[tex]\begin{gathered} The\text{ following function } \\ N=N_oe^{\lambda t}\text{, N=}amount\text{ of radium-226} \\ t=\text{ time} \\ \lambda\text{= constant} \\ \text{Radium -226} \\ \text{half life = 1640 years} \\ U\sin g\text{ last information, we can find the value of }\lambda \\ \frac{N}{N_o}=e^{\lambda t} \\ \\ \ln (\frac{N}{N_o})=\ln (e^{\lambda t}) \\ \\ \ln (\frac{N}{N_o})=\lambda t \\ \frac{N}{N_o}=\frac{1}{2}=0.5 \\ \ln (0.5)=\lambda t \\ \lambda=\frac{\ln(0.5)}{t} \\ \\ \lambda=\frac{\ln(0.5)}{1640} \\ \\ \lambda=-0.000423 \\ Now,\text{ on the case of 200mg of radium -226} \\ N=200e^{-0.000423t} \\ t=4000\text{ years} \\ \\ N=200e^{-0.000423\cdot(4000)} \\ N=36.83 \\ \text{After 4000 years the sample will contain }36.83mg\text{ of radium -226} \end{gathered}[/tex]

please help me work through this if you can, thank you!

Given:

[tex]C=\frac{x^4}{4}-\frac{4}{3}x^3-\frac{35}{2}x^2+150x[/tex]Let's solve for the following:

• (a). Find C'(x).

Here, we are to find the derivative of C(x).

Apply the sum rule:

[tex]\begin{gathered} C^{\prime}=\frac{d}{dx}(\frac{x^4}{4})+\frac{d}{dx}(-\frac{4}{3}x^3)+\frac{d}{dx}(-\frac{35}{2}x^2)+\frac{d}{dx}(150x) \\ \\ C^{\prime}=x^3-4x^2-35x+150 \end{gathered}[/tex]• (b). The critical numbers of C(x).

The critical numbers will be the points where the graph changes direction.

Using the derivative, let's solve for x.

[tex]x^3-4x^2-35x+150=0[/tex]Factor the left side using the rational root test:

[tex](x-5)(x^2+x-30)=0[/tex]Now factor using the AC method:

[tex]\begin{gathered} (x-5)((x-5)(x+6))=0 \\ \\ (x-5)(x-5)(x+6)=0 \end{gathered}[/tex]Equate each factor to zero and solve for x:

[tex]\begin{gathered} x-5=0 \\ \text{ Add 5 to both sides:} \\ x-5+5=0+5 \\ x=5 \\ \\ \\ x-5=0 \\ x-5+5=0+5 \\ x=5 \\ \\ \\ x+6=0 \\ x+6-6=0-6 \\ x=-6 \end{gathered}[/tex]**Therefore, the critical numbers of C(x) are:**

**x = -6, 5**

• (C). Increasing interval.

Use the critical points to find the increasing and decreasing intervals.

Using interval notation, the increasing interval is:

[tex](-6,5)\cup(5,\infty)[/tex]• D. Decreasing interval:

Using interval notation, the decreasing interval is:

[tex](-\infty,-6)[/tex]**ANSWER:**

**(a). C'(x) = x³ - 4x² - 35x + 150**

**(b). x = -6, 5**

**(c). Increasing: (-6, 5) U (5,∞)**

** Decreasing: (-∞, -6)**

Solve the problems fill in the blanks show you work I NEED HELP!

The fill in the blanks (that uses** division**) can be filled as -1.2([tex](\frac{2}{5} )[/tex] ÷ -2, -0.48÷-2 and 0.24.

According to the question,

We have the following information:

-1.2 [tex](\frac{2}{5} )[/tex] ÷ -2

Now, we have to solve this **problem** and fill the blanks at the same time.

Now, the first step would be to write the question.

So, for the first blank we have:

-1.2 [tex](\frac{2}{5} )[/tex] ÷ -2

Now, the required **number** is (2/5).

Now, the next step is to multiply -1.2 with 2/5:

-2.4/5

-0.48

So, the second blank can be filled by -0.48.

Now, the third** step** would be to divide -0.48 by -2:

-0.48÷-2

0.24

So, the next blank can be filled by 0.24.

Hence, the numbers in the blanks are [tex]\frac{2}{5}[/tex], -0.48 and 0.24 respectively.

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what is the probability that a card drawn randomly from a standard deck of 52 cards is a nine? express your answer as a fraction

Given the question, we need to know that a standard deck contains 52 cards out of which there are 4 cards showing nine.

[tex]\text{Probability = number of possible outcomes/number of total outcomes}[/tex]To get the probability of randomly drawing a nine will be:

[tex]\begin{gathered} \frac{n\text{ (nine)}}{n\text{ (total cards)}} \\ pr(nine)=\frac{4}{52}=\frac{1}{13} \end{gathered}[/tex]Hence, **the probability of drawing nine is 1/13**

111 pointFind cot(1.570/7.3). Round to 3 decimal places.Type your answer...

[tex]\cot (\frac{1.5\pi}{7.3})=\tan ^{-1}(\frac{1.5\pi}{7.3})\text{ = }32.844[/tex]

how do I put 5/7-i in standard form?

To put

[tex]\frac{5}{7-i}[/tex]In the standard form, we must multiply the numerator and denominator by the complex conjugate of (7-i), it means

[tex]\frac{5}{7-i}\cdot\frac{7+i}{7+i}[/tex]And now we solve it, therefore

[tex]\frac{5}{7-i}\cdot\frac{7+i}{7+i}=\frac{5(7+i)}{7^2-i^2}[/tex]Remember that

[tex]i^2=-1[/tex]Then

[tex]\begin{gathered} \frac{5(7+i)}{7^2-i^2}=\frac{5(7+i)}{49+1} \\ \\ \frac{5(7+i)}{49+1}=\frac{5(7+i)}{50} \end{gathered}[/tex]Now we can simplify it

[tex]\frac{5(7+i)}{50}=\frac{7+i}{10}[/tex]**And we have it in the standard form**

How many gallons of gas will Justin need to drive 120 miles? Use a table or proportionas

As you can see, both variables are directly proportional.

Thus, the given line should have the form:

[tex]y=kx[/tex]Where k is the proportionality constant.

To find the number of gallons of gas needed to drive 120 miles, we should find k first.

Notice that we need 1 gallon of gas to drive 25 miles and 2 gallons of gas to drive 50 miles. The rate of change (k) between these variables is:

[tex]\frac{50-25}{2-1}=25\frac{gallons}{\text{mile}}[/tex]This means that it takes 25 gallons to drive each mile.

Now, we could replace:

[tex]\begin{gathered} 120=25x \\ x=\frac{120}{25}=4.8\text{gallons} \end{gathered}[/tex]Therefore, Justin needs 4.8 gallons to drive 120 miles.

I don’t know how to find the median and the mode HELPP

First, let's write the data set in crescent order:

[tex]35,35,44,50,50,50,56,60,65,70,86,90,110[/tex]The mean is given by the sum of all values divided by the number of values:

[tex]\begin{gathered} mean=\frac{35+35+44+50+50+50+56+60+65+70+86+90+110}{13}\\ \\ mean=\frac{801}{13}\\ \\ mean=61.62 \end{gathered}[/tex]The median is given by the central value of the set in crescent order. Since this set has 13 values, the median is the 7th value:

[tex]median=56[/tex]The mode is the value that repeats the most. Looking at the set, the value that repeats the most (three times) is 50, so **the mode is 50.**

What is the answer to this question?

The **probability** of drawing a blue socks and white socks is 1/12.

Given that, a drawer contains 10 red socks, 6 white socks and 8 blue socks.

What is probability of an event?**Probability** is a type of **ratio** where we **compare** how many times an **outcome** can occur compared to all **possible** outcomes.

As we know the probability of an event = Number of **favorable** **outcomes**/**Total** number of outcomes

Total number of outcomes = 10+6+8=24

Probability of getting blue socks = 8/24 = 1/3

**Probability** of getting white socks = 6/24 =1/4

Now, probability of an **event** = 1/3 × 1/4 = 1/12

Therefore, the **probability** of drawing a blue socks and white socks is 1/12.

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11 to the power of 2

When a number is raised to a power, it simply refers to the number of times that number is multiplied by itself.

Applying that logic here, 11 to the power of 2 is simply

[tex]11\times11[/tex]and the answer is

[tex]=121[/tex]**Therefore, 11 to the power of 2 is ****121.**

If translation T maps point A(-3, 1) onto point A'(5, 5),

which is the translation T?

**Answer: **[tex](x,y) \to (x+8, y+4)[/tex]

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

Explanation:

The x coordinate of A and A' are -3 and 5 respectively.

This is an increase of +8 since -3+8 = 5

Therefore, we shift the preimage 8 units to the right to get the image point.

That explains how x turns into x+8.

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

The y coordinates of A and A' are 1 and 5 respectively.

This is an increase of +4, so we shift the preimage up 4 units.

This signals that y turns into y+4

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

To recap we found that,

x becomes x+8y becomes y+4Therefore we get to the answer [tex](x,y) \to (x+8, y+4)[/tex]

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

To check the answer, let's apply this translation to point A(-3,1) and we should get (5,5) as the result.

[tex](\text{x},\text{y})\to(\text{x}+8,\text{y}+4)\\\\(-3,1)\to(-3+8,1+4)\\\\(-3,1)\to(5,5)\\\\[/tex]

This confirms the answer is correct.

The investment you make into a start-up company is also known as ____.

The investment you make into a start-up company is also known as **Venture capital**.

**Answer: Venture capital**

Alexander Litvinenko was poisoned with 10 micrograms of the radioactive substance Polonium-210. Since radioactive decay follows a compounded continuously model, we can determine the amount of substance left in Alexander Litvinenko's body at any given time. If Polonium-210 has a decay rate of .502%, then determine the amount of Polonium-210 left in his body after 190 days. Provide 3 decimal places and units in your answer.

We have

Initial mass of Polonium-210

[tex]N_0=10\text{ micrograms}[/tex]Decay rate, r = 0.502% = 0.00502

Time, t = 190 days

Then, We know that left amount is given by

[tex]N=N_0e^{-rt}[/tex]Solving

[tex]N=10e^{-0.00502(190)}=10e^{-0.9538}=3.853[/tex]**Answer:** 3.853 micrograms

Find the volume of the following triangular prism. *15 points5 in6,1 in13 in7 in215.5 cubic inches220.5 cubic inches225.5 cubic inches227.5 cubic inches

Given:

The length of the base of the triangular base of the prism, b=7 in.

The altitude of the triangular base, l=5 in.

The height of the prism, h=13 in.

Now, the volume of the prism can be calculated as,

[tex]\begin{gathered} V=\text{ Base Area}\times height \\ =\frac{1}{2}bl\times h \\ =\frac{1}{2}\times7\times5\times13 \\ =227.5\text{ cu. in} \end{gathered}[/tex]Therefore, the volume of the triangular prism is **227.5 cu. in**.

7. 200 is the ? th term of 24, 35, 46, 57, ..

200 is the 17th term of the **progression**

**What is arithmetic progression?**

Arithmetic **progression **or arithmetic sequence (AP) is a numerical series in which the difference between subsequent terms is constant. The sequence 5, 7, 9, 11, 13, 15,..., for example, is an arithmetic **progression **with a common difference of 2. A finite arithmetic **progression**, or simply an arithmetic **progression**, is a finite section of an arithmetic **progression**. An arithmetic series is the sum of a finite arithmetic **progression**. An arithmetic series is the sum of the elements of a finite arithmetic **progression**. A closed expression determines the product of the members of a finite arithmetic **progression **with a beginning element a1, common differences d, and n elements in total.

This can be solved using arithmetic** progression**

Aₙ = a + (n - 1)d

where, Aₙ = 24, d = 35 - 24 = 11

so, n = (Aₙ - a)/d + 1 = (200-24)/11 + 1 = 17

Hence, 200 is the 17th term of the **progression**

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What is 3+2? and add 7 then subtract 4

the answer is 8

add 3+2, that gives u 5, add 7+5 which is 12, do 12-4 its equal to 8

Is -36/6 a integer number

**Answer:**

Yes

**Step-by-step explanation:**

-36/6 = -6

Integers are the whole numbers (0.1.2.3.4.5....) and their opposites ...-4,-3,-2,-1

I need help with this questionif a certain piece of music is written in 3/4 time, how many eighth notes are required per measure of music?

**Solution**

For this case we need to take in count that in music 3/4 time can be grouped into 3 groups of 2 eight notes and then the answer for this case would be:

**6/8**

. Which units can be used to measure distance?

A. seconds

B. cubic centimeters

C. meters

D. liters

**Answer:**

c

**Step-by-step explanation:**

The metre or meter, symbol m, is the primary unit of length in the International System of Units, though its prefixed forms are also used relatively frequently.

Why did the turkey volenteer to be the drummer in the popular bird band? only 2.1 section

**Answer:**

I eat birds tasty

A boy gets #2.00 per week as pocket money. His sister gets only #1.60 per week

Find the ratio of the boy's allowance to his sister's. If his sister gets 20 k more per week,what will be the new ratio?

The R**atio **of the boy's allowance to his sister's is 5 : 4.

The new R**atio **is 1 : 10000.

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;

Give the standard form of the given equation below. If it is a quadratic equation, then give the a, b, and c coefficients. 3x2-2x-5x(x-7)=(x-2)(x+4)+1

**Answer:**

**Step-by-step explanation:**

The quadratic equation in standard form is represented by:

[tex]ax^2+bx+c=0[/tex]For the following equation;

[tex]\begin{gathered} 3x^2-2x-5x(x-7)=(x-2)(x+4)+1 \\ 3x^2-2x-5x^2+35x=x^2+4x-2x-8+1 \\ -2x^2-2x+35x=x^2+2x-7 \end{gathered}[/tex]Compute all the terms on the left side, equalizing 0:

[tex]\begin{gathered} -3x^2+31x+7=0 \\ \text{Then, the a,b and c coefficeints would be:} \\ a=-3 \\ b=31 \\ c=7 \end{gathered}[/tex]Need some guidance and reassurance because I am not getting full points on my homework. Thanks.

Given the parent function:

[tex]y=x^3[/tex]If y is **shifted right by 7 units**, we obtain:

If y is **compressed horizontally by a factor of 7**, we have:

If y is **stretched** **vertically by the factor 7, **we obtain:

A **shift downwards by 7 units** gives:

Which expression is equal to log(xy / z) ?

The expression that can be said to as equal to log(xy / z) is the expression logx + logy - logz and this can be found out through the **logarithmic identities.**

The “basic”** logarithmic function **can be seen as the the function, y=logbx, where x, b>0 and b≠1.

In the kind of complex analysis that we come across three types of **logarithms **namely ln, log and Log are used.

In mathematics log tends to always mean the natural log.

Log is often seen as the main and principal branch that comes under the **complex logarithm.**

Logarithmic functions are said as important only largely because of their relationship to the kind of exponential functions. **Logarithms **can also be used to solve any kind of exponential equations and thus help to explore the properties of** exponential functions.**

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A line passes through the point (4,-7) and has a slope of -6 write an equation in slope intercept form for this line

y=-6x+b

(4,-7) substitution to equation

-7=-24+b

b=17

(4,-7) substitution to equation

-7=-24+b

b=17

I don’t know the rest of the definition of my vocabulary

**Step 1**

Given;

**Step 2**

Two angles are called** supplementary** when their measures add up to 180 degrees.

Two angles are called** complementary **when their measures add to 90 degrees.

The two angles are said to be **adjacent angles** when they share the common vertex and side. The endpoint of the rays, forming the sides of an angle, is called the vertex of an angle. Adjacent angles can be complementary angles or supplementary angles when they share the common vertex and side.

**Answer;**

**Two angles are called supplementary when their measures add up to 180 degrees.**

**Two angles are called complementary when their measures add to 90 degrees.**

**The two angles are said to be adjacent angles when they share the common vertex and side.**

(a) Describe in words a sequence of transformations that maps AABC to

AA"B"C".

(b) Write an ordered-pair rule for each transformation in the sequence.

**Answer:**

**Step-by-step explanation:**

You want a **sequence of transformations** that maps **∆ABC to ∆A'B'C'** where the vertices are **A(-5, 3), B(-2, 3), C(-4, 1), A'(8, -2), B'(5, -2), and C'(7, 0)**.

Segment AB is a 3-unit line segment directed to the right. Segment A'B' is a 3-unit line segment directed to the left. This means there is no dilation involved in the transformation. At least, the figure has been **reflected left-to-right**.

Point C is below segment AB, while point C' is above segment A'B'. This means the figure has also been **reflected top-to-bottom**.

Together these reflections can be accomplished by either of **reflection across the origin**, or rotation 180° about the origin.

The reflected figure would leave A' at (5, -3). Its location at (8, -2) means the figure has also been translated to the right and up.

Translation to the **right** has been by 8 -5 = 3 units.

Translation **up** has been by -2 -(-3) = 1 unit.

Triangle ABC can be transformed to triangle A'B'C' by ...

The corresponding ordered-pair rules for these transformations are ...

__

*Additional comment*

The single transformation that will accomplish the mapping is ...

(x, y) ⇒ (3 -x, 1 -y) . . . . . reflection across the point (3/2, 1/2)

Find the solution.7 · x = 8412139177

We need to find the value for x using the inverse operation:

Then:

[tex]\begin{gathered} 7-x=84 \\ \end{gathered}[/tex]Let us solve for x:

The x is subtracting, so it will add up to the other side:

[tex]7=84+x[/tex]Now, the 84 is positive, so it will be negative on the other side:

[tex]\begin{gathered} 7-84=x \\ x=-77 \end{gathered}[/tex]Hence, **the solution for x is -77**.

If the proportion of the total disposable income spent on consumer goods and services is 91.6 percent and if consumers spend 82.0

percent of each additional dollar, what is

Instructions: Round your responses to three decimal places.

a. the APC?

b. the APS?

c. the MPC?

d. the MPS?

a) The **Average Propensity to Consume (APC)**, rounded to three **decimal places**, is 0.916.

b) The **Average Propensity to Save (APS)**, rounded to three **decimal places**, is 0.084.

c) The **Marginal Propensity to Consume (MPC)**, rounded to three **decimal places**, is 0.820.

d) The **Marginal Propensity to Save (MPS)**, rounded to three **decimal places**, is 0.180.

The **average propensity to consume** is the **ratio** of **consumption expenditures** to the total consumers' **disposable income**.

The **average propensity to save** is the inverse of the **APC**. It shows the rate of or total disposable income reserved instead of being consumed.

The **marginal propensity to consume** is the ratio of consumption based on additional income.

The **marginal propensity to save** is the inverse of the **MPC**, showing the percentage of the **additional income** saved instead of being consumed.

APC = 91.6%

= 0.916

APS = 1 - APC

= 0.084 (1 - 0.916)

MPC = 82%

= 0.820

MPS = 1 - MPC

= 0.180 (1 - 0.820)

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which anwser show the best approximation of [tex] \sqrt[]{53} [/tex]1. 7.72. 8.13. 7.34. 7.1

**The Solution:**

The given number is

[tex]\sqrt[]{53}[/tex]Finding the best approximation for the above number, we have

[tex]\sqrt[]{53}=7.28\approx7.3[/tex]Therefore, **the best approximation of the given number is 7.3 (option 3)**

Which of the following hypotheses is not a valid null hypothesis?a.H0: µ = 0b.H0: µ ≥ 0c.H0: µ ≤ 0d.H0: µ < 0

**Given:**

a. H0: µ = 0

b. H0: µ ≥ 0

c. H0: µ ≤ 0

d. H0: µ

**Required:**

To choose the hypotheses that is not valid.

**Explanation:**

The null hypothesis: It is a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

Here the option d is not valid.

**Final Answer:**

The option d is not valid.

H0: µ < 0

en performing a multiproduct cvp analysis, the weighted-average contribution margin per unit is computed using the mix, and the mix is used to compute the weighted-average contribution margin ratio. (enter only one word per blank.)
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what is the bond quote for a $1,000 face value bond with an 8% coupon rate (paid semiannually) and a required rate of return of 7.5%, if the bond is 6.48574, 8.47148, 10.519, and 14.87875 years from maturity?
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