Let ​f(x) = 1 / 2+3x and point P = (2 , 1/8)

a.) Use the following definition of the slope of the tangent line at x = a to find the slope of the line tangent to the graph of f at P.

m = lim f(a+h) - f(a) / h
tan h --> 0

b.) Determine an equation of the tangent line at P.

Answer:

===================================

Continuous compounding equation:

Now find the time t for the balance to grow to $51806.67:

The time required is about 9 years.

Answer:

9 years

Step-by-step explanation:

Continuous Compounding Formula

[tex]\large \text{$ \sf A=Pe^{rt} $}[/tex]

where:

Given values:

Substitute the given values into the formula and solve for r to find the annual interest rate:

[tex]\implies \sf 39786.15=36000 \cdot e^{2.5r}[/tex]

[tex]\implies \sf \dfrac{39786.15}{36000}= e^{2.5r}[/tex]

[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=\ln e^{2.5r}[/tex]

[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=2.5r[/tex]

[tex]\implies \sf \dfrac{1}{2.5}\ln \left(\dfrac{39786.15}{36000}\right)=r[/tex]

[tex]\implies \sf r=\dfrac{2}{5}\ln \left(\dfrac{39786.15}{36000}\right)[/tex]

[tex]\implies \sf r=0.03999996933...[/tex]

[tex]\implies \sf r=0.04\;(2\;d.p.)[/tex]

Therefore, the annual interest rate is 0.04 = 4%.

To calculate how many years it will take for $36,000 to grow to $51,806.67, substitute the values into the formula and solve for t:

[tex]\implies \sf 51806.67=36000 \cdot e^{0.04t}[/tex]

[tex]\implies \sf \dfrac{51806.67}{36000}= e^{0.04t}[/tex]

[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= \ln e^{0.04t}[/tex]

[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= 0.04t[/tex]

[tex]\implies \sf \dfrac{\ln\left(\dfrac{51806.67}{36000}\right)}{0.04}=t[/tex]

[tex]\implies \sf t=9.10 \; (2\;d.p.)[/tex]

Therefore, it will take about 9 years for $36,000 to grow to $51,806,67.

Answer:

36.147580645161

Step-by-step explanation:

Hope it helps you

The geometric sequence in which the 3rd term is of 18 and the 7th term is of 29/8 is defined as follows:

[tex]a_n = 40(0.67)^{n - 1}[/tex]

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

The nth term of a geometric sequence is given by the rule presented as follows:

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

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

The rule for the nth term is also called the general rule.

The general takes the first term as the reference term, however, the 7th term can be written as a function of the 3rd term, as follows:

[tex]a_7 = a_3q^{4}[/tex]

The fraction that represents the mixed number 3 and 5/8 is:

(3 x 8 + 5)/8 = 29/8.

Then the common ratio of the sequence is calculated as follows:

[tex]\frac{29}{8} = 18q^{4}[/tex]

[tex]q^4 = \frac{29}{144}[/tex]

[tex]q = \sqrt[4]{\frac{29}{144}}[/tex]

[tex]q = \left(\frac{29}{144}\right)^{\frac{1}{4}}[/tex]

q = 0.67.

Then the first term of the sequence is obtained as follows:

[tex]a_3 = a_1q^2[/tex]

[tex]a_1 = \frac{a_3}{q^2}[/tex]

[tex]a_1 = \frac{18}{(0.67)^2}[/tex]

[tex]a_1 = 40[/tex]

Then the definition of the geometric sequence is:

[tex]a_n = 40(0.67)^{n - 1}[/tex]

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The previous population for the city is 23460.

Given that the population of a city has increased by 15% since it was last measured, the previous population will be:

= 1 - 15%

= 85%

Therefore, the previous population will be:

= Percentage × New population

= 85% × 27600

= 0.85 × 27600

= 23460

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By applying the distributive property of multiplication, this mathematical expression can be rewritten as -3(-4x + 5) = 12x - 15.

The distributive property of multiplication states that when the sum of two (2) or more addends are multiplied by a particular numerical value, the same result and output would be obtained as when each addend is multiplied respectively by the same numerical value, and the products are added together.

Mathematically, the distributive property of multiplication can be represented by this mathematical expression:

a(b + c) = ab + ac.

By applying the distributive property of multiplication to the given mathematical expression, we have the following:

-3(-4x + 5) = -3(-4x) + [-3(5)]

-3(-4x + 5) = 12x - 15.

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Answer:

  (-12, 10)

Step-by-step explanation:

You want to solve by elimination the system of equations ...

The point of elimination is to make one of the variables disappear from the equation.

Here, we observe that the x-coefficients are opposites in the two equations. When we add them, the result is 0: the x-term disappears. Adding the two equations, we have ...

  (2x +4y) +(-2x -3y) = (16) +(-6)

  y = 10 . . . . . . . . simplified

Using this value for y, we can find x. Substituting into the first equation gives ...

  2x +4(10) = 16

  2x = 16 -40 = -24 . . . . . subtract 40 and simplify

  x = -12 . . . . divide by 2

The ordered pair that is the solution is (x, y) = (-12, 10).

The equation for the situation is written as y = 35x + 10

Information gotten from the question include

a tank is filing with petrol at rate of 35 liter per minute

the tank already had 10 liters in it before the filling began

A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.

y = mx + c

Definition of variables to suit the problem to be solved

y = total petrol in  the tank in  liters

m = rate of filling liter per minute = 35

x = number of liters

c = quantity prior to filling = 10

The equation is written as

y = 35x + 10

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The number of cars that Tim has in all is 215.

The boxcars that Tim has is written in standard form. While the boxcars that Tim's grandfather gave him is written in scientific form. The first step is to convert the scientific form to standard form.

Number of boxcars that Tim's grandfather gave him: 1.62 x 10²

= 1.62 x 100

= 162

The next step is to add the boxcars that Tim's grandfather gave him to the number of boxcars that Tim has.

Total boxcars that Tim has = 162 + 53 = 215 box cars

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Answer: 12,000pd=6 tons

Step-by-step explanation:

The area of triangle XYZ in the given graph with points X(-6,-5), Z(-4,-2) and Y(-2,-6) is 4 units square.

If the triangle's three vertices are specified on the coordinate plane, the area of a triangle in coordinate geometry may be determined. In coordinate geometry, a triangle's area is the area or space it occupies in the 2-D coordinate plane. The formula for finding the area of a triangle with three coordinates [tex](x_1,y_1), (x_2,y_2),[/tex] and [tex](x_3,y_3)[/tex]  is as follows:

Area of triangle [tex]=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]

Now, finding the area of trianlge with points X(-6,-5), Z(-4,-2) and Y(-2,-6).

Area of triangle [tex]=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]

[tex]=\frac{1}{2}|-6(-2+5)-4(-6+5)+-2(-5+2)|\\=\frac{1}{2}\times|-18+4+6|\\ =4[/tex]

Therefore, the area of triangle XYZ in the given graph is 4 units square.

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The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10

Given that,

The function is f(x)= -2x² + 3x + 10

We have to find f(1) and f(3).

The core concept of mathematics' calculus is functions. The functions are special types of relations. In mathematics, a function is represented as a rule that produces a distinct result for each input x.

Take the function

f(x)= -2x² + 3x + 10

Take x as 1

f(1)= -2(1)² + 3(1) +10

f(1)= -2+3+10

f(1) = 11

Now, Take x as 3

f(3)= -2(3)² + 3(3) +10

f(3)= -18+9+10

f(3) = 1

Therefore, The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10

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The probability is 0.5881.

From the question, we have

n = 20

P(X[tex]\geq[/tex]10) = 1-P(X<10)

P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)

P(X[tex]\leq[/tex]9) = P(X=0)+P(X=1)+..................+P(X=9)

=0.4119

P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)

=1-0.4119

=0.5881

Probability:

Probability refers to possibility. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is.

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Yes, Dan productivity is constant and slope is given as 3/2.

What is slope?

The slope of a line is also called its gradient or rate of change. The slope formula is the vertical change in y divided by the horizontal change in x, sometimes called rise over run. The slope formula uses two points, (x1, y1) and (x2, y2), to calculate the change in y over the change in x.

Let we take one pair of point (4,6) and (2,3) and other pair of point is (6,9) and (0,0)

The slope between  (4,6) and (2,3)  is

m = y2-y1/x2-x1

m = 3-6/2-4

=3/2

The slope between (6,9) and (0,0)

m = 0-9/0-6

=3/2

Hence, both the slopes are equal, therefore his work is constant.

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The probability of more than 5 people is 0.4729

Since 49% of the people in a community use the emergency room at a hospital in one year, a sample of 11 people use the emergency room. To find the probability of more than 5 people use the emergency room, we use binomial probability.

The binomial probability of x is given by  P(X = x) = ⁿCₓpˣ(1 - p)ⁿ⁻ˣ

where

Since

We determine the probability of x ≤ 5.

So, P(X ≤ 5) = ¹¹C₀p⁰(1 - p)¹¹⁻⁰ + ¹¹C₁p¹(1 - p)¹¹⁻¹ + ¹¹C₂p²(1 - p)¹¹⁻² + ¹¹C₃p³(1 - p)¹¹⁻³

+ + ¹¹C₄p²(1 - p)¹¹⁻⁴ + ¹¹C₅p⁵(1 - p)¹¹⁻⁵

= ¹¹C₀(0.49)⁰(1 - 0.49)¹¹ + ¹¹C₁(0.49)¹(1 - 0.49)¹⁰ + ¹¹C₂(0.49)²(1 - 0.49)⁹ + ¹¹C₃(0.49)³(1 - 0.49)⁸

+ ¹¹C₄(0.49)⁴(1 - p)⁷ + ¹¹C₅(0.49)⁵(1 - 0.49)⁶

= (1)(1)(0.51)¹¹ + (11)(0.49)(0.51)¹⁰ + (55)(0.49)²(0.51)⁹ + (165)(0.49)³(0.51)⁸

+ (330)(0.49)⁴(0.51)⁷ + (462)(0.49)⁵(0.51)⁶

= 0.0006071 + (11)(0.49)(0.0011904) + (55)(0.2401)(0.0023341) + (165)(0.117649)(0.0045768)

+ (330)(0.0576480)(0.0089741) + (462)(0.0282475)(0.0175963)

= 0.0006071 + 0.0064163 + 0.0308230 + 0.0888452

+ 0.1707218 + 0.2296378

= 0.5270512

≅ 0.5271

So, P(X ≤ 5) = 0.5271, the probability of more than 5 people is P(X ≥ 5) = 1 - P(X ≤ 5)

= 1 - 0.5271

= 0.4729

The probability is 0.4729

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Answer:

y = -2/3 x + 8

Step-by-step explanation:

here is the answer in the image above

The probabilities are given as follows:

a) P(both marbles are red) = 0.167 = 16.7%.

b) P(red then green) = 0.111 = 11.1%.

c) P(green then blue) =  0.083 = 8.3%.

d) P(blue and red, any order) = 0.333 = 33.3%.

The probability of an event in an experiment is calculated as the number of desired outcomes in the context of the experiment divided by the number of total outcomes in the context of the experiment.

In this problem, the are nine marbles, of which four are red, hence the probability of both red is:

p = 4/9 x 3/8 = 0.167 = 16.7%.

(the marble is not replaced, hence for the second marble there will be eight possible marbles, of which 3 are red).

For red then green, the probability is:

p = 4/9 x 2/8 = 0.111 = 11.1%.

For green then blue, the probability is:

p = 2/9 x 3/8 = 0.083 = 8.3%.

For blue or red, there are two possible outcomes, blue then red or red then blue, hence:

p = 3/9 x 4/8 + 4/9 x 3/8 = 0.333 = 33.3%.

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The number of packages that need to be delivered daily by UpEx to make a profit of $32 is 115 packages

First, find the contribution margin of each package:

= Charge to deliver package - Cost to deliver package

= 3.50 - 1.50

= $2.00

To find the number of packages needed to make a profit of $32, the formula is:

= (Fixed cost + projected profit) / Contribution margin

= ( 198 + 32) / 2

= 230 / 2

= 115 packages

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Answer: not a function

Step-by-step explanation: one input has more than one output and to be a function it must have one output for every input.

I hope it’s right, best of luck :)

The mean of the given scores is 15.43

It is the average value of the set given.

It is calculated as:

Mean = Sum of all the values of the set given / Number of values in the set

We have,

Score       Mid-values       Frequency      Total score

1 - 7                  4                  19                 19 x 4 = 76

8 - 10               9                  16                  16 x 9 = 144

11 - 15              13                  6                   13 x 6 = 78

16 - 20            18                  14                  18 x 14 = 252

21 - 35            28                 15                  28 x 15 = 420  

36 - 50           43                  4                   43 x 4 = 172

Total                                     74                  1142

Mean = 1142 / 74 = 15.43

Thus,

The mean of the given scores is 15.43

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Answer:

3

Step-by-step explanation:

[tex]27^{\frac{1}{3}}[/tex]

1 ) Factor the number [tex]27=3^{3}[/tex]

[tex]=\left(3^3\right)^{\frac{1}{3}}[/tex]

2 ) Simplify...

3

Hope this helps! :)

The sales tax percent when Ava went shopping for a new pair of pants is 3%.

From the information, Ava went shopping for a new pair of pants the listed price of the pair of pants was $21 but the price with tax came out to $21.63.

The sales tax will be:

= $21.63 - $21

= $0.63

The percentage of sales tax will be:

= Sales tax / Original price × 100

= 0.63 / 21 × 100

= 3%

The sales tax is 3%.

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The exact interval notation of given equation is  x > - 8.

What is interval notation for compound inequalities?

The numbers that bound a continuous set of real numbers can be used to express the continuous set in interval notation. Intervals have many characteristics in common with ordered pairs when written.

We have,

            3x+1 > 2x-3 > x-11

Apply the rule that if m > v > n, then m > v and v > n to the given compound inequality to find the solution.

∴ m = 3x + 1,

   v = 2x - 3,

   n = x - 11.

Hence, by applying the rule of compound inequality we get,

          3x + 1 > 2x - 3 and 2x - 3 > x - 11

3x + 1 > 2x - 3

3x + 1 - 1 > 2x - 3 -1

3x > 2x - 4

3x - 2x > 2x - 4 - 2x

x > - 4

2x - 3 > x - 11

2x - 3 + 3 > x - 11 + 3

2x  > x - 8

2x  - x > x - 8 - x

x > - 8

The two computations above indicate that the acquired intervals, x > - 4 and x > - 8, will be combined to yield the answer to the given inequality.

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Answer: function

Step-by-step explanation: for ever input the is one output, even though -10,5 showed up multiple times it had the same input and output. So it’s a function …if I remember correctly.

If the 5 is negative on just one, it’s not a function.

so sorry if it’s wrong, best of luck :)

The equation has a unit rate of 3/4

From the question, we have the following parameters

16y = 12x

The above equation is a linear equation

In fact, it is a linear equation that shows a proportional relationship

Note that all linear equations have a constant rate of change

So, the proportional relationship is given as

16y = 12x

Make y the subject

y = 12/16x

This gives

y = 3/4x

As a general rule

A proportional relationship is represented as y = kx

Where k represents the unit rate

So, we have the following representation

y = 3/4x and y = kx

By comparing the equations

k = 3/4

Hence, the unit rate of the equation is 3/4

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Possible question

Find the unit rate.

16y = 12x

The domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is

[ -4 , ∞ ).

Given, a graph of the function,

{ x | x ≥ -4 }

{ y l y ≥ 2 }

Now, we have to find the domain of the function

As, we know that the domain of a function is the set of real values on which the given function is defined.

so, it is clear that the function is defined for all the real values greater than or equal to -4.

Hence, the domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is

[ -4 , ∞ ).

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Answer: f(x) = x³ + 7x² -8x

Step-by-step explanation:

Zeros means that at that x-value, the graph crosses the x-axis and y = 0. Hence, to meet those criteria, the function would be:

[tex]f(x)=x(x+8)(x-1)[/tex]

At those three x-values, the corresponding y-values would equal 0.

Multiply them out to find the polynomial function:

[tex]f(x)=x(x+8)(x-1)=x(x^{2} -x+8x-8)=x^{3} +7x^{2} -8x[/tex]

Step-by-step explanation:

your intercepts are:

y-intercept:

y = - 7/2x + 2

y = - 7/20 + 2

y = 2

( 0 , 2 )

x-intercept:

y = - 7/2x + 2

0 = - 7/2x + 2

-2 = - 7/2x

-7/2x = -2

2x = 2/7

x = 4/7

( 4/7 , 0 )

Using the distance formula, the distance between A to point Z to the nearest tenth is 4.1 .

The 2D distance formula gives the shortest distance between two points in a two-dimensional plane.

Using the distance formula,

d = √(y₂ - y₁) + (x₂ - x₁)²

let's use the distance formula, to find the distance between point A and XZ.

Using the coordinates,

A(3, 3) and XZ(4, -1)

Therefore,

x₁ = 3

x₂ = 4

y₁ = 3

y₂ = -1

d = √(4 - 3)² + (-1 - 3)²

d = √1² + (-4)²

d = √1 + 16

d = √17

Therefore,

d = 4.12310562562

d = 4.1

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Answer:

a)  Line r

b)  Line q

c)  Line s

d)  Line q

e)  Line p

Step-by-step explanation:

A transversal is a line that crosses two other lines in the same plane at two distinct points.

a) The transversal connecting ∠1 and ∠5 is line r.

b) The transversal connecting ∠7 and ∠14 is line q.

c) The transversal connecting ∠8 and ∠11 is line s.

d) The transversal connecting ∠6 and ∠15 is line q.

e) The transversal connecting ∠3 and ∠9 is line p.