Find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→0 e3x − 1 − 3x x2

The domain of the function y = √(x + 6) - 7 is x > -6

The equation of the function is given as

y = √(x + 6) - 7

Set the radical greater than 0

x + 6 > 0

Subtract 6 from both sides of the equation

x > -6

Hence, the domain of the function y = √(x + 6) - 7 is x > -6

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The domain of the function in discuss described as; y = √x+6 -7 is; x >= -6.

According to the task content, it follows that the domain of.the function can be evaluated by means of the characteristics associated with the square root.

The function given is; y = √x+6 -7

Since, the square root of a negative number renders a complex number as it's results, it follows that; x+6 >= 0.

Hence, x >= -6.

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For function g(x, y) = 9x² + 6y²,

the absolute minimum is 15 and the absolute maximum is 303

For given question,

We have been given a function g(x, y) = 9x² + 6y² subject to the constraint −1≤x≤1 and −1≤y≤7

We need to find the absolute maximum and minimum values of the function.

First we find the partial derivative of function g(x, y) with respect to x.

⇒ [tex]g_x=18x[/tex]

Now, we find the partial derivative of function g(x, y) with respect to x.

⇒ [tex]g_y=12y[/tex]

To find the critical point:

consider [tex]g_x=0[/tex]        and      [tex]g_y=0[/tex]

⇒         18x = 0         and      12y = 0

⇒          x = 0           and       y = 0

This means, the critical point of function is (0, 0)

We have been given constraints −1 ≤ x ≤ 1 and −1 ≤ y ≤7

Consider g(-1, -1)

⇒ g(-1, -1) = 9(-1)² + 6(-1)²

⇒ g(-1, -1) =  9 + 6

⇒ g(-1, -1) = 15

And g(1, 7)

⇒ g(1, 7) =  9(1)² + 6(7)²

⇒ g(1, 7) = 9 + 294

⇒ g(1, 7) = 303

Therefore, for function g(x, y) = 9x² + 6y²,

the absolute minimum is 15 and the absolute maximum is 303

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Triangle ABC was reflected over the y axis and translated 3 units down to form triangle A'B'C'.

Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, translation, rotation and dilation.

Rigid transformation preserves the shape and size of the figure. Reflection, translation, rotation are rigid transformations.

Triangle ABC was reflected over the y axis and translated 3 units down to form triangle A'B'C'.

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Check the picture below.

The removable discontinuity of the given function is (-3, -3).

What are the discontinuities of the function?

The given function is  [tex]f(x) = \frac{x^{2}+ 12x + 27 }{x^{2} + 4x +3}[/tex]

f(x)=(x²+12x+27)/(x²+4x+3)=(x²+9x+3x+27)/(x²+3x+x+3+

=(x+9)(x+3)/(x+3)(x+1)=(x+9)/(x+1)

The holes in the graph by factoring and cancelling are (-3, -3).

Therefore, the removable discontinuity of the given function is (-3, -3).

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Answer: first answer is -3 for both and second is x=-1

Step-by-step explanation:

The 17th term of the geometric sequence given in the problem is:

[tex]a_{17} = \frac{13}{8}[/tex]

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

The nth term of a geometric sequence is given by:

[tex]a_n = a_1q^{n-1}[/tex]

In which [tex]a_1[/tex] is the first term.

As a function of the mth term, the nth term can also be given as follows:

[tex]a_n = a_mq^{n - m}[/tex]

In this problem, we have that:

[tex]a_{13} = 26, q = \frac{1}{2}[/tex]

Hence the 17th term is:

[tex]a_{17} = a_{13}q^{4}[/tex]

[tex]a_{17} = 26 \times \frac{1}{16}[/tex]

[tex]a_{17} = \frac{13}{8}[/tex]

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Answer:

37 child tickets / 74 adult tickets

Step-by-step explanation:

I randomly picked a number and increased or decreased whether the solution was too high or low (guess and check)

The number of child tickets sold that day is 37.

We have,

Let's assume the number of child tickets sold is "C" and the number of adult tickets sold is "A."

The cost of a child ticket: $6.80

The cost of an adult ticket: $9.90

The total sales for the day: $984.20

The number of adult tickets sold is twice the number of child tickets sold:

A = 2C

To find the number of child tickets sold, set up an equation based on the total sales:

6.80C + 9.90A = 984.20

Substituting the value of A from equation 4:

6.80C + 9.90(2C) = 984.20

Simplifying the equation:

6.80C + 19.80C = 984.20

26.60C = 984.20

C = 984.20 / 26.60

C ≈ 37

Therefore,

37 child tickets were sold that day.

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Answer:

Step-by-step explanation:

Area of circle:

area = π · r · r

Radius=  [tex]\frac{16}{2}[/tex]= 8

[tex]3.14\times { 8 }^{ 2 }[/tex] = 200.96 [tex]cm^2\\[/tex]

Answer: (5/2, 3)

Step-by-step explanation:

Substituting in x=5/2,

[tex]2(5/2)+5y=20\\\\5+5y=20\\\\5y=15\\\\y=3[/tex]

So, the point is (5/2, 3)

The required answers are:

1) The center of the circle = (4, 8)

2) The radius of the circle = 2.5 units

3) The equation of the circle = (x - 4)² + (y - 8)² = 6.25

The equation of the circle which has a center at (h, k) and a radius of 'r' units is (x - h)² + (y - k)² = r²

To calculate radius 'r', we have r = sqrt( (x1 - h)² + (y1 - k)²)

Where (x1, y1) is the point that lies on the circle.

Given that,

The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5)

We know that the longest chord on a circle is nothing but the diameter of the circle.

So, the center is the midpoint of the diameter. I.e.,

(h, k) = ([tex]\frac{4+4}{2}[/tex], [tex]\frac{5.5+10.5}{2}[/tex])

⇒ (h, k) = (4, 8)

Therefore, the center of the circle is (4, 8)

Then, the radius is calculated by

r = sqrt( (x1 - h)² + (y1 - k)²)

⇒ r = [tex]\sqrt{(4-4)^2+(5.5-8)^2}[/tex]

⇒ r = 2.5 units

Thus, the radius of the circle is 2.5 units.

So, the equation of the circle with center (4, 8) and radius of 2.5 units is,

(x - h)² + (y - k)² = r²

⇒ (x - 4)² + (y - 8)² = 2.5²

⇒ (x - 4)² + (y - 8)² = 6.25

Thus, the equation of the circle is x - 4)² + (y - 8)² = 6.25.

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Answer:

x - 14.95 = 12.48

x = 27.43

Step-by-step explanation:

The null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% of the significance level is [tex]H_{0}[/tex]:μ<0.06.

Given that the significance level is 0.10.

We are required to form the null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% the significance level.

Hypothesis is a statement which is tested for its validity. Null hypothesis is the statement which is accepted or not by z test,t test,f test ,chi-square test or any other test.

We have to take opposite of the statement to form a null hypothesis. Since we have to check whether the proportion of people who owns cats is larger than 60% of the significance level, we have to assume that it is smaller than 60% of the significance level.

60% of the significance level=0.60*0.10=0.06.

Null hypothesis is [tex]H_{0}[/tex]:μ<0.06

Hence the null hypothesis to test the claim that the proportion of people who owns cats is larger than 60% of the significance level is [tex]H_{0}[/tex]:μ<0.06.

Question is incomplete.The question should include the following:

Find the null hypothesis for the testing.

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Answer:

$3,500 labor and $3,500 materials

Step-by-step explanation:

furnishings + labor + materials = 10,000

furnishings = 3000

3000 + labor + materials = 10,000

labor = materials

3000 + labor + labor = 10,000

2(labor) = 7,000

labor = 7,000/2

labor = 3,500

labor = materials = 3,500

[tex] \huge\mathbb{ \underline{SOLUTION :}}[/tex]

[tex]\\[/tex]

The [tex]\mathrm{distributive \: property}[/tex] states that an expression that is given in the form of [tex]\small\sf{ A(B + C)}[/tex] can be solved as [tex]\small\sf{A \times (B + C) = AB + AC}[/tex] . So:

[tex]\small\longrightarrow\sf{-24-18-2+4}[/tex]

[tex]\small\longrightarrow\sf{-42+2}[/tex]

▪ [tex]\large\tt{All \: \: options \: \: are \: \: wrong}[/tex]

[tex]\\[/tex]

[tex]\huge \mathbb{ \underline{ANSWER:}}[/tex]

[tex]\small\longrightarrow\sf{−6 (2^2+3) − 2 (1^2 - 2) = \underline{-6(4+3)}}[/tex]

Answer:

○ [tex]h = \frac{2A}{b}[/tex]  

Step-by-step explanation:

We are given:

[tex]A = \frac{1}{2} b h[/tex]

To solve for [tex]h[/tex], we have to rearrange the equation to make [tex]h[/tex] the subject:

[tex]A = \frac{1}{2} b h[/tex]

⇒ [tex]2A = bh[/tex]     [multiplying both sides by 2]

⇒ [tex]\frac{2A}{b} = h[/tex]       [dividing both sides by b]

⇒  [tex]h = \frac{2A}{b}[/tex]      [swapping sides]

Answer:

4 is correct answer.

Step-by-step explanation:

That because it contains property of communicative. That is a×b=b×a.

Answer:

A function that is positive in the entire interval [-3, -2] is -x2 - 5x - 5.

Answer:

The second function (second graph and choice)

Step-by-step explanation:

If you look at the second function you will see that within the closed interval [-3,-2] the graph y  values are positive

First choice is incorrect since at x = -2  the y value is negative

Third choice incorrect since at x = -2, y value is negative

Fourth choice incorrect since y value is negative for x = -2

[tex]\qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ \displaystyle S=\sum\limits_{i=0}^{\infty}\ a_1\cdot r^i\implies S=\cfrac{a_1}{1-r}\quad \begin{cases} a_1=\stackrel{\textit{first term}}{\frac{1}{8}}\\ r=\stackrel{\textit{common ratio}}{\frac{2}{3}}\\ \qquad -1 < r < 1 \end{cases}[/tex]

[tex]\displaystyle\sum_{k=0}^{\infty} ~~ \underset{a_1}{\frac{1}{8}}\underset{r}{\left( \frac{2}{3} \right)}^k\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{1-\frac{2}{3}}\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{\frac{1}{3}}\implies S=\cfrac{3}{8}[/tex]

When graphing, the intersections of the graphs represent the solutions of the system.

When graphing a system of equations, you just need to graph both equations in the same coordinate axis.

The solutions of the system are all the points where the graphs of the two equations intersect.

This means that if there is only one intersection, there is one solution.

But we can have more than one intersection, like in the case where at least one of the equations is a polynomial of degree 2 or more.

And there is also the case that the graphs never intersect, like in parallel lines, in these cases we have no solutions.

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Work Shown:

k = scale factor

k = (A'B')/(AB)

k = 8/12

k = (4*2)/(4*3)

k = 2/3

Triangle A'B'C' (image) has side lengths that are 2/3 as long compared to the side lengths of triangle ABC (preimage).

The probability that the jackpot prize will be won in each of four consecutive draws is (1/60)⁴.

The number of consecutive draws needed will be, n = 233

Probability is the likelihood or chance of an event happening or not.

From the given question, the probability of the jackpot being won in any draw is 1/60.

The probability that the jackpot prize will be won in each of four consecutive draws will be:

1/60 * 1/60 * 1/60 * 1/60 = (1/60)⁴

b. The number of consecutive draws that needs to be made for there to be a greater than 98% chance that at least one jackpot prize will have been won is calculated as follows:

There is a 100% - 98% chance that that none has been won = 2% that none has been won.

Also, the probability of the jackpot not being won in a draw is = 1 1/60 = 59/60

The number of consecutive draws needed will be (59/60)ⁿ ≤ 0.02

Solving for n by taking logarithms of both sides:

n = 233

In conclusion, probability measures chances of an event occurring or not.

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Using relations in a right triangle, it is found that the angle of depression is of θ = 56.31º.

The relations in a right triangle are given as follows:

For this problem, we have that:

Hence, considering θ as the depression angle, we have that:

tan(θ) = 9000/6000

tan(θ) = 1.5

θ = arctan(1.5)

θ = 56.31º.

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Answer:

Step-by-step explanation:

The minimum is at the vertex of this parabola because it opens up.

Now if (-4, 2) is the minimum then  all the y values on the parabola must be > 2,

But we are given that  y = -3 is on the graph ( the point (6,-3) - that is y < 2 here,

Therefore (-4, 2)  cannot be the vertex .

Substitute [tex]y=4+5\sqrt x[/tex] and [tex]dy=\frac5{2\sqrt x}\,dx[/tex]. Then the integral is

[tex]\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = 2 \int \frac{1}{(4+5\sqrtx)^2} \frac{5}{2\sqrt x} \, dx = 2 \int y^{-2} \, dy[/tex]

By the power rule,

[tex]\displaystyle \int y^{-2} \, dy = -y^{-1} + C[/tex]

so that

[tex]\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = \boxed{-\frac2{4+5\sqrt x} + C}[/tex]

The correct answer is option (C) median.

The median will help Emily to find the number that appears in the middle of the 25 numbers that are arranged in ascending order.

The mean, median, and mode are the three most commonly used measures of central tendency for populations that do not have much data, that is, they do not need to be grouped.

The mean, also known as average, is the value obtained by dividing the sum of a cluster of numbers by the number of them.

When arranging the numbers from least to largest, the median sits exactly in the middle of the values that are above. The median is a set that is a value that is in the middle of the other values.

The number that appears most frequently in a set of numbers is called the mode.

So, The median will help Emily to find the number that appears in the middle of the 25 numbers that are arranged in ascending order.

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Assuming you mean [tex]f(x,y) = xy^2[/tex] over the domain

[tex]D = \left\{(x,y) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } x^2 + y^2 \le 3\right\}[/tex]

we first observe that [tex]f(x,y) = 0[/tex] for all [tex](x,y)[/tex] on the coordinate axes.

There are no critical points elsewhere in the interior of [tex]D[/tex], since

[tex]\dfrac{\partial f}{\partial x} = y^2 = 0 \implies y=0[/tex]

[tex]\dfrac{\partial f}{\partial y} = 2xy = 0 \implies x = 0 \text{ or } y = 0[/tex]

Parameterize the circular arc boundary by [tex]x=\sqrt3\cos(t)[/tex] and [tex]y=\sqrt3\sin(t)[/tex], where [tex]0\le t\le\frac\pi2[/tex]. Then

[tex]f(x(t), y(t)) = g(t) = 3\sqrt3 \cos(t) \sin^2(t) = 3\sqrt 3 (\cos(t) - \cos^3(t))[/tex]

Find the critical points of [tex]g[/tex].

[tex]g'(t) = -3\sqrt3 \sin(t) + 9\sqrt3 \cos^2(t) \sin(t) = 0[/tex]

[tex]-3 \sin(t) (1 - 3 \cos^2(t)) = 0[/tex]

[tex]\sin(t) = 0 \text{ or } 1 - 3 \cos^2(t) = 0[/tex]

[tex]\sin(t) = 0 \text{ or } \cos^2(t) = \dfrac13[/tex]

[tex]\sin(t) = 0 \text{ or } \cos(t) = \pm\dfrac1{\sqrt3}[/tex]

In the first case, we get

[tex]t = \sin^{-1}(0) + 2n\pi \text{ or } t = \pi - \sin^{-1}(0) + 2n\pi[/tex]

where [tex]n[/tex] is an integer; the only solution on the boundary of [tex]D[/tex] is [tex]t=0[/tex] corresponding to the point [tex](\sqrt3,0)[/tex].

In the second case, we get

[tex]t = \cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi \text{ or } t = -\cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi[/tex]

with only one relevant solution at [tex]t=\cos^{-1}\left(\frac1{\sqrt3}\right)[/tex]  corresponding to [tex](1,\sqrt2)[/tex].

In the third case, we get

[tex]t = \cos^{-1}\left(-\dfrac1{\sqrt3}\right) + 2n\pi \text{ or } t = -\cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi[/tex]

but there is no [tex]t[/tex] in this family of solutions such that [tex]0\le t\le\frac\pi2[/tex].

So, we find

[tex]\min\left\{xy^2 \mid (x,y) \in D\right\} = 0 \text{ at } (0,0)[/tex]

(but really any point on either axis works)

[tex]\max \left\{xy^2 \mid (x,y) \in D\right\} = 2 \text{ at } (1,\sqrt2)[/tex]

Answer: 104 degrees farenheit

Step-by-step explanation:    H =  record high temperature. -5 + 109 = H. -5+109 = 109 + (-5) = 109-5 = 104. H = 104.

Answer:

104

Step-by-step explanation:

Let x = record high and y = record low temperature in Charlotte. The difference between the records high and low, x and y, is 109 degrees Fahrenheit, so x - y = 109. Record low is -5, so x - (-5) = 109.

x + 5 = 109

x = 104

Answer:

d

Step-by-step explanation:

the opposite angles of a cyclic quadrilateral sum to 180° , that is

∠ ABD + ∠ ACD = 180°

∠ ABD + 48° = 180° ( subtract 48° from both sides )

∠ ABD = 132°