Given equation is x" - 2x' + x = 24t^2e’
Auxiliary equation is r^2 - 2r +1 =0 (r-1)^2 = 0 => r = 1 is a double root.

If p varies inversely as the square of q and inversely as r if p equals 36 and Q equals to 3 and R equal to 4, then when q = 5 and r = 4, p = 1.

The relationship between p, q, and r can be represented as:

p ∝ 1/q^2 * 1/r

where ∝ means "varies inversely as".

Substituting the given values into this equation, we get:

36 ∝ 1/3^2 * 1/4

36 ∝ 1/36

Multiplying both sides by 36, we get:

p = 1

So, when q = 3 and r = 4, p = 1.

To find p when q = 5 and r = 4, we use the same equation and solve for p:

p ∝ 1/5^2 * 1/4

p ∝ 1/100

Multiplying both sides by 100, we get:

p = 1

So, when q = 5 and r = 4, p = 1.

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Answer: Triangle 1 and 3. The right triangles can be identified using the Pythagorean theorem. Add the squares of 2 sides together and check if it's equal to square of the 3rd side.

Step-by-step explanation:

In triangle 1,

Side 1= 20

Side 2= \(\sqrt{425}\)

Side 3= 5

For a right angled triangle,

A²+B²= C²

where A,B,C are sides of the triangle

Since, 20² + 5² = 400 + 25 = 425 = C²

Therefore, C= \(\sqrt{425}\) = Side 2.

Hence, it is a right-angled triangle.

In triangle 2,

Side 1= 14

Side 2= 21

Side 3= 10

For a right angled triangle,

A²+B²= C²

where A,B,C are sides of the triangle

Since, 14² + 10²  = 196 + 100 = 296 = C²

Therefore, C= \(\sqrt{296}\) ≠ 21.

Hence, it is not a right-angled triangle.

In triangle 3,

Side 1= 6/11

Side 2= 8/11

Side 3= 10/11

For a right angled triangle,

A²+B²= C²

where A,B,C are sides of the triangle

Since, (6/11)² + (8/11)²  = 36/121 + 64/121 = 100/121 = C²

Therefore, C= \(\sqrt{100/121}\) = 10/11 = Side 3 .

Hence, it is a right-angled triangle.

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Answer: the square root of 4 is a real number while 1/2 is a rational number

Step-by-step explanation:

To find the factors of the expression 12x^2 - 4x - 1, we can use the factor theorem. The factor theorem states that if a polynomial expression evaluates to zero when a certain value is substituted for the variable, then that value is a factor of the polynomial.

To determine if a certain value is a factor of the expression, we can use synthetic division. Synthetic division involves dividing the polynomial expression by the possible factor to check if the remainder is zero. If the remainder is zero, then the value is a factor.

Let's start by checking if x = 1 is a factor of the expression:

    1 | 12 - 4 - 1
      |     12   8
      |____________
       12   8   7

Since the remainder is not zero, x = 1 is not a factor of the expression.

Next, let's check if x = -1 is a factor of the expression:

   -1 | 12 - 4 - 1
      |    -12  16
      |____________
       12 -16  15

Again, the remainder is not zero, so x = -1 is not a factor of the expression.

Now, let's check if x = 2 is a factor of the expression:

    2 | 12 - 4 - 1
      |     24 40
      |____________
       12  20 39

Once again, the remainder is not zero, so x = 2 is not a factor of the expression.

Finally, let's check if x = -2 is a factor of the expression:

   -2 | 12 - 4 - 1
      |    -24 56
      |____________
       12 -28 55

As we can see, the remainder is not zero, so x = -2 is not a factor of the expression.

After checking all the possible factors, we can conclude that none of the values tested (1, -1, 2, -2) are factors of the expression 12x^2 - 4x - 1.

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To use the normal approximation for a test of two proportions, n1 p1, n1 (1 - p1), n2 p2, and n2 (1 - p2) must all be greater than or equal to 5.

This is because the normal distribution assumes that the sample size is large enough to approximate the binomial distribution, and the rule of thumb is that each cell (n1 p1, n1 (1 - p1), n2 p2, and n2 (1 - p2)) should have at least 5 expected counts. If any of these cells have fewer than 5 expected counts, then the normal approximation is not reliable and a more appropriate test should be used. It is important to note that this is just a rule of thumb, and other factors such as the size of the effect and the desired level of significance should also be considered when deciding on a statistical test.

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The range just means the y-output of the function, ie the y values.

You can see that the function curves up to y=2 and then curves back down, but doesn't go above y = 2.

So the second option is the correct one, because the y-values of the function go from infinity up to y=2, but not above y=2 hence the ≤ sign.

Eliminate x by adding the two equations, yielding \(y = 6\). Then, substitute y into either equation to solve for x, obtaining \(x = -3\). The solution is \((-3,6)\).

An equation is a mathematical statement that uses symbols, numbers, and variables to show that two expressions are equal, typically with an equal sign between them.

A variable is a quantity or symbol that can take on different values in a mathematical expression, equation, or formula, typically represented by a letter.

To eliminate a variable in a system of equations, we want to find a way to add or subtract the two equations so that one of the variables is eliminated.

In this case, we can eliminate the variable x by adding the two equations

\(x- 7y = -45\)

\((-x +8y = 51)\)

\(y = 6\)

Now that we have found the value of y, we can substitute it into either of the original equations to solve for x. Let's use the first equation

\(x-7y = -45\\x- 7(6) = -45\\x = -45+42\\x = -3\)

Therefore the solution to the system of equations is \(x= -3\) and\(y= 6\)

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the solution to the system of equations is (x, y) = (-3, 6).

What is an Equations?

Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.

To eliminate the variable x, we can add the two equations together. This will result in the x variable being eliminated, leaving us with an equation in terms of y:

(x - 7y) + (-x + 8y) = -45 + 51

Simplifying and solving for y, we get:

y = 6

Now that we know the value of y, we can substitute it into one of the original equations to solve for x. Using the first equation:

x - 7(6) = -45

Simplifying and solving for x, we get:

x = -3

Therefore, the solution to the system of equations is (x, y) = (-3, 6).

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Answer:

212121.

Step-by-step explanation:

It is odd number so it ends in 1.

There are 3 2's and 3 1's which alternate.

Diivisible by 3 and 9

It is 212121.

Answer:

the answer is A

Step-by-step explanation:

Answer:

A. - ∞ < x < ∞

Step-by-step explanation:

Answer:

The correct equation to use would be : 4(2+5)

(Whether you use the distributive property or not you will get the same answer in this equation)

with the equation 4 × 2 + 5 Mihini would end up multiplying 4 × 2 which is not what she wants. (She wants to find the number that is 4 TIMES the SUM of 2 + 5

Answer:

Angle k= 105

Angle n= 40

Angle m= 65

Answer:

m∡k = 105°

m∡n = 40°

m∡m = 65°

Step-by-step explanation:

m∡k = 105° because this angle forms a linear pair with a 75° angle

m∡n = 40° because this angle forms a linear pair with a 140° angle

m∡m = 65° because there are 180° in a triangle and one angle is 75° and the other angle is a vertical angle to angle 'n' which means they are equal

Using proportions, it is found that the total distance covered by him in 6 days is of 28.5 km.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

The daily distance is of 2 x 2.375 km, as 1000 m = 1 km and he walks both ways. Hence, the total distance covered by him in 6 days is given by:

D = 6 x 2 x 2.375 = 28.5 km.

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Answer:

(-2,-3)

Step-by-step explanation:

this is because when any point is flipped over y=-x, it makes both y and x negative.

hope this helps :)

a. the equation \(\(x - \pi - \frac{1}{2}\cos\left(\frac{1}{3}x\right) = 0\)\) has only one real zero \(p\) within the interval \(\([0, 2\pi]\)\). b. the iterates \(\(p_{n+1} = g(p_n)\)\) converge to the fixed point \(p\) for any initial approximation \(\(p_0 \in [0, 2\pi]\)\). c. This iterative process allows us to approximate the zero \(p\) of the function \(f(x)\) with the desired error tolerance of \(10^{-9}\).

(a) To show that the equation \(x - \pi - \frac{1}{2}\cos\left(\frac{1}{3}x\right) = 0\) has only one real zero \(p\), we can analyze the behavior of the function \(f(x) = x - \pi - \frac{1}{2}\cos\left(\frac{1}{3}x\right)\).

Let's consider the interval \([0, 2\pi]\) since the initial approximation \(p_0\) is stated to be within this range. We observe that \(f(x)\) is a continuous function on this interval. Furthermore, \(f(x)\) is a monotonically increasing function on this interval since the derivative of \(\frac{1}{2}\cos\left(\frac{1}{3}x\right)\) is always positive.

Now, let's examine the behavior of \(f(x)\) at the endpoints of the interval. We have \(f(0) = -\pi - \frac{1}{2}\cos(0) < 0\) and \(f(2\pi) = \pi - \frac{1}{2}\cos\left(\frac{2}{3}\pi\right) > 0\). Since \(f(x)\) is continuous and monotonically increasing, by the Intermediate Value Theorem, there exists at least one real zero \(p\) in the interval \([0, 2\pi]\).

To show that there is only one real zero in this interval, we can demonstrate that the derivative of \(f(x)\) is always positive. The derivative of \(f(x)\) is \(1 + \frac{1}{6}\sin\left(\frac{1}{3}x\right)\), which is always positive for any value of \(x\). Thus, there can be no local maximum or minimum points where the function crosses the x-axis again.

Therefore, the equation \(x - \pi - \frac{1}{2}\cos\left(\frac{1}{3}x\right) = 0\) has only one real zero \(p\) within the interval \([0, 2\pi]\).

(b) We are given \(g(x) = \pi + \frac{1}{2}\cos\left(\frac{1}{3}x\right)\) and we want to show that the iterates \(p_{n+1} = g(p_n)\) converge to the zero \(p\) of \(f(x)\) for any initial approximation \(p_0 \in [0, 2\pi]\).

First, let's analyze the function \(g(x)\). \(g(x)\) is a continuous function on \([0, 2\pi]\) since \(\cos\left(\frac{1}{3}x\right)\) is continuous. We can also observe that \(g(x)\) maps the interval \([0, 2\pi]\) into itself.

Next, we need to show that \(g(x)\) has a fixed point \(p\) where \(g(p) = p\). By substituting \(x = p\) into \(g(x)\), we get:

\(g(p) = \pi + \frac{1}{2}\cos\left(\frac{1}{3}p\right) = p\)

Now, we need to show that \(|g'(x)| < 1\) for all \(x \in [0, 2\pi]\). Taking the derivative of \(g(x)\), we have:

\(g'(x) = -\frac{1}{6}\

sin\left(\frac{1}{3}x\right)\)

Since \(\sin\left(\frac{1}{3}x\right)\) is bounded between -1 and 1, we have \(|g'(x)| < \frac{1}{6} < 1\) for all \(x \in [0, 2\pi]\).

By the Contraction Mapping Theorem, since \(g(x)\) is continuous, maps the interval into itself, has a fixed point \(p\), and satisfies \(|g'(x)| < 1\) for all \(x \in [0, 2\pi]\), the iterates \(p_{n+1} = g(p_n)\) converge to the fixed point \(p\) for any initial approximation \(p_0 \in [0, 2\pi]\).

(c) To compute the zero \(p\) using the fixed point method with an error tolerance of \(10^{-9}\), we can start with an initial approximation \(p_0\) and iterate using the formula \(p_{n+1} = g(p_n)\) until the difference between consecutive iterates is less than the error tolerance.

Let's choose an initial approximation \(p_0\) within the interval \([0, 2\pi]\). For example, we can take \(p_0 = 2\pi\).

Using the formula \(p_{n+1} = g(p_n)\), we can iteratively compute the next iterates until the difference between consecutive iterates is less than \(10^{-9}\).

Let's denote \(p_n\) as the \(n\)th iterate of \(p\) and \(p_{n+1}\) as the \((n+1)\)th iterate of \(p\). We can continue the iteration process until \(|p_{n+1} - p_n| < 10^{-9}\).

This iterative process allows us to approximate the zero \(p\) of the function \(f(x)\) with the desired error tolerance of \(10^{-9}\).

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Answer:

6

Step-by-step explanation:

18-12=6

Ans=6 candies

Answer: Trinomial

Step-by-step explanation: This expression has 3 terms (2a^2, -5a, and 7), so it is a trinomial. An easy way to remember this is that 'tri' = 3, 'bi' = 2, and 'mono' = 1. Hope this helped :)

Answer:

3 yards^2

Step-by-step explanation: Area of a rectangle=Length*Width. 6 yards*0.5 yards=3 yards^2

Answer:

The problem was that she omitted 0 in-between 3 and 2

Answer & Step-by-step explanation:

The student did not write a number that was in the thousands. Since there was no hundreds in their number, they need a zero in the hundreds place:

3 thousand

0 hundred

2 tens

7 ones

that is, 3027

Answer:

here's your answer.........

Answer:

13a) E is (a,3b)

13b) I is (4a,0)

Step-by-step explanation:

Answer:

The answers that I got from solvng it:

13a) E is (a,3b)

13b) I is (4a,0)

Sorry, if it isnt right...

Answer:

\( m\angle SXW =113\degree \)

Step-by-step explanation:

In circle with center X, VS is diameter.

So \( \widehat {SWV} \) is a semicircular arc.

\( \therefore m\widehat{SWV} = 180\degree \)

\( \therefore (12x+7)\degree + (21x +8)\degree = 180\degree \)

\( \therefore (33x+15)\degree = 180\degree \)

\( \therefore 33x+15 = 180 \)

\( \therefore 33x = 180-15 \)

\( \therefore 33x = 165 \)

\( \therefore x =\frac{165}{33} \)

\( \therefore x =5 \)

\( m\angle SXW =m\widehat{SW} \)

(Measure of central angle is equal to the measure of its corresponding minor arc)

\( m\angle SXW =(21x + 8)\degree \)

\( m\angle SXW =(21\times 5+ 8)\degree \)

\( m\angle SXW =(105+ 8)\degree \)

\( m\angle SXW =113\degree \)

Assume that jessie and lester have formed a contract whereby jessie agrees to deliver 10,000 dozen grade a large eggs to be shipped in paper cartons. a shortage of paper makes paper cartons much more expensive, so jessie uses styrofoam cartons and ships the eggs. lester is entitled to cancel the contract based on a material breach of the contract: False.

A carton is a box or container that is typically composed of paperboard, corrugated fiberboard, or liquid packaging board. For packing, a variety of carton types are employed. A box may also be used to describe a carton.

A carton is a food or beverage container that is made of cardboard or plastic. A carton is a large, durable cardboard box used for storage and transportation of products.

The answer is false since Lester lacks the authority to terminate the agreement due to a substantial breach.

Thus, the given statement is false.

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(a) The points where the tangent to curve C is horizontal or vertical can be found by analyzing the derivatives of the parametric equations. (b) To find the equations of the tangents to C at a given point, we need to find the derivative of the parametric equations and use it to determine the slope of the tangent line. (c) The concavity of the curve C can be determined by analyzing the second derivative of the parametric equations.

(a) To find points where the tangent is horizontal or vertical, we need to find values of t that make the derivative of y (dy/dt) equal to zero or undefined. Taking the derivative of y with respect to t:

dy/dt = 15t² - 1

To find where the tangent is horizontal, we set dy/dt equal to zero and solve for t:

15t² - 1 = 0

15t² = 1

t² = 1/15

t = ±√(1/15)

To find where the tangent is vertical, we need to find values of t that make the derivative undefined. In this case, there are no such values since dy/dt is defined for all t.

(b) To find the equations of tangents at a given point, we need to find the slope of the tangent at that point, which is given by dy/dt. Let's consider the point (t₀, 0). The slope of the tangent at this point is:

dy/dt = 15t₀² - 1

Using the point-slope form of a line, the equation of the tangent line is:

y - 0 = (15t₀² - 1)(t - t₀)

Simplifying, we get:

y = (15t₀² - 1)t - 15t₀³ + t₀

(c) To determine where the curve is concave upward or downward, we need to find the second derivative of y (d²y/dt²) and analyze its sign. Taking the derivative of dy/dt with respect to t:

d²y/dt² = 30t

The sign of d²y/dt² indicates concavity. Positive values indicate concave upward regions, while negative values indicate concave downward regions. Since d²y/dt² = 30t, the curve is concave upward for t > 0 and concave downward for t < 0.

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Feedback: With p-value <e; α our decision is to reject the null hypothesis.

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The probability that exactly two of the students are juniors is given as follows:

0.270 = 27%.

The mass probability formula is presented as follows:

\(P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}\)

\(C_{n,x} = \frac{n!}{x!(n-x)!}\)

The parameters are:

The values of these parameters for this problem are given as follows:

\(N = 60, n = 4, k = 18\)

Hence the probability that exactly two of the students are juniors is calculated as follows:

\(P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}\)

\(P(X = 2) = h(2,60,4,18) = \frac{C_{18,2}C_{42,2}}{C_{60,4}} = 0.270\)

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Answer:

Kenneth: 12 years old

Randy: 48 years old

Step-by-step explanation:

For this problem, we can use system of equations to solve. Let's use r for Randy and k for Kenneth.

Equation 1

r=4k

This equation comes from the first part, where Randy is four times as old as Kenneth.

Equation 2

r=36+k

This equation comes from the second part, where Randy is 36 years older than Kenneth.

Since we have r=, we can use substitution method.

4k=36+k                   [subtract both sides by k]

3k=36                       [divide both sides by 3]

k=12

Now that we have found k, we can plug that into any equation to find r.

r=4(12)                      [multiply]

r=48

Now, we know that Randy is 48 years old and Kenneth is 12 years old.

Answer:

Step-by-step explanation:

70% or more 70% is most likely a b-

The order would be DE < FD < EF. To order the sides of triangle ΔDEF from shortest to longest, you need to compare the lengths of the sides.

1. Determine the lengths of sides DE, EF, and FD.
2. Compare the lengths to find the shortest side.
3. Identify the next shortest side.
4. The remaining side will be the longest.

Once you have the side lengths, simply arrange them in ascending order (from shortest to longest) to answer your question. For example, if DE is 3 units, EF is 5 units, and FD is 4 units, the order would be DE < FD < EF.
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Answer:

x= 15/4

Step-by-step explanation:

cross multiply then divide both sides by 16

Answer:

x=15/4

Step-by-step explanation: