lim
x→ pi/4
(sec x)/(4x^2 -5x)

[tex]{\Large \begin{array}{llll} w=t^4 e^{\frac{2-t}{5+5t}} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ 4t^3e^{\frac{2-t}{5+5t}}~~ + ~~\stackrel{\textit{\LARGE product rule}}{t^4 \stackrel{\textit{\large chain rule}}{e^{\frac{2-t}{5+5t}}\cdot \stackrel{quotient~rule}{\cfrac{(-1)(5+5t)~~ - ~~(2-t)(5)}{(5+5t)^2}}}} \\\\\\ 4t^3e^{\frac{2-t}{5+5t}}~~ + ~~t^4 e^{\frac{2-t}{5+5t}}\cdot \cfrac{(-5-5t)~~ - ~~(10-5t)}{(5+5t)^2}[/tex]

[tex]4t^3e^{\frac{2-t}{5+5t}}~~ + ~~t^4 e^{\frac{2-t}{5+5t}}\cdot \cfrac{-15}{(5+5t)^2} \implies t^3 e^{\frac{2-t}{5+5t}}\left[4- \cfrac{15t}{(5+5t)^2} \right] \\\\\\ t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{(100t^2+200t+100)-15t}{(5+5t)^2} \right]\implies t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{100t^2+185t+100}{(5+5t)^2} \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} \cfrac{dw}{dt}=t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{100t^2+185t+100}{(5+5t)^2} \right] \end{array}}~\hfill[/tex]

a = 252, x = 7; y = 5925.13

Equation is given by the formula:

where:

yo = the initial population; r = growth rate, x = time

In your case, yo = 252. And the grow rate is 57%, therefore the r = 0.57. Finally, the exponential equation has this form:

Then, if we want to know what is the population after 7 days, we can evaluate this function:

Solve the inequality for x.

Answer:

x > 2

Step-by-step explanation:

10x+5 > 25

Solve

Subtract 5 from each side

10x+5-5 > 25-5

10x > 20

Divide each side by 10

10x/10 > 20/10

x > 2

-2

The range of the function is a set of all the values of y on the graph

The lowest point of the graph on the y-axis is -2

Therefore, the lowest value of the range of the function shown on the graph is -2

The solution of the system of equation is the point where the 2 lines cross each other.

By looking at the graph, we can see that they cross each other at (-3,-8)

Correct option: 2

The simplified  complex fraction is

120/11

This is further explained below.

Generally, the complex fraction  is

[tex]\frac{\left(\frac{16}{3}\right)}{\left(\frac{22}{45}\right)}[/tex]

Multiply the numerator by the reciprocal of the denominator.

[tex]\frac{16}{3} \cdot \frac{45}{22}[/tex]

Cancel the common factor of 2

[tex]$\frac{8}{3} \cdot \frac{45}{11}$[/tex]

Combine 8 and 15/11

=8*15/11

=120/11

In conclusion, The result can be shown as.

120/11

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Answer:

(-∞, -12), (2, ∞)

or

(-∞, -12) U (2, ∞)

Step-by-step explanation:

|3x + 15| > 21, |3x + 15| < -21

3x + 15 > 21,  3x + 15 < - 21

     -15    -15         -15     -15

------------------   -------------------

3x > 6               3x < - 36

÷3  ÷3               ÷3     ÷3

------------         ------------------

x > 2                  x < -12

(-∞, -12), (2, ∞)

or

(-∞, -12) U (2, ∞)

I hope this helps!

Since a sample is a subset of the population the sample mean varies around the mean of the population.

An average of a group of data is called a sample mean. A data set's central tendency, standard deviation, and variance can all be determined using the sample mean. A population average can be determined using the sample mean, among other things.

The statistic known as the sample mean is created by arithmetically averaging the values of a variable in a sample. The sample mean is an estimator of the expected value if the sample is taken from probability distributions with a common expected value.

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The measure of the angle having the question mark, found using the law of sines is ? ≈ 140°

The law of sines states that the sine of an interior angle of a triangle is proportional to the the length of the side opposite to the angle.

The given parameters are;

The radius of circle A = 6.4

The radius of circle B = 4.0

The radius of circle C = 3.0

The measure of angle at C = 24°

The length of the segment AB = 6.4 + 4 = 10.4

According to the law of sines, we have;

[tex]\dfrac{sin(A)}{a} = \dfrac{sin(B)}{b} = \dfrac{sin(C)}{c}[/tex]

Where;

a = The length of the side opposite angle ∠A

b = The length of the side opposite angle ∠B

c = the length of the side opposite angle ∠C

From the diagram, the side opposite angle ∠C, is AB, which by comparison indicates that when side c = AB = 10.4

c = 10.4

Similarly, the side opposite angle ∠A = a = BC = 4 + 3 = 7

a = 7

The known information are therefore; ∠C = 24°, a = 7, c = 10.4

[tex]\dfrac{sin(A)}{a} = \dfrac{sin(C)}{c}[/tex]

[tex]\dfrac{sin(A)}{7} = \dfrac{sin(24^{\circ})}{10.4}[/tex]

[tex]sin(A) = \dfrac{sin(24^{\circ})}{10.4}\times 7[/tex]

[tex]\angle A = arcsine \left( \dfrac{sin(24^{\circ})}{10.4}\times 7\right)[/tex]

∠A + ∠B + ∠C = 180° (angle sum property of a triangle)

∠B = 180° - (∠A + ∠C)

Which gives;

[tex]\angle B = 180^{\circ} -\left(24^{\circ} + arcsine \left( \dfrac{sin(24^{\circ})}{10.4}\times 7\right)\right) \approx 140^{\circ}[/tex]

The measure of the angle is ∠B ≈ 140°

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The mean number of unoccupied seats per flight during the past year will lie between 11.15 and 12.05.

What is mean in mathematics?

There are many mean types in maths, particularly in statistics. Each mean serves to condense a specific set of data, frequently so that the general magnitude and sign of the data set can be better understood.

The arithmetic mean, also referred to as "arithmetic average," is a calculation that divides the total number of values in a data collection by their sum to get the central tendency of that set of numbers.

a) Calculations:

Degrees of freedom = n - 1 = 225 - 1 = 224

For 224 degrees of freedom and 90% confidence interval,

Critical t value = 1.652

Hence,

90% confidence interval will be:

[tex]11.6 \pm 1.652*4.1/\sqrt{225}[/tex]

[tex]11.6 \pm 0.45[/tex]

(11.15, 12.05)

b) Interpretation:

We are 90% confident that the population mean number of unoccupied seats will lie between 11.15 and 12.05

c) Caution with interpretation:

In this scenario, individual confidence interval should be calculated rather than mean number of unoccupied seats since we can't generalize this interpretation for individual flights.

d) To narrow a confidence interval, we can either increase the sample size or can decrease the confidence level.

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Complete question

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected, and the number of unoccupied seats is noted for each of the sampled flights (from text, p. 306, example 6.3; use NOSHOW excel file). Given the mean number of unoccupied seats for the 225 random flights is 11.6 and the standard deviation is 4.1, estimate μ, the mean number of unoccupied seats per flight during the past year, using a 90% confidence interval.

a. Calculations

b.Interpretation

c.Caution with interpretation

d.How to narrow a confidence interval

The image of the transformation are (2, -9), (-9, -7), (-7, -4)

The coordinates of the triangle are given as

(-2, -9), (9, -7), (7, -4)

The transformation is given as

Reflection across the y-axis

This is represented as

(x, y) ⇒ (-x, y)

This means that the transformation is (-x, y)

When this rule is applied on the coordinates, we have

(2, -9), (-9, -7), (-7, -4)

Next, we plot the image of the transformation on a coordinate plane

See attachment for the image of the transformation

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Answer:

The polygon has 14 sides.

Explanation:

Given that the sum of the interior angles of the polygon is 2,160.

Recall that the sum of interior angles of a polygon can be calculated using the formula;

Substituting the value of T, to solve for n;

Therefore, the polygon has 14 sides.

Answer:Explanation:

C. 75

n cthe oomparison of variables, a negative correlation means that as the value of one of the variables increases, the value of the other variable decreases.

Given the variables k and l.

• L=150 when k = 7

• L=? when k=14

The variable k has increased.

Thus, to satisfy the condition for a negative correlation, the variable, l must decrease.

Thus, the only likely possible value of l is 75.

Option C is correct.

Answer:

r = 5%

Step-by-step explanation:

Formula : I=PRT

GIVEN:

I = 125

P = 5000

R = R

T = 6

125= (5000)(R)(0.5)

125=2500R

0.05=R

Interest rate = 0.05(100)

                    = 5%

The intersection point of the circle and the line is at-(2.68, 7.36).

For the given question;

The equation of the line is; y = 2x + 2   ......eq 1

The centre of the circle (h,k) = (0,2)

The radius of the circle = 6 units.

This is the equation's standard form.

(x - h)² + (y - k)² = r²

Put the values in the formula;

(x - 0)² + (y - 2)² = r²

(x)² + (y - 2)² = r²

Put the value of y from eq 1.

(x)² + (2x + 2 - 2)² = 6²

(x)² + (2x)² = 36

Simplifying;

x = ±6/√5

But the intersection must be in first quadrant.

x = 6/√5 = 2.68

Put the value of x in eq 1.

y = 2x + 2

y = 2×2.68 + 2

y = 7.36

Thus, the coordinates of the intersection of the line and the circle are (2.68, 7.36).

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Answer:

x = 30

Step-by-step explanation:

2x + 15 = 3x - 15

--------------------------

2x + 15 = 3x - 15

-3x          -3x

----------------------

-x + 15 = -15

    -15     -15

-------------------

-x = -30

÷-1   ÷-1

----------------

x = 30

I hope this helps!

Answer:

A) Reflexive Property of Equality

Step-by-step explanation:

If you look at your reflection in a mirror, you see yourself! Likewise, by the Reflexive Property, any number is its own mirror image. Any number (such as a real number) is equal to itself!

What this means is that any number is the answer as no number is given. It could be 3m = 3n, 10m = 10n, or 85m - 85n as long as the number of "m" is the same to the number of "n."

The property of a(b+(-b))=(b+(-b))a is called as Commutative Property of Multiplication.

The Commutative Property in mathematics is a rule that allows interchange the order of terms or factors without any change in the end result.

In multiplication, we can interchange the factors without affecting the end result of multiplication. For example, 2 x 3 = 3 x 2 = 6

In addition, we can interchange the terms without affecting the summation result. For example, 2+3 = 3+2 = 5

But Commutative Property does not apply to subtraction or division.

For example, 2-3 != 3-2

or, 2/3 != 3/2

Hence the answer to the question is Commutative Property of Multiplication.

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q + r = p is the equation formed by using the details in the question

In an algebraic equation of the form y=mx+b, where m denotes the slope and b the y-intercept, a linear equation is a first-order (linear) term and a constant. Sometimes the phrase "linear equation with two variables" is used to describe the previous equation, which contains the variables y and x.

Ax+By=C is the standard form for two-variable linear equations. Standard form is used, for instance, in the linear equation 2x+3y=5. This method makes it simple to find an equation's two intercepts (x and y).

p the stove is hot

q the soup is cold

r the food is done

equation to be created was

the food is done and the soup is cold ,  then the stove is hot.

and means +

therefore equation formed by plugging in the values will be ,

q + r = p

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Answer:

Step-by-step explanation:

You want the factors of 5y² +13y +6, and the solutions to 5y² +13y +6 = 0.

The quadratic ax²+bx+c with a leading coefficient a≠1 can be factored by considering the factors of the product 'ac' that have a sum of 'b'.

  5·6 = 30 = (1)(30) = (2)(15) = (3)(10) = (5)(6)

The sums of these factor pairs are, respectively, 31, 17, 13, 11. The factor pair with a sum of 13 is (3)(10).

There are a couple of methods that can be used to apply this knowledge to the factorization of the quadratic. Perhaps the most straightforward is to write the quadratic as 4 terms, using the found values to rewrite the linear term as their sum:

  5y² +13y +6 = 5y² +3y +10y +6 . . . . . . . . . 13 ⇒ 3 +10

Now, pairs of terms can be factored:

  = y(5y +3) +2(5y +3)

  = (y +2)(5y +3) . . . . . . . fully factored expression

The solutions that makes the product of factors be zero will be the values of x that make the factors be zero. A product is only zero if one of the factors is.

  y +2 = 0   ⇒   y = -2

  5y +3 = 0   ⇒   y = -3/5

The solutions to the quadratic equation are y = {-2, -3/5}.

__

Additional comment

Another way to find the factors is this. For factors p, q that have a product of 'ac' and a sum of 'b', the factorization can be ...

  (ay +p)(ay +q)/a

For the values in this problem, this would be ...

  (5y +3)(5y +10)/5

The latter of these factors can be divided by 5, so this can be simplified to ...

  (5y +3)(y +2) . . . . as above

The result of the approximation of -16 + square root of 40 is -3.35.

An approximation refers to anything which is similar to but not identical to something else.

Rounding can be used to approximate a number. Before performing the operations, an estimate can be approximated by rounding this same values within it.

For the given number;  

-16 + square root of 40 can be written as.

= -16 + √40

Find the factors of 40 by prime factorization;

40 = 2×2×2×5

Put the value in expression.

= -16 + √2×2×2×5

Take out square root of 2.

= -16 + 2√2×5

= -16 + 2√10

Fine the approximate value of √10 as this is near to √9.

Thus, √10 = 3.12

Put the value in expression.

= -16 + 2×3.12

= -16 + 12.649

= -3.35

Thus, the approximation of -16 + square root of 40 is -3.35.

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The coordinates along the directed line segment is (-1/8, 1/4)

The points are given as:

A  = (3, 4)

B = (-2, -2)

The ratio on the segment can be represented as;

m : n = 5 : 3

So, we have the coordinates of the point Q that lies along the directed line segment to be

Q = 1/(m + n) * (mx2 + nx1, my2 + ny1)

So, we have:

Q = 1/(5 + 3) * (5 * -2 + 3 * 3, 5 * -2 + 3 * 4)

Evaluate

Q = (-1/8, 1/4)

Hence, the coordinates of the point Q is (-1/8, 1/4)

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Answer:

(-12, 21)

Explanation:

3 times a number decreased by 5 means 3x -5.

This number is between -41 and 58; therefore,

Now we solve for the unknown number x.

Adding 5 to the three sides gives

finally, dividing the inequality by 3 gives

the parenthesis indicates that the -12 and 21 are not included in the solution interval =.

Answer:

I think its 64 please let me know if its wrong

After simplified the equation we get
a=54
b= 540
c=1800
d=2000

What is simplified expression?

Solving a math problem is the same thing as simplifying an expression. You basically try to write an expression as simply as you can when you simplify it. At the conclusion, there shouldn't be any more multiplying, dividing, adding, or removing to do. Take this formula, for instance: 4 + 6 + 5. To simplify an expression, create an equivalent expression without any terms that are similar. The expression will then be rewritten using the fewest terms possible. When confronted with the phrase 4x+5(3x12),

Sol-

2(3x+10)^3

=2(Bx)^3+3.(3x)^2.10+3.3x.10^2+10^3

=2(27x^3+270x^2+900x+1000)

=54x^3+540x^2+1800x+2000

So,

a=54

b=540

c=1800

d=2000

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The vertical and horizontal asymptotes for the graphed function are given as follows:

a) The vertical asymptotes are given at  x = -1 and x = 2.

b) The horizontal asymptote is given at y = -1.

In the context of this problem, we have that the asymptotes can be extracted from the graph as follows:

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Answer:

[tex]{ \tt{ { \sin}^{2} x + \cos {}^{2}x = 1 }}[/tex]

- Divide through by cos²x ;

[tex]{ \tt{ \frac{ { \sin }^{2}x }{ \cos {}^{2} x} + \frac{ \cos {}^{2} x }{ { \cos }^{2} x} = \frac{1}{ { \cos }^{2}x } }} \\ \\ { \boxed{ \tt{ { \tan}^{2} x + 1 = { \sec }^{2} x}}}[/tex]

- Divide tgrough by sin²x;

[tex]{ \tt{ \frac{ { \sin }^{2}x }{ { \sin}^{2}x } + \frac{ \cos { }^{2}x }{ { \sin}^{2}x } = \frac{1}{ { \sin}^{2} x} }} \\ \\ { \boxed{ \tt{1 + { \cot }^{2}x = \csc {}^{2} x}}}[/tex]

Answer:

See below for proof.

Step-by-step explanation:

Given trigonometric identity:

[tex]\large\boxed{\sin^2x+\cos^2x =1}[/tex]

To prove the identity 1 + cot²x = cosec²x :

[tex]\boxed{\begin{aligned}\sin^2x+\cos^2x &=1\\\textsf{Divide by $\sin^2x$}\implies \dfrac{\sin^2x}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x}& =\dfrac{1}{\sin^2x}\\1+\left(\dfrac{\cos x}{\sin x}\right)^2&=\left(\dfrac{1}{\sin x}\right)^2\\1+(\cot x)^2&=(\text{cosec}\: x)^2\\1+\cot^2x&=\text{cosec}\: ^2x\end{aligned}}[/tex]

To prove the identity tan²x + 1 = sec²x :

[tex]\boxed{\begin{aligned}\sin^2x+\cos^2x &=1\\\textsf{Divide by $\cos^2x$}\implies \dfrac{\sin^2x}{\cos^2x}+\dfrac{\cos^2x}{\cos^2x}& =\dfrac{1}{\cos^2x}\\\left(\dfrac{\sin x}{\cos x}\right)^2+1&=\left(\dfrac{1}{\cos x}\right)^2\\(\tan x)^2+1&=(\sec x)^2\\\tan^2x+1&=\sec ^2x\end{aligned}}[/tex]

Additional identities used:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Trigonometric Identities}\\\\$\tan \theta=\dfrac{\sin \theta}{\cos \theta}$\\\\$\cot \theta=\dfrac{\cos \theta}{\sin \theta}$\\\\$\csc \theta=\dfrac{1}{\sin \theta}$\\\\$\sec \theta=\dfrac{1}{\cos \theta}$\\\end{minipage}}[/tex]