A sample of 200g of an isotope decays to another isotope according to the function A(t)=200 e⁰.⁰⁵⁴⁺
(a) How much of the initial sample will be left in the sample after 25 years?
(b) How long will it take the initial sample to decay to half of its original amount?

The values of x in the given equation, x² - 5x = -4, are x = 4 and x = 1

From the question, we are to solve the given equation by factoring

The given equation is

x² - 5x = -4

To solve the equation by factoring, transfer the constant to the other side of the equation

x² - 5x + 4 = 0

Factorize the left hand side expression

x² -x -4x + 4 = 0

x(x - 1) -4(x -1) = 0

(x - 4)(x - 1) = 0

x - 4 = 0 OR x - 1 = 0

x = 4 OR x = 1

Hence, the values of x are 4 and 1

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Answer:

6(4)-4=3(4)+8

24-4=12+8

20=20

Step-by-step explanation:

To prove x=4, substitute four into the equation

6(4)-4=3(4)+8

24-4=12+8

20=20

Therefore x=4 is proved

Answer:

(Answer in Image)

Step-by-step explanation:

6x-4=3x+8

6x-4-3x=3x+8-3x                    subtract -3x on both sides

3x-4=8                                    

3x-4+4=8+4                               add four on both sides

3x=12                                      simplify

3/3x=12/3                                       divide both sides by 3

x=4

The correct inequality which has no solution will be;

⇒ x ≤ –2 and 2x ≥ 6

⇒ x ≤ –1 and 5x ≤ 5

What is Inequality?

A relation by which we can compare two or more mathematical expression is called an inequality.

Given that;

There are some inequality is shown.

Now,

Find the inequality which has no solution as;

a) x ≤ –2 and 2x ≥ 6

⇒ x ≤ - 2

⇒ 2x ≥ 6 = x ≥ 6/2 = x ≥ 3

We get;

⇒ x ≤ - 2 and x ≥ 3

Which is not possible.

Hence, it has no solution.

b) x ≤ –1 and 5x ≤ 5

⇒ x ≤ - 1

⇒ 5x ≤ 5 = x ≤ 1

We get;

⇒ x ≤ - 1 and x ≤ 1

Which is not possible.

Hence, it has no solution.

c) x ≤ –1 and 3x ≥ –3

⇒ x ≤ - 1

⇒ 3x ≥ - 3 = x ≥ - 1

We get;

⇒ x ≤ - 1 and x ≥ - 1 ⇒ x = - 1

Thus, It has solution.

d)  x ≤ –2 and 4x ≤ –8

⇒  x ≤ –2 and

⇒ 4x ≤ - 8 = x ≤ - 2

Thus, It has solution.

Therefore, The correct inequality which has no solution will be;

⇒ x ≤ –2 and 2x ≥ 6

⇒ x ≤ –1 and 5x ≤ 5

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Answer:

x = -2

Step-by-step explanation:

The first thing we can notice about this figure is that the two small outer triangles are congruent because of the given side congruence markings as well as the fact that the large triangle ABC is isosceles, thus angle A is congruent to angle C.

Using the above information, we can construct the equation:

3x + 2 = 2x

and solve it for x.

x = -2

The equation we constructed simply equates the corresponding sides of the two small outer triangles, as they are congruent by the CPCTC theorem.

Answer:

4/3

Step-by-step explanation:

you use the slope formula y^1 - y^2 over x^1 over x^2, substitute the said values and are left with the slope of 4/3 in its improper form.

Hope this helps!!! :3

The first function is an odd and the second one is neither even nor odd. The correct options are (1) and (2).

A function y = f(x) is a one to one relationship between two sets X and Y where the set X is called the domain and Y the range of function f(x) ans x ∈ X and y ∈ Y.

The given problem can be solved as follows,

(2) The graph of the given function shows the following,

The value of the function is positive for positive x and negative for negative x.

It can be mathematically written as,

f(-x) = -f(x).

Which shows the given graph is of an odd function.

(3) The given function is f(x) = 4x³ + 2x² - x⁵.

In order to check the type of the function, find its value at the same value of x with different signs as,

f(1) = 4 × 1³ + 2 × 1² - 1⁵ = 5

And, f(-1) = 4 × -1³ + 2 × -1² - -1⁵ = -1

Since, f(1) ≠ f(-1), the function is not even.

And f(-1)  ≠ -f(1), the function is not odd.

Thus, the given function is neither even nor odd.

Hence, the first and the second functions of the given problem are odd and neither even nor odd respectively.

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By solving the system of linear equations by elimination we get the value of x= 2 and y = 1.

What is the system of linear equations?

A grouping of one or more linear equation containing the same variables is known as a system of linear equations.

Here, we have

Given:  8x - 5y = 11....(1), 4x - 3y = 5...(2)

We solve equation(2) and find the value of x

4x = 5+ 3y

x = 5/4 + 3y/4

Now, we put the value of x in equation (1) and we get

8(5/4 + 3y/4) - 5y = 11

5×2 + 2×3y - 5y = 11

10 + y = 11

y = 1

Now, we put the value of y in equation(2) and we get

4x = 5+ 3(1)

4x = 8

x = 2

Hence, by solving the system of linear equations by elimination we get the value of x= 2 and y = 1.

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The perimeter of the polygon is 57 meters.

A polygon is an enclosed two-dimensional shape with three or more straight sides that are connected together. It can be regular or irregular and can be made up of triangles, squares, rectangles, and other shapes. Polygons are classified by the number of sides they have, such as triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), and hexagons (6 sides). Polygons can also be classified by their internal angles, such as acute (less than 90 degrees), right (90 degrees), obtuse (greater than 90 degrees), and reflex (greater than 180 degrees). Polygons are used in many mathematical problems and can be used to represent a variety of real-world objects, such as buildings, mountains, and stars.

The perimeter of the polygon is 57 meters.

The perimeter is calculated by adding up the lengths of each side. 7m + 16m + 9cm + 15m = 57m.

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Answer:

11.9988

Step-by-step explanation:

Use a calculator.

Change the equations to inequalities  to modify the graph and the solutions are (-1.871, 1.5) and (1.871, 1.5)

The graph is the given parameter

From the graph, we have the following equations

f(x) = x² - 2

g(x) = -x² + 5

To modify the graph, we simply change the equations to inequalities

After the modifications in (a), we have:

f(x) >  x² - 2

g(x) ≥  -x² + 5

From the graph of the inequalities, the curves of the inequalities intersect at (-1.871, 1.5) and (1.871, 1.5)

Hence, the solutions are (-1.871, 1.5) and (1.871, 1.5)

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The value of y is 15.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

87 + 87 y = 1218 + 174

Now, Solving

87+ 87y= 1392

87y= 1392- 87

87y= 1305

y= 1305/ 87

y= 15

Hence, the value of y is 15.

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2x+5y=18 equ(1)

x-3y=-2 equ(2)

from equ (2) make x the subject of the formula

x=-2+3y

substitute x=2+3y in equ (1)

2+(-2+3y)+5y=18

-4+6y+5y=18

6y+5y=18+4

11y/11=22/11

y=2

substitute y=2 in equ(2)

x+3(2)=-2

x-6=-2

x=-2+6

x=4

x=4 and y=2

Answer:

Line 2

Step-by-step explanation:

This would represent a negative slope.  A negative slope goes down if you start on the left side and go down.

Part a: P(A) = 1/2; P(B) =  2/3; P(C) = 1/12; P(A ∩ B ∩ C) = 1/36.

Part b: Part b: A, B, and C independent events.

Rolling a dice twice has 36 possible results, hence the following is the sample space:

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

For the stated question-

Part a:  P(A), P(B),P(C), and P(A ∩ B ∩ C).

P(A) = 18/36 = 1/2

P(B) = 24/36 = 2/3

P(C) = 3/36 = 1/12

P(A ∩ B ∩ C) = 1/36

Part b: A, B, and C independent or not.

For the events to be independent,

P(A ∩ B ∩ C) = P(A).P(B).P(C)

P(A).P(B).P(C) = 1/2*2/3*1/12 = 1/36

As, P(A ∩ B ∩ C) = P(A).P(B).P(C).

Thus, the events are independent.

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Answer:

(0,-5) , (3, -6) , (6, -7) ,(9, -8)

Step-by-step explanation:

I apologize for being unable to graph, but here are some of the points.

(0,-5) <-- The "y intercept"

slope is y2-y1/x2-x1, so "-1" is added to the "y" of the coordinate, and 3 is added to the "x"

(3,-6)

(6,-7)

(9,-8)

Hope this helps :)

Answer:

it is 3/5

Step-by-step explanation:

cos is the adjuacent devided by hypotenuse so it is 27/45 and that is 3/5

Answer:

Check out the image shown above/below

Yes, it is a proportional relationship. The relationship is distance (d) is directly proportional to time(t). The equation of the relationship is d= 65t

Direct Variation is said to be the relationship between two variables in which one is a constant multiple of the other. For example, when one variable changes the other, then they are said to be in proportion. If d( distance) is directly proportional to t( time) the equation is of the form d= kt (where k is a constant).

When d =65miles and t= 1hour, k =d/t

k= 65/1= 65

for other values of d and t, k = 65

Therefore the relationship between the distance and time is a proportional relationship. And the equation of the relationship is d= 65t

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The system of equations that Tristan needs to solve is:

x*0.05 + y*0.10 = 1.40

x = 2*y

Let's define the variables:

x = number of nickels.

y = number of dimes.

Knowing that the net is $1.40 and that he has twice as many nickels as dimes, we can write:

x*0.05 + y*0.10 = 1.40

x = 2*y

That is our system of equations that Tristan can use to find the numbers of nickels and dimes.

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We can substitute the number in the inequality to find if the inequality holds true. If the inequality holds true, it is a valid solution. This is very similar to substituting a number in an equation, in order to check if it holds.

The same holds true for others.

The number of notebook is 36.

An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario.

Let the notebook be n.

Let the t shirt be t.

The equation will be:

n + t = 60 ..... i

7n + 12t = 540 ..... ii

From I n = 60 - t ..... iii

Put equation III into ii

7n + + 12t = 540 .

7(60 - t) + 12t = 540

420 - 7t + 12t = 540

5t = 540 - 420

5t = 120

Divide

t = 120 / 5

t = 24

Notebook will be:

= 60 - 24

= 36

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a) Histogram (figure attached)

b) The expected number of defective is equal to 0.4 syringes.

c) Probability that the batch will be accepted is equal to 0.94275.

d) The standard deviation is 0.6164.

It is a binomial distribution "DISCRETE probability distribution that encapsulates the likelihood that a value will take one of two independent values when a particular set of conditions is met. The assumptions for the binomial distribution are that each trial has a single possible outcome, each trial has an equal chance of success, and each trial is independent of the others or mutually exclusive ".

Let X be the number of syringes being defective as a binomial random variable.

We are given that in the batch, there can be assumed to be 5% defective syringes.

∴ n = 8

Probability of syringes being defective, p = 0.05

Probability of syringes being not defective , q = 1-p = 1-0.05 = 0.95

a) Histogram (figure attached)

b) We know that the formula for Expectation in binomial distribution is;

μ =n×p = 8×0.05

=0.4 syringes

Therefore, the inspector will find the expected number of defective is equal to 0.4 syringes.

c) We have to find binomial probability for that we will use the binomial probability distribution is given by:

P(X=x) = ⁿCₓpˣ (1-p)^n-x, x = 0,1,2,3 ..8

Therefore,

P(x<2) = P(X=0) + P(X=1)

= ⁸C₀(0.05)⁰(1-0.05)⁸ + ⁸C₁(0.05)¹(1-0.05)⁷

=  0.66342 + 8×0.05×0.6983

= 0.66342 + 0.27933

= 0.94275

Therefore, probability that the batch will be accepted is equal to 0.94275.

d) We first find the variance:

σ² = Var(X) = n×p×(1-p)

= 8 × 0.05 ×(1-0.05)

= 0.38

Thus, the standard deviation is

σ = [tex]\sqrt{0.38}[/tex] = 0.6164

Thus,

a) Histogram (figure attached)

b) The expected number of defective is equal to 0.4 syringes.

c) Probability that the batch will be accepted is equal to 0.94275.

d) The standard deviation is 0.6164.

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Step-by-step explanation:

chances of rolling 1= 1/6

chances of rolling 2=1/6

chances of rolling anything else=2/3

Hello,

I hope you and your family are doing well!

(a) The exponential growth model is given by the equation P(t) = P0 * r^t, where P(t) is the population t years after 2010, P0 is the population in 2010 (7420), and r is the annual growth rate (2.3%).

Substituting the given values into the equation gives us:

P(t) = 7420 * 1.023^t

This is the equation that estimates the population t years after 2010.

(b) To estimate the population of the town in 2022, we can substitute t = 12 into the equation:

P(12) = 7420 * 1.023^12

Calculating this gives us:

P(12) = 7420 * 1.023^12 = 7420 * 1.291891 = 9593.45

So the estimated population of the town in 2022 is approximately 9593

Happy to help!

Hello,

I hope you and your family are doing well!

If a ticket to the circus was 10% less on a Tuesday than it was on a Saturday, then it was 90% of the Saturday ticket price. If you paid $27 each to attend Tuesday's show, then the Saturday ticket price was $27 / 0.9 = $30.

Since you bought two tickets, the total cost of attending the circus on Saturday would have been 2 * $30 =60

Therefore, your total cost would have been $60 had you attended the circus on Saturday.

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Happy Holidays and New Year!

The tringles ΔKLM, ΔXYZ, and ΔUVW are similar to the ΔHIJ.

What is the similarity of the triangles?

When it comes to Euclidean geometry, two things are said to be comparable if they have the same shape or the same shape as each other's mirror image. One can be created from the other more precisely by evenly scaling, possibly with the inclusion of further translation, rotation, and reflection.

We have,

In ΔHIJ

∠HIJ = 64°

∠JHI = 96°

and the side are

HI = 2 unit

IJ = 5.5 unit

HJ = 5 unit

So, the third angle will according to the sum of the angles of a triangle:

∠IJH = 20°

the triangles similar to ΔHIJ are:

A) It is similar to ΔHIJ cause,

∠HIJ ≅ ∠KLM

∠IJH ≅ ∠LMK

so, the third also will be equal:

∠JHI ≅ ∠LKM

D) It is similar to ΔHIJ cause,

∠IJH ≅ ∠XZY

HI/YX = 2/4 = 1/2

HJ/YZ = 5/10 = 1/2

C) It is similar to ΔHIJ cause,

HI/UV = 2/4 = 1/2

HJ/UW = 5/10 = 1/2

IJ/VW = 5.5/11 = 1/2

Hence, the tringles ΔKLM, ΔXYZ, and ΔUVW are similar to the ΔHIJ.

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Answer:

Ann must pay $800 in income tax for the first $8,000 of her taxable income because 8000 * .1 = 800

Ann must pay $4,600 in income tax for the amount of her taxable income over $8,000 because (35000 - 8000) * .16 = 4600

Therefore, Ann must pay a total of $800 + $4,600 = $<<800+4600=5400>>5,400 in income tax for last year.

Answer:

$5,120 in taxes, total

Step-by-step explanation:

Ann made $35,000 in taxable income last year. Suppose the income tax rate is 10% for the first $8,000 plus 16% for the amount over $8,000. How much must Ann pay in income tax for last year?

8,000 * 10% = 800 in taxes for first $8,000

remaining: 35,000 - 8,000 = $27,000 to be taxed at 16%

27,000 * 16% = $4,320

$800 + $4,320 = $5,120 in taxes, total

The number of acres owned for which the average intake is 2200 calories per day is 2 acres obtained by the algebraic expression that relates the acres of land and the calories per day.

It is given that, the number of calories consumed daily by a person owing x acres of land is given by the algebraic expression,

C(x) = 270 ln(x+1) + 1900

The average intake of calories consumed per day is given by C(x) = 2200

Here x is the number of acres of land. Then, the required number of acres of land is

C(x) = 270 ln(x+1) + 1900

⇒ 2200 = 270 ln(x+1) + 1900

⇒ 270 ln(x+1) = 2200 - 1900

⇒ 270 ln(x+1) = 300

⇒ ln(x+1) = 300/270 = 10/9

Applying base 'e' on both sides, we get

⇒ (x+1) = [tex]e^\frac{10}{9}[/tex]

⇒ x+1 = 3.0377

⇒ x = 3.0377 - 1 = 2.0377

When rounding the final answer to the nearest tenths, we get x = 2 acres.

When rounding the final answer to the nearest thousand, we get x = 2.038 acres.

Thus, for an average intake of 2200 calories per day, the number of acres of land owned is approximately 2 acres.

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