The height (in feet) of a rocket launched from the ground is given by the function f(t) = -16t2 160t. match each value of time elapsed (in seconds) after the rocket’s launch to the rocket's corresponding instantaneous velocity (in feet/second).

We can comment that the maximum value of the given data is 0.2, maximum value is 42. The interquartile range is 15 to 37, first and third quartile values are 15 and 37 respectively. It can also be inferred from the box plot that there are no outliers. The median of the given data, as shown in the box plot, is 28.

What is a box plot?

A box and whisker plot, often known as a box plot, shows a data set's five-number summary. A box is drawn from the first quartile to the third quartile in a box plot. At the median, a vertical line passes through the box. The five-number summary of a box plot includes the following:

It also tells if there are any outliers in the data.

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Using proportions, it is found that it will take Voyager 2 82,900 years from its location to travel to the second closest star.

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.

In this problem, the Voyager 2 travels 15.4 km/s. In kilometers per year, considering that one hour has 3600 seconds, the measure is given by:

15.4 x 365 x 24 x 3600 = 485,654,400 km/year.

The distance to Proxima Centauri in km is given by:

d = |1.93 x 10^10 km - 4.24 x 9.5 x 10^12 km| = 4.02607 x 10^13 km.

Hence the time in years is given by:

t = d/v =  4.02607 x 10^13/485,654,400 = 82,900.

It will take Voyager 2 82,900 years from its location to travel to the second closest star.

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Answer:

Step-by-step explanation:

let the ratio of men women and children be 13x 5x and 7x

now

13x + 5x + 7x = 4152

25x = 4152

x = 4152/25

x = 166.08

The total number of integral solutions of x + y + z = 12, where −3 ≤ x ≤ 4, 2 ≤ y ≤ 11, and z ≥ 3 is; 59 integer solutions

We are given the condition that;

−3 ≤ x ≤ 4

Thus, we will use x values of -3, -2, -1, 0, 1, 2, 3, 4

When;

x = -3 and y = 2, in x + y + z = 12, solving for z gives z = 13

x = -3 and y = 3,  in x + y + z = 12, solving for z gives z = 12

x = -3 and y = 4,  in x + y + z = 12, solving for z gives z =11

x = -3 and y = 5,  in x + y + z = 12, solving for z gives z = 10

x = -3 and y = 6,  in x + y + z = 12, solving for z gives z = 9

x = -3 and y = 7,  in x + y + z = 12, solving for z gives z = 8

x = -3 and y = 8,  in x + y + z = 12, solving for z gives z = 7

x = -3 and y = 9,  in x + y + z = 12, solving for z gives z = 6

x = -3 and y = 10,  in x + y + z = 12, solving for z gives z = 5

x = -3 and y = 11,  in x + y + z = 12, solving for z gives z = 4

Thus, there are 10 solutions with x =-3

Repeating the above with x = -2, we will have 10 solutions

Similarly, with x = -1, we will have 9 more solutions

Similarly, with x = 0, we will have 8 more solutions.

Similarly, with x = 1, we will have 7 more solutions.

Similarly with x = 2, we will have 6 more solutions.

Similarly with x = 3, we will have 5 more solutions.

Similarly with x = 4, we will have 4 more solutions.

Thus,

Total number of solutions = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 10

= 59 integer solutions

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The AD's length of 3m will enable the angle at B to be divided in half.

Angle Bisector Theorem: What is it?

The angle bisector of a triangle divides the opposing side into two portions that are proportional to the other two sides, according to the angle bisector theorem, in simpler words the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

The triangle has sides of 5, 7, and (x+4) m.

Angle B's angle bisector will only be the BD if

x/4 = 5/7

x = 5 *4 / 7

x = 20/7 = 2.85 ≈ 3m

Thus if AD has length of 3m then it will enable the angle at B to be divided in half.

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Answer:

AD = 3m

Hope this helps!

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

Im going to be quick on this since your in a hurry, is AB is 3 and QR = 1  and their similar then BC would be 3 times the amount of RS which is 1.5 so the Answer is 4.5

Answer:

C

Step-by-step explanation:

　　＿＿＿　 AA

*～/■　■　⊂ P

　 |　■　■.（＿）

　 U U￣￣U U

a sampling method is dependent when the individuals selected for one sample are used to determine the individuals in the second sample

Sampling can be defined as a technique of selecting a subset of the population or  individual members in order to make statistical inferences from them and estimate the entire characteristics of the whole population.

Sampling methods are used by researchers in market research to reduce the workload and also to research the entire population

It is time-friendly, cost-effective method and forms the basis of research design.

Some sampling methods are;

It can be said to be dependent when selecting from one sample affects another sample.

Thus, a sampling method is dependent when the individuals selected for one sample are used to determine the individuals in the second sample.

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Answer:

[tex]\dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x[/tex]

Take partial fractions of the given fraction by writing out the fraction as an identity:

[tex]\begin{aligned}\dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A}{x-4}+\dfrac{B}{x+2}\\\\\implies \dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A(x+2)}{(x-4)(x+2)}+\dfrac{B(x-4)}{(x-4)(x+2)}\\\\\implies x+5 & \equiv A(x+2)+B(x-4)\end{aligned}[/tex]

Calculate the values of A and B using substitution:

[tex]\textsf{when }x=4 \implies 9 = A(6)+B(0) \implies A=\dfrac{3}{2}[/tex]

[tex]\textsf{when }x=-2 \implies 3 = A(0)+B(-6) \implies B=-\dfrac{1}{2}[/tex]

Substitute the found values of A and B:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x = \int \dfrac{3}{2(x-4)}-\dfrac{1}{2(x+2)}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ax^n\:\text{d}x=a \int x^n \:\text{d}x$\end{minipage}}[/tex]

If the terms are multiplied by constants, take them outside the integral:

[tex]\implies \displaystyle \dfrac{3}{2} \int \dfrac{1}{x-4}- \dfrac{1}{2} \int \dfrac{1}{x+2}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating}\\\\$\displaystyle \int \dfrac{f'(x)}{f(x)}\:\text{d}x=\ln |f(x)| \:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\implies \dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}[/tex]

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For an alternative approach, expand and complete the square in the denominator to write

[tex](x-4)(x+2) = x^2 - 2x - 8 = (x - 1)^2 - 9[/tex]

In the integral, substitute [tex]x - 1 = 3 \sin(u)[/tex] and [tex]dx=3\cos(u)\,du[/tex] to transform it to

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \int \frac{3\sin(u) + 6}{9 \sin^2(u) - 9} 3\cos(u) \, du \\\\ ~~~~~~~~~~~~ = - \int \frac{\sin(u) + 2}{\cos(u)} \, du \\\\ ~~~~~~~~~~~~ = - \int (\tan(u) + 2 \sec(u)) \, du[/tex]

Using the known antiderivatives

[tex]\displaystyle \int \tan(x) \, dx = - \ln|\cos(x)| + C[/tex]

[tex]\displaystyle \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C[/tex]

we get

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \ln|\cos(u)| - 2 \ln|\sec(u) + \tan(u)| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|\frac{(\sec(u) + \tan(u))^2}{\cos(u)}\right|[/tex]

Now, for [tex]n\in\Bbb Z[/tex],

[tex]\sin(u) = \dfrac{x-1}3 \implies u = \sin^{-1}\left(\dfrac{x-1}3\right) + 2n\pi[/tex]

so that

[tex]\cos(u) = \sqrt{1 - \dfrac{(x-1)^2}9} = \dfrac{\sqrt{-(x-4)(x+2)}}3 \implies \sec(u) = \dfrac3{\sqrt{-(x-4)(x+2)}}[/tex]

and

[tex]\tan(u) = \dfrac{\sin(u)}{\cos(u)} = -\dfrac{x-1}{\sqrt{-(x-4)(x+2)}}[/tex]

Then the antiderivative we found is equivalent to

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = - \ln\left|-\frac{3(x+2)}{(x-4) \sqrt{-(x-4)(x+2)}}\right| + C[/tex]

and can be expanded as

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = -\ln\left| \frac{3(x+2)^{1/2}}{(x-4)^{3/2}}\right| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|(x+2)^{1/2}\right| + \ln\left|(x-4)^{3/2}\right| + C \\\\ ~~~~~~~~~~~~ = \boxed{\frac32 \ln|x-4| - \frac12 \ln|x+2| + C}[/tex]

Answer:

y = 1/5x -3

Step-by-step explanation:

Use y = mx +b as your model.  We plug in our slope for me and our y-intercept for b.

Answer:

2

Step-by-step explanation:

given x=1 and y=7

now, given expression ,

x+y/4

by putting the values of the x and y ,we get

x+y/4

= 1+7/4

= 8/4

= 2 (Ans.)

Answer:

0

Step-by-step explanation:

54m² - 54m² = 0

Water is 0

Answer:

$25.60

Step-by-step explanation:

1 kg = 1000 grams

peanuts = 6.40 per kg = 0.064 per grams

0.064*400 = 25.6

Answer:

[tex]\displaystyle{\sin \theta = \dfrac{2ab}{a^2+b^2}}\\\\\displaystyle{\cos \theta = \dfrac{a^2-b^2}{a^2+b^2}}\\\\\displaystyle{\csc \theta = \dfrac{a^2+b^2}{2ab}}\\\\\displaystyle{\sec \theta = \dfrac{a^2+b^2}{a^2-b^2}}\\\\\displaystyle{\cot \theta = \dfrac{a^2-b^2}{2ab}}[/tex]

Step-by-step explanation:

We are given that:

[tex]\displaystyle{\tan \theta = \dfrac{2ab}{a^2-b^2}}[/tex]

To find other trigonometric ratios, first, we have to know that there are total 6 trigonometric ratios:

[tex]\displaystyle{\sin \theta = \sf \dfrac{opposite}{hypotenuse} = \dfrac{y}{r}}\\\\\displaystyle{\cos \theta = \sf \dfrac{adjacent}{hypotenuse} = \dfrac{x}{r}}\\\\\displaystyle{\tan \theta = \sf \dfrac{opposite}{adjacent} = \dfrac{y}{x}}\\\\\displaystyle{\csc \theta = \sf \dfrac{hypotenuse}{opposite} = \dfrac{r}{y}}\\\\\displaystyle{\sec \theta = \sf \dfrac{hypotenuse}{adjacent} = \dfrac{r}{x}}\\\\\displaystyle{\cot \theta = \sf \dfrac{adjacent}{opposite} = \dfrac{x}{y}}[/tex]

Since we are given tangent relation, we know that [tex]\displaystyle{y = 2ab}[/tex] and [tex]\displaystyle{x = a^2-b^2}[/tex], all we have to do is to find hypotenuse or radius (r) which you can find by applying Pythagoras Theorem.

[tex]\displaystyle{r=\sqrt{x^2+y^2}}[/tex]

Therefore:

[tex]\displaystyle{r=\sqrt{(a^2-b^2)^2+(2ab)^2}}\\\\\displaystyle{r=\sqrt{a^4-2a^2b^2+b^4+4a^2b^2}}\\\\\displaystyle{r=\sqrt{a^4+2a^2b^2+b^4}}\\\\\displaystyle{r=\sqrt{(a^2+b^2)^2}}\\\\\displaystyle{r=a^2+b^2}[/tex]

Now we can find other trigonometric ratios by simply substituting the given information below:

Hence:

[tex]\displaystyle{\sin \theta = \dfrac{y}{r} = \dfrac{2ab}{a^2+b^2}}\\\\\displaystyle{\cos \theta = \dfrac{x}{r} = \dfrac{a^2-b^2}{a^2+b^2}}\\\\\displaystyle{\csc \theta = \dfrac{r}{y} = \dfrac{a^2+b^2}{2ab}}\\\\\displaystyle{\sec \theta = \dfrac{r}{x} = \dfrac{a^2+b^2}{a^2-b^2}}\\\\\displaystyle{\cot \theta = \dfrac{x}{y} = \dfrac{a^2-b^2}{2ab}}[/tex]

will be other trigonometric ratios.

Answer:

i'll give you answer.Dont worry. Since i came back from school

If T is the midpoint of PQ and PT = 3x+3, TQ = 7x-9, then PT = 12 units.

Determining the Value of PT

It is given that,

T is the midpoint of PQ ........ (1)

PT=3x+3 ......... (2)

TQ=7x-9 .......... (3)

From (1), the distance from P to T and the distance from T to Q will be equal.

⇒ PT = TQ [Since, a midpoint divides a line into two equal segments]

Hence, equating the equations of PT and TQ given in (2) and (3) respectively, equal, we get the following,

3x + 3 = 7x - 9

or 7x - 9 = 3x + 3

or 7x - 3x = 9 + 3

or 4x = 12

or x = 12/4

⇒ x = 3

Substitute this obtained value of x in equation (2)

PT = 3(3) + 3

PT = 9 + 3

PT = 12 units

Thus, if T is the midpoint of PQ, then the measure of PT and TQ is equal to 12 units.

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The mass of the sample should be reported as 3.54 g

The amount of matter in an object is expressed in terms of mass.

The most frequent way to determine mass is to weigh something.

The units of mass are grams, kilograms, tonnes (in metric units), or ounces and pounds (US units).

According to the question,

A label on an empty sample container reads 10.000 g

A sample of a compound is added and the mass of the sample container is found to be 13.54 g.

The mass of the sample should thus be reported as,

= 13.54 - 10.000

= 3.54 g

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Answer:

[tex]\displaystyle x=\frac{5}{4},\;\;1\frac{1}{4}, \;\; or \;\; 1.25[/tex]

Step-by-step explanation:

    To solve for x, we need to isolate the x variable.

    Given:

[tex]\displaystyle x+\frac{1}{2} =\frac{7}{4}[/tex]

    Subtract [tex]\frac{1}{2}[/tex] from both sides of the equation:

[tex]\displaystyle (x+\frac{1}{2})-\frac{1}{2} =(\frac{7}{4})-\frac{1}{2}[/tex]

[tex]\displaystyle x=\frac{7}{4}-\frac{1}{2}[/tex]

    Now, we will create common denominators to simplify.

[tex]\displaystyle x=\frac{7}{4}-\frac{2}{4}[/tex]

[tex]\displaystyle x=\frac{5}{4}[/tex]

The answer is x < 1.

Bring the constant to the other side.

Divide by 4 on both sides.

[tex]\Large\texttt{Answer}[/tex]

[tex]\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\space\space\qquad\qquad\qquad}}[/tex]

[tex]\Large\texttt{Process}[/tex]

[tex]\rm{4x-6 < -2}[/tex]

Do you remember that we need to get x by itself to find its value?

We should do this:

⇨ Add 6 to both sides

[tex]\rm{4x-6+6 < -2+6}[/tex]

On the left hand side (lhs), the 6s add up to zero; on the right hand side (rhs), the -2 and 6 result in 4. Hence

[tex]\rm{4x < 4}[/tex]

Now divide both sides by 4

[tex]\rm{\cfrac{4x}{4} < \cfrac{4}{4}}[/tex]

Simplifying fractions gives us

[tex]\rm{x < 1}[/tex]

* what this means is: numbers less than 1 will make the statement true

[tex]\Large\texttt{Verification}[/tex]

Substitute 1 into the original inequality [tex]\boxed{4x-6 < -2}[/tex]

[tex]\rm{4(1)-6 < -2}[/tex]

[tex]\rm{4-6 < -2}[/tex]

Do the arithmetic

[tex]\rm{-2 < -2}[/tex]

Hope that helped

The number of terms, 'n' is 8

Let's determine the common ratio;

common ratio, r = 3/1 = 3

The formula for sum of geometric series with 'r' greater than 1 is given as;  Sn = a( r^n - 1) / (r - 1)

n is unknown

Sn = 3280

Substitute the value

3280 = 1 ( 3^n - 1) / 3- 1

3280 = 3^n -1 /2

Cross multiply

3280 × 2 = 3^n - 1

6560 + 1 = 3^n

6561 = 3^n

This could be represented as;

3^8 = 3^n

like coefficient cancels out

n = 8

Thus, the number of terms, 'n' is 8

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The number which is express in each of the models as given in the image attached to the task content are as follows;

a). 1.37

b). 1.37

c). 1.37.

It follows from the task content that the models describe that One flat represents 1 whole, One rod represents 1 tenth and one unit represents 1 hundredth.

It therefore follows from the task content that in each of the models, the algebraic sum of flat(s), rods and units as the case may be results in the value; 1.37 as the utmost number represented by the models.

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Total charges from bank A exists $15.

Total charges on Bank B spending exists $45.

Given: Bank A charges a $15 monthly fee for a checking account and debit card, with unlimited transactions.

Shade 25 transactions.

Total charges from bank A = $15 monthly

Bank B charged a $35 monthly fee for a checking account and debit card, plus $ 0.70 for each transaction.

She made 25 transactions.

Total charges on bank B = $35 + (0.40)25

Total charges on bank B = $35+10

Total charges on bank B = $45

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Answer:

21

Step-by-step explanation:

In the diagram, the three tangents (segment touching a circle at one point) have equal length.

6y - 3 = 29 - 2y

8y = 32

y = 4

Since the lengths of segments AM and AN are equivalent, we can substitute the value of y into the expression, 6y - 3, to find AN.

6y - 3 = 6*4 - 3 = 24 - 3 = 21

No,the population mean is not equal to 9.2 and the value in t statistic is -1.02.

Given sample size of 49,sample mean of 8.5,standard deviation of 1.5, significance level of 0.01.

We are required to find out whether the population mean is equal to 9.2 and the value of t in test statistic.

We have to first make the hypothesis for this.

[tex]H_{0}[/tex]:μ≠9.2

[tex]H_{1}[/tex]:μ=9.2

We have to use z statistic because the sample size is more than 30.

Z=(X-μ)/σ

We have been given sample mean but we require population mean in the formula so we will use sample mean.

Z=(8.5-9.2)/1.5

=-0.7/1.5

=-0.467

P value of -0.467 is 0.67975.

P value is greater than 0.01 so we will accept the hypothesis means population mean is not equal to 9.2.

t=(X-μ)/s/[tex]\sqrt{n}[/tex]

=(8.5-9.2)/1.5/[tex]\sqrt{49}[/tex]

=-0.7/0.21

=-1.02

Hence it is concluded that no,the population mean is not equal to 9.2 and the value in t statistic is -1.02.

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Answer:

Option D

Step-by-step explanation:

Using the surd law :

[tex]\sqrt{ab} = \sqrt{a}\sqrt{b}[/tex]

We can find the largest square number that goes into 75 :

Let's write the multiples of 75 :

1 , 75

3 , 25

5 , 15

The only square number is 25

So using the law mentioned above we split √75 into :

√25√3

The square root of 25 is 5

Now we have our final answer of 5√3

Hope this helped and have a good day

The simplified form of expression √75 is 5√3.

Option D is the correct answer.

We have,

To simplify √75, we can factor it into its prime factors and then take the square root:

√75 = √(3 * 5 * 5)

= √(3 x 5²)

Take out the perfect square factor from under the square root:

= √3 x √5²

= √3 x 5

= 5√3

Thus,

The simplified form of expression √75 is 5√3 which is option D.

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The value of the function g(-3) from the given piecewise function is 1

A piecewise-defined function is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain.

From the given piecewise function, we are to find the value of the function when x is -3 that is g(-3)

In order to determine the equivalent function, we need to determine the function where x = -3

The equivalent function is g(x) = x+ 4

Substitute x = -3 into the resulting function

g(x) = x + 4

g(-3) = -3 + 4

g(-3) = 1

Hence the value of the function g(-3) from the given piecewise function is 1

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Answer:

5,250 km

Step-by-step explanation:

[tex]\frac{hours}{miles}[/tex] = [tex]\frac{hours}{miles}[/tex]

[tex]\frac{3}{3500}[/tex] = [tex]\frac{4.5}{x}[/tex]  Cross Multiply

3x = (4.5)(3500)

3x = 15750 Divide both sides of the equation by 3

x = 5,250

Answer:

(0, - 3 )

Step-by-step explanation:

- 5x - 3y - 9 = 0 → (1)

4x - 18y - 54 = 0 → (2)

multiplying (1) by - 6 and adding to (2) will eliminate y

30x + 18y + 54 = 0 → (3)

add (2) and (3) term by term to eliminate y

34x + 0 + 0 = 0

34x = 0 ⇒ x = 0

substitute x = 0 into either of the 2 equations and solve for y

substituting into (2)

4(0) - 18y - 54 = 0

- 18y - 54 = 0 ( add 54 to both sides )

- 18y = 54 ( divide both sides by - 18 )

y = - 3

solution is (0, - 3 )

Answer:

(0, -3)

Step-by-step explanation:

This system of equations consists of two equations. There are 3 main ways to solve a system of equations:

First, start by having the variables on one side.

[tex]-5x-3y-9=0 \Rightarrow \text{Add 9 to both sides} \Rightarrow -5x-3y=9\\4x-18y-54=0 \Rightarrow \text{Add 54 to both sides} \Rightarrow 4x-18y=54 \Rightarrow \text{Simplify} \Rightarrow 2x-9y=27[/tex]

This method is the easiest to use in this situation.

In this method, we increase equations by a certain factor in order to eliminate one variable.

We can see that 3y in the first equation can be multiplied by 6 in order to obtain the 18y in the second equation. Therefore, we can multiply the whole first equation by 6:

[tex]-30x-18y=54\\4x-18y=54[/tex]

Now, subtract the two equations to eliminate y.

[tex]-34x=0\\x=0[/tex]

Plug in 0 to x in either of the equations to solve for y:

[tex]-5(0)-3y=9\\0-3y=9\\-3y=9\\ \text{Divide both sides by -3}\\y=-3[/tex]

OR

[tex]4(0)-18y=54\\0-18y=54\\-18y=54\\\text{Divide both sides by -18}\\y=-3[/tex]

Therefore:

(x, y) = (0, -3)

We need at least 7 terms of the Maclaurin series for ln(1 + x)  to estimate ln 1.4 to within 0.0001

For given question,

We have been given a function f(x) = ln(1 + x)

We need to find  the estimate of In(1.4) within 0.001 by applying the function of the Maclaurin series for f(x) = In (1 + x)

The expansion of ln(1 + x) about zero is:

[tex]ln(1+x)=x-\frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4} +\frac{x^5}{5} -\frac{x^6}{6} +.~.~.[/tex]

where -1 ≤ x ≤ 1

To estimate the value of In(1.4), let's replace x with 0.4

[tex]\Rightarrow ln(1+0.4)=0.4-\frac{0.4^2}{2} + \frac{0.4^3}{3} -\frac{0.4^4}{4} +\frac{0.4^5}{5} -\frac{0.4^6}{6} +.~.~.[/tex]

From the above calculations, we will realize that the value of  [tex]\frac{0.4^5}{5}=0.002048[/tex] and [tex]\frac{0.4^6}{6}=0.000683[/tex]  which are approximately equal to 0.001

Hence, the estimate of In(1.4) to the term [tex]\frac{0.4^6}{6}[/tex]  is enough to justify our claim.

Therefore,  we need at least 7 terms of the Maclaurin series for function ln(1 + x)  to estimate ln 1.4 to within 0.0001

Learn more about the Maclaurin series here:

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Answer:  [tex]\\ \lim\limits_{k \to \infty} (1+\frac{4}{k})^k =e^4.[/tex]

Step-by-step explanation:

[tex]\displaystyle\\ \lim_{k \to \infty} (1+\frac{4}{k})^k \\x=\frac{x}{4} *4\\So,\ \lim_{k \to \infty} (1+\frac{4}{k})^\frac{k}{4}*4 \\ \lim_{k \to \infty} ((1+\frac{4}{k})^\frac{k}{4} )^4.\\Use\ the\ second\ wonderful\ limit:\\\boxed { \lim_{x \to \infty} (1+\frac{1}{x})^x=e },\\\\So,\\ \lim_{k \to \infty} (1+\frac{4}{k})^k =e^4.[/tex]

We can also apply l'Hôpital's rule by first rewriting the limit as

[tex]\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp\left(\ln \left(1 + \frac4k\right)^k\right) = \exp\left(\lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k}\right)[/tex]

Applying the rule gives

[tex]\displaystyle \lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k} = \lim_{k\to\infty} \frac{\left(-\frac4{k^2}\right)/\left(1+\frac4k\right)}{-\frac1{k^2}} = 4 \lim_{k\to\infty} \frac1{1 + 4k} = 4[/tex]

so that the overall limit is

[tex]\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp(4) = \boxed{e^4}[/tex]