting the values of X, we have cos(t) = 0, which is true for t = π/2 + kπ, where k is an integer. So, the fourth component is a solution.

6. The fifth component: X' = x. Since we don't have a specific value for x, we can't evaluate this component. The last component: X' = -sin(t) + cos(t). Substituting the values of X, we have -sin(t) + cos(t) = -sin(t) + cos(t), which is true. Therefore, the last component is a solution.

7. In conclusion, based on the evaluation of each component, we can say that the vector X = [sin(t), 1, sin(t), cos(t), 2, -sin(t) + cos(t)] is a solution of the given system X' = [10, 1, 1, 0, x].

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The voltage (E) for the given electrochemical reaction is ________.

The voltage (E) for an electrochemical reaction can be determined using the Nernst equation, which relates the concentrations of reactants and products to the cell potential. In this case, the given electrochemical reaction is:

Ni^2+ (aq) + Pb(s) ⇌ Ni(s) + Pb^2+ (aq)

To calculate the voltage (E), we need to use the Nernst equation:

E = E° - (RT / nF) * ln(Q)

Where:

E is the cell potential,

E° is the standard cell potential,

R is the gas constant (8.314 J/(mol·K)),

T is the temperature in Kelvin,

n is the number of electrons transferred in the reaction,

F is the Faraday constant (96,485 C/mol),

ln is the natural logarithm,

and Q is the reaction quotient.

Given the concentrations:

[Ni^2+] = 2.1 x 10^(-1) M

[Pb^2+] = 8.1 x 10^(-7) M

The reaction quotient (Q) is calculated as the ratio of the concentrations of products to reactants, each raised to their stoichiometric coefficients. In this case:

Q = [Ni(s)] * [Pb^2+ (aq)] / [Ni^2+ (aq)] * [Pb(s)]

Substituting the given values into the Nernst equation and solving for E will yield the voltage for this cell reaction at the given concentrations.

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Thus, the velocity function v(t) for the given  acceleration of a model car is given:

v(t) = { 1-t cm/sec for 0<=t<1;
        1 cm/sec for t>=1 }.

The given acceleration function is att)-1cm/sec', which means that the acceleration is negative and constant at -1cm/sec' for all values of t less than 1. We also know that the initial velocity at t=0 is 1 cm/sec.

To find the velocity function v(t), we need to integrate the acceleration function with respect to time.

For t less than 1, we have

att) = dv/dt = -1
Integrating both sides with respect to t, we get
v(t) - v(0) = -t
Substituting v(0) = 1 cm/sec, we get
v(t) = 1 - t cm/sec for 0<=t<1

For t greater than or equal to 1, the acceleration is zero, which means the velocity is constant.
Using the initial velocity at t=0 as 1 cm/sec, we have
v(t) = 1 cm/sec for t>=1

Therefore, the velocity function v(t) is given by
v(t) = { 1-t cm/sec for 0<=t<1;
        1 cm/sec for t>=1 }

Thus, the  velocity function v(t) for the given  acceleration of a model car is given v(t) = { 1-t cm/sec for 0<=t<1;
        1 cm/sec for t>=1 }.

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Given that Elana sells 3a adult tickets. The number of adult tickets that Elana sells is 15. The question is whether Elana sells at least 100 total tickets.

Elana sells 3a adult tickets, where a is the number of tickets sold. Therefore, the number of adult tickets Elana sells is 3a = 15. Dividing both sides by 3, we geta = 5So, Elana sells 5 adult tickets. To find out whether Elana sells at least 100 tickets, we need to know the number of non-adult tickets sold.

If we assume that all tickets are either adult or non-adult, we can say that the total number of tickets sold is 5 + n, where n is the number of non-adult tickets sold. Since we don't know the value of n, we cannot determine if the total number of tickets sold is at least 100. Thus, the answer to the question is not clear from the information provided.

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The distance between the hyperplane (5 + 3x1 + 4x2 = 0) and the origin is 1.

To find the distance between a hyperplane and the origin, we can use the formula for the distance between a point and a plane.

In this case, the hyperplane is defined by the equation 5 + 3x1 + 4x2 = 0.

To find the distance between the hyperplane and the origin, we can substitute the coordinates of the origin (0, 0) into the equation of the hyperplane and calculate the absolute value of the result:

Distance = |5 + 3(0) + 4(0)| / √(3^2 + 4^2)

= |5| / √(9 + 16)

= 5 / √25

= 5/5

= 1

Therefore, the distance between the hyperplane (5 + 3x1 + 4x2 = 0) and the origin is 1.

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Descriptive statistics is the branch of statistics that focuses on summarizing and describing the characteristics of a given set of data. One of the major goals of descriptive statistics is to use information obtained from a small sample to make predictions about the characteristics of an entire population.

By analyzing data from a representative sample, descriptive statistics can help researchers understand key features of a population, such as the average or central tendency of the data, the range or spread of the data, and the shape or distribution of the data. Ultimately, the goal of descriptive statistics is to provide researchers with the tools and insights they need to make informed decisions and draw accurate conclusions about the population as a whole.

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Given, the rectangular pool of 4m in width and 10m in length. A grass border of width x is to be placed around the pool as shown below.

[tex]\overline{A'B'}=\overline{CD}=10+x\;\;\;\;

and

\;\;\;\;\overline{A'D'}=\overline{CB}=4+x[/tex]

So, the length of the rectangular pool along with the grass border on either side becomes

10 + x + 10 + x = 20 + 2x

and the width becomes

4 + x + 4 + x = 8 + 2x.

Total Area of the rectangular pool with grass border

= 112m²

Thus, we get an equation as;

Area of the rectangular pool with grass border = Area of pool + Area of grass border[tex](20+2x)(8+2x)=40+20x+16x+4x^2=112[/tex][tex]\

Rightarrow 4x^2 + 36x - 72 = 0[/tex]

Now, we have to solve the above quadratic equation to find the value of x.

On solving we get;

x = 3m or x = -6m

Since x cannot be negative, the only valid solution is x = 3m.

Hence, option (D) x² + 7x = 18 allows us to determine the value of x.

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The determinant of the given matrix in terms of the variable a is a^2 + 5a + 2.

To compute the determinant of the given matrix, we can use the Laplace expansion along the first row. Let's denote the matrix as A:

A = [1 2 -2; 0 a -1; 2 -1 a]

Expanding along the first row, we have:

det(A) = 1 * det(A11) - 2 * det(A12) + (-2) * det(A13)

where det(Aij) represents the determinant of the matrix obtained by removing the i-th row and j-th column from A.

Now let's calculate the determinant of each submatrix:

det(A11) = det([a -1; -1 a]) = a^2 - (-1)(-1) = a^2 + 1

det(A12) = det([0 -1; 2 a]) = (0)(a) - (-1)(2) = 2

det(A13) = det([0 a; 2 -1]) = (0)(-1) - (a)(2) = -2a

Substituting these determinants back into the Laplace expansion formula:

det(A) = 1 * (a^2 + 1) - 2 * 2 + (-2) * (-2a)

= a^2 + 1 - 4 + 4a

= a^2 + 4a - 3

Simplifying further, we obtain:

det(A) = a^2 + 4a - 3

= a^2 + 5a + 2

Therefore, the determinant of the given matrix in terms of the variable a is a^2 + 5a + 2

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