Opportunity cost and production possibilities
Shen is a talented artist who sells hand-crafted goods on his website. Shen currently crafts and sells both crochet hats and baskets. He spends 8 hours
a day working on crafts. The following table gives different daily output scenarios depending on how much of his time is spent on each good.

The equation used to model the amount of money that Jada would have in the account after x years is [tex]y= 200(1 + {0.03})^{x}[/tex].

Jada would have in her account $218 after 36 months. The solution has been obtained by using the concept of compound interest.

What is compound interest?

When interest is calculated, it is done so using both the principal and the interest that has accumulated over the preceding period. Unlike simple interest, which does not take the principal into account when computing the interest for the following month, compound interest takes the principal into account. Compound interest is commonly represented by the letter C.I. in mathematics.

We know that formula for compound interest is

[tex]A= P(1 + \frac{r}{n} )^{nt}[/tex]

So, here A= y, P = $200 , n=1 and t=x and r = 0.03

Thus, we get

[tex]y= 200(1 + \frac{0.03}{1} )^{x}[/tex]

[tex]y= 200(1 + {0.03})^{x}[/tex]

So, the second equation is the correct one.

The amount of money Jada would have after 36 months means the amount after 3 years.

So, x=3

Putting this in the equation, we get

[tex]y= 200(1 + {0.03})^{3}[/tex]

y = 200 (1.09)

y = $218

Hence, the equation used to model the amount of money that Jada would have in the account after x years is [tex]y= 200(1 + {0.03})^{x}[/tex] and Jada would have in her account $218 after 36 months.

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30 is the fast surface area of the balloon expanding when the radius of the balloon is 4 inches

Now, According to the question:

A spherical balloon is expanding at the rate of 60 pie cubic inches /sec.

The radius of the balloon is 4 inches.

Now, Solving the problem:

By using formula:

The formula for the volume of a sphere is V = 4/3 π r³,

where V = volume and r = radius.

Volume = [tex]\frac{4}{3}\pi r^3[/tex]

We have that:

Differentiate the formula:

[tex]\frac{dV}{dt}=4\pi r^2.\frac{dr}{dt}= 60[/tex]

=> [tex]\frac{dr}{dt}=\frac{60}{4\pi r^2}[/tex]

We know that [tex]A=4\pi r^2,[/tex] and so we also have

[tex]\frac{dA}{dt}=8\pi r.\frac{dr}{dt}[/tex]

Plug in what we found for dr/dt and plug in r=4 to get:

[tex]\frac{dA}{dt}=8\pi r.\frac{60}{4\pi r^2}[/tex]

[tex]\frac{dA}{dt}=\frac{120}{r}=\frac{120}{4}[/tex]

[tex]\frac{dA}{dt}=30[/tex]

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 (√7)^3  is equal to 7^(3/2)

The Power of a Power Property

The Power of a Power Property states that when you have a base raised to an exponent, and that base is also raised to an exponent, you can simplify the expression by multiplying the exponents.

For example, (a^x)^y = a^(xy)

In the case of square root of 7^3, the expression (√7)^3 can be written in the form of base and exponent, where the base is √7 and the exponent is 3.

Now, we can use the Power of a Power Property to simplify the expression, by multiplying the exponents.

So, (√7)^3 = (7^(1/2))^3 = 7^(1/2 * 3) = 7^(3/2)

This means that the square root of 7^3 is equal to 7^(3/2) because the Power of a Power Property allows us to simplify the expression by multiplying the exponents.

It's worth noting that the square root of 7^3 is the same as taking the cube root of 7^3, which is 7^(3/3) = 7^1 = 7.

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At x =6 the function has a positive slope.

A function is a mathematical expression, rule, or law that describes the relationship between an independent variable and a dependent variable. At x = 2 and x = 6, the function y(x) is zero, and it has a value of 1 for all other values of x. the dependent variable. Functions are essential for building physical relationships in both math and science.

A function, which generates one output from a single input, is an illustration of a rule. origin of the image. An illustration of this is Y=X2.

But

At x = 2, the function starts to decline and has a negative slope. negative derivative

while

At x = 6, the function increases with a positive slope, producing a positive derivative.

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Answer:

Step-by-step explanation:

ans = x^2 - 3x + 5

Mathematically, Aaliyah needs to read 8.75 hours (or 8 hours, 45 minutes) to fulfill her promise of reading 50 more pages than Yohance.

The mathematical operations used to determine the hours that Aaliyah needs to read this weekend to meet her target include multiplication, addition, and division.

In the first instance, we must determine the total pages read by Yohance this weekend using multiplication as follows: 300 (7.5 x 40).

Secondly, the additional pages Aaliyah will read are added to Yohance's to show the total pages read by Aaliyah.

Lastly, the total pages are divided by Aaliyah's reading speed or rate to determine the hours she requires.

                                           Aaliyah    Yohance

Average pages per hour       40             40

Additional page                     50

Yohance weekend reading time = 7.5 hours

Total pages read by Yohance this weekend = 300 (7.5 x 40)

The total pages to be read by Aaliyah this weekend = 350 (300 + 50)

Aaliyah's total weekend reading time 8.75 hours (350/40)

Thus, for Aaliyah to read 50 more pages than Yohance, she needs 8.75 hours to fulfill her promise.

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Answer:  In the month of October, the animal shelter saved 92 animals.

Step-by-step explanation: In the month of October, the animal shelter saw an increase of 15% of animals being saved from the previous month. To calculate this increase, we can use the formula:

increase = (original number) x (percent increase)

In this case, the original number is 80 (the number of animals saved in September) and the percent increase is 15%. So:

increase = 80 x 0.15 = 12

This means that in October, the animal shelter saved an additional 12 animals beyond the 80 animals that were saved in September. To find the total number of animals saved in October, we add this increase to the original number:

animals saved in October = 80 + 12 = 92

So, in the month of October, the animal shelter saved 92 animals.

The series for this problem is absolutely convergent, as the limit assumes a value lower than 1.

The infinite series for this problem is defined as follows:

[tex]\sum_{i = 1}^{\infty} \frac{1}{n!}[/tex]

Hence the general term is given as follows:

[tex]a_n = \frac{1}{n!}[/tex]

The limit is given as follows:

[tex]L = \lim_{n \rightarrow \infty} \left|\frac{a_{n + 1}}{a_n}\right|[/tex]

The (n + 1)th term is given as follows:

[tex]a_{n + 1} = \frac{1}{(n + 1)!}[/tex]

The factorial can be simplified as follows:

(n + 1)! = (n + 1) x n!.

Hence the limit will be calculated of:

1/[(n + 1) x n!] x n! = 1/(n + 1).

The result of the limit is given as follows:

[tex]L = \lim_{n \rightarrow \infty} \frac{1}{n + 1} = 0[/tex]

As the limit assumes a value of zero, the series is absolutely convergent.

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Answer:

20.80 kg

Step-by-step explanation:

45 lb + 12 oz = 45 lb + 12/16 lb = 45.75 lb

45.75 lb × (1 kg)/(2.2 lb) = 20.80 kg

Answer: Fall Out Boy's service is cheaper than Five Seconds of Summer's service.

Step-by-step explanation: The cost of Five Seconds of Summer's tutoring service can be represented by the equation:

C = 5 + 5h

where C is the total cost and h is the number of hours.

The cost of Fall Out Boy's tutoring service can be represented by the equation:

C = 9 + 3h

To compare the cost of the two services, we can use the substitution method. We can set the number of hours (h) equal to a variable and substitute that variable into the equation for both services.

For example, if we set h = 4 (4 hours of tutoring), we get:

C(Five Seconds of Summer) = 5 + 5(4) = 25

C(Fall Out Boy) = 9 + 3(4) = 21

The cost of Fall Out Boy's service is $21 and the cost of Five Seconds of Summer's service is $25. Therefore, Fall Out Boy's service is cheaper for 4 hours of tutoring.

We can also graph the cost of the two services on a coordinate plane. The x-axis represents the number of hours and the y-axis represents the cost. The equation for Five Seconds of Summer's service would be y = 5 + 5x and the equation for Fall Out Boy's service would be y = 9 + 3x

Service | Equation

Five Seconds of Summer y = 5 + 5x

Fall Out Boy |                        y = 9 + 3x

Answer:

2) Acute triangle, isosceles

3) Both are 65 degrees

4) (diagram)

5) a) both 25 degrees

b) midpoint

Step-by-step explanation:

2) If you're struggling search up "draw a triangle" and you'll find a website that will draw it for you. It will help you get an idea of it looks like and its nature.

3) We know that sides AB and AC are 5cm. So this triangle is an isosceles which makes the 2 remaining angles equal (if you draw a diagram it will be clear)

So:

180-50/2 = 65

ABC and ACB = 65 degrees each

4) Construct the line AH down the middle like in the diagram I drew.

5) a) Since AH is a straight line, we know that the angle AHB will be 90 degrees. After question 3 we also know that angle ABH is 65 degrees.

BAH = 180-90-65

BAH = 25 degrees

CAH = 50 - 25

CAH = 25 degrees

b) AH is the midpoint for BAC because the 2 segments formed are both 25 degrees.

The result of the calculation is 680.

The number 3.115 has four significant figures, the number 0.2281 has four significant figures, the number 712.5 has three significant figures and the number 45 has two significant figures. This means that when performing calculations with these numbers, the answer must be rounded to the least number of significant figures, which is two.

The calculation is 3.115 + 0.2281 + 712.5 - 45. Using the rules of significant figures, we can calculate this as 3.1 + 0.2 + 713 - 45.

The result of the calculation is 675.3 and this must be rounded to the least number of significant figures, which is two. Therefore, the result of the calculation is 680.

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Las pesas de menor masa son listadas como sigue:

En esta pregunta tenemos seis casos de tres pesas, entre las cuales debemos determinar cuál tiene la menor masa. Esto implica el uso de la conversión entre unidades de masa a una única unica, tal que la pesa de menor capacidad será la de menor cantidad. El procedimiento se resume a continuación:

Caso 1

32 hectogramos - 3200 gramos

4 kilogramos - 4000 gramos

500 decagramos - 5000 gramos

La pesa de 32 hectogramos reporta la menor masa.

Caso 2

35 gramos

4500 miligramos - 4.5 gramos

80 decigramos - 8 gramos

La pesa de 80 decigramos reporta la menor masa.

Caso 3

9500 decigramos - 950 gramos

6 kilogramos - 6000 gramos

70 hectogramos - 7000 gramos

La pesa de 9500 decigramos reporta la menor masa.

Caso 4

365 decigramos - 3650 gramos

90 gramos

30 decagramos - 300 gramos

La pesa de 90 gramos reporta la menor masa.

Case 5

75 hectogramos - 7500 gramos

5 kilogramos - 5000 gramos

6200 gramos

La pesa de 5 kilogramos reporta la menor masa.

Case 6

8500 hectogramos - 850000 gramos

30 kilogramos - 30000 gramos

900 decagramos - 9000 gramos

La pesa de 8500 hectogramos reporta la menor masa.

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The angles converted to degrees are: 615°, 165°, 112.5° with their respective quadrants of their terminal side as: 3rd, 2nd, and 2nd.

We simplify each angles of the angle in the radian to a single fraction and then convert them to degree by multiplying by 180°/π as follows:

11π/4 + 2π/3 = (33π + 8π)/12 {12 is the LCM of the denominators}

11π/4 + 2π/3 = 41π/12

11π/4 + 2π/3 = 41π/12 × 180°/π

11π/4 + 2π/3 = 615°

The terminal side of 615° is in the 3rd quadrant

π/6 - 5π/4 = (2π - 15π)/12 {12 is the LCM of the denominators}

π/6 - 5π/4 = -13π/12

π/6 - 5π/4 = -13π/12 × 180°/π

π/6 - 5π/4 = -195°

π/6 - 5π/4 = 165° {changing -195° to positive angle by adding 360°}

The terminal side of 165° is in the 2nd quadrant.

9π/8 - π/2 = (9π - 4π)/8 {8 is the LCM of the denominators}

9π/8 - π/2 = 5π/8

9π/8 - π/2 = 5π/8 × 180°/π

9π/8 - π/2 = 225°/2

9π/8 - π/2 = 112.5°

The terminal side of 112.5° is in the 2nd quadrant.

Therefore, the angles converted to degrees are: 615°, 165°, 112.5° with their respective quadrants of their terminal side as: 3rd, 2nd, and 2nd.

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Answer:a

Step-by-step explanation:

Answer:

A school year is 36 weeks long, so 3/4 of the school year is 36 * (3/4) = 27 weeks.

If Sunny works at the library for three weeks longer than three over four of the school year, he works for 27 + 3 = 30 weeks.

So, Sunny works at the library for 30 weeks in the school year.

The number of weeks in the school year that Sonny worked at the library is; 30 weeks

We are told that Sunny works at the library for three weeks longer than three over four of the school year a school year is 36 weeks long.

Now, since he works three weeks longer than three over four of the school year, then the expression is;

³/₄(36) + 3

= 27 + 3

= 30 weeks

That figure represents the number of weeks in the school year that Sonny worked at the library.

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The most clearly violated principle of ethical research in this scenario is individual information should be kept private, only summary results should be made public.

Ethical research requires that the privacy and confidentiality of study participants be protected. This is because the information collected from participants is often sensitive and personal, and the participants may have trusted the researcher to keep their information private.

In this scenario, the researcher has published the raw data, including personal information such as each participant's name, residence, and medical history, which could potentially compromise the privacy of the participants.

This is a clear violation of the principle of confidentiality, as the participants may not have agreed to have their personal information made publicly available.

It is important for researchers to be transparent and honest in their reporting of results, but they also have a responsibility to protect the privacy of their participants.

The publication of raw data, including personal information, should only be done with the explicit consent of the participants or under special circumstances where it is necessary for scientific advancement or the public good.

In conclusion, while the researcher's intention of being transparent in their reporting of results is commendable, the publication of the raw data, including personal information, is a clear violation of the principle of ethical research and could have serious consequences for the privacy and confidentiality of the study participants.

Hence, the correct answer is D.

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--The given question is incomplete; the complete question is

"A researcher collects various pieces of information from volunteers for a study into the health effects of drinking and smoking. The information collected includes each volunteer's name, residence, age, height, weight, medical history and their history of smoking cigarettes and drinking alcohol. All participants agreed to provide their information after the researcher explained how it would be used.

When publishing their results, the researcher feels that it would be unethical not to include the raw data which they have used to support their findings. So the researcher publishes the findings of the study, including all the supporting data on all the study participants, in a well-known journal.

The principle of ethical research most clearly violated (if any) is:

A. the researcher has falsified results

B. the researcher has failed to report negative results

C. no ethical principles have been violated; there was no informed consent

D. individual information should be kept private, only summary results should be made public"--

The number of tens that he regrouped as ones is equal to 2 tens.

The number of books that are on each shelf is equal to 66 books.

In Mathematics, a quotient simply refers to a mathematical expression that is typically used for the representation of the division of a number by another number.

Based on the information provided above, the number of books on each shelf is given by this quotient:

Quotient = 264/4

Quotient = 66

For the number of tens that Mr. Kent regrouped as ones on the shelf, we have the following:

26/4 = 24/4 + 2 = 2 tens.

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Answer:

c < 10 = The bag contains fewer than 10 carrots.

c > 10 = Catherine biked farther than 10 miles.

c < 20 = The chair is shorter than 20 inches

c > 20 = Carlos scored over 20 points

Answer: 12/4

Step-by-step explanation: 12/4 = 3/1 = 3

The absolute value of the given expression is 8.

Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

Given that an expression |10| - |‐5| + |‐3| we need to find its absolute value,

|10| - |‐5| + |‐3|

= 10 - 5 + 3

= 5 + 3

= 8

Hence, the absolute value of the given expression is 8.

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The cost of the one baseball card will be $45

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the cost of one dozen baseball cards is $540. The cost of one baseball card will be calculated by dividing the total cost by 12.

Cost = 540 / 12

Cost = $45

Hence, the cost of one card will be $45.

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The correct solution to the system of equation is (-2, -3)

From the question, we have the following parameters that can be used in our computation:

6g-3h=-3

7=h-5g

Divide through 6g-3h=-3  by 3

so, we have

2g-h=-1

Make h the subject

h = 2g + 1

So, we have

7 = 2g + 1 - 5g

Evaluate the equation

-3g = 6

So, we have

g = -2

This becomes

h = 2 * -2 + 1

Evaluate

h = -3

Hence, the solution is (-2, -3)

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The end behavior for f(x)=x^5 is as x approaches infinity, f(x) approaches infinity and as x approaches negative infinity, f(x) approaches negative infinity

From the question, we have the following parameters that can be used in our computation:

f(x) = x^5

Calculate f(oo) and f(-oo) i.e the function at infinity and negative infiniyt

So , we have

f(x) = (oo)^5 = +oo

f(x) = (-oo)^5 = -oo

This means that x increases with f(x)

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The solutions for the inequality "y is greater than one fifth times x plus 3" lies in both quadrant I and quadrant II.

In Mathematics, a quadrant can be defined as the area that is occupied by the values on the x-coordinate and y-coordinate of a cartesian coordinate.

Generally speaking, there are four (4) major quadrants in any graph and these include the following:

y is greater than one fifth times x plus 3 ⇒ y > x/5 + 3.

Next, we would use an online graphing calculator to plot the given inequality. Therefore, the required solution is the shaded area above the dash line as shown in the graph attached below.

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Answer:

e

Step-by-step explanation:

e

There are 30 boys and 20 girls in the classroom.

Let the variable p represent the number of people. Next, we would translate the given word problem into an algebraic equation by using a ratio as follows;

Number of girls + Number of boys = Total number of people

2p + 3p = 50

5p = 50

Dividing both sides of the equation by 5, we have the following:

p = 50/5

p = 10 people.

For the number of girls, we have:

2g = 2(10)

2g = 20 girls.

For the number of boys, we have:

3g = 3(10)

3g = 30 boys.

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The function would be [tex]f(x)=\frac{1}{2}x + 1 \ and\ f(x)=\frac{1}{2}x - \frac{1}{2}[/tex]

What is linear function?

A linear function is one with one or two variables that does not have exponents. It is a function that has a straight line graph.

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. To find the equation of the linear function given the values of x and f(x), we can use two points and use the slope-intercept form of the equation.

A way to find the equation of the linear function is to use the point-slope form: f(x) = m(x-x1) + f(x1), where (x1,f(x1)) is a point on the line.

If we use the points (-4,-2) and (0,0) we can find the slope:

m = (0 - (-2)) / (0 - (-4)) = 2/4 = 1/2

then using point-slope form:

f(x) = 1/2(x - (-4)) + (-2)

f(x) = 1/2x + 1

so the equation of the linear function is f(x) = 1/2x + 1

Alternatively, we can use the two points (-2,-1) and (0,0) we can find the slope:

m = (0 - (-1)) / (0 - (-2)) = 1/2

then using point-slope form:

f(x) = 1/2(x - (-2)) + (-1)

f(x) = 1/2x - 1/2

so the equation of the linear function is f(x) = 1/2x - 1/2.

Therefore, equation f(x) = 1/2x+1 and f(x) = 1/2x - 1/2 represents the given values of x and f(x)

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18) The expression with greater value are a and b. = 256  (19) the density of the object is X⁶ (20) The mass of the object is Y¹⁴  (21) The simplified expression is (3x)²  (22) Tobe uses the multiplication law for the exponent

The given parameters are

18) (a)   2³ · 2⁵ Using the multiplication law of indices, the two numbers are in the same base therefore add their powers

This implies 2³⁺⁵ = 2⁸ = 256

(b) (4)⁷/(4)³

Using the multiplication law of indices, the two numbers are in the same base therefore add their powers

4⁷⁻³ = 4⁴ = 256

(19)  Mass = X¹⁵ grams

Volume = X³ cm₁

Density = Mass/Volume

= X¹⁵/X³³

X¹⁵⁻³ = X⁶

Density = X⁶

(20) Density = Y¹⁰ grams/cm³

Volume = Y⁴ cm³

Mass = Density*volume

Mass = Y¹⁰ · Y⁴

= Using the multiplication law of indices, the two numbers are in the same base therefore add their powers

Y¹⁰⁺⁴

= Y¹⁴

(21) [ (3x)¹/³ · (3x)²/³)] / (3x)²/³

Using both multiplication and division laws of indices

(3x)²⁻¹

This is equal to  (3x)¹

(22) 8⁷· 8³ · 8²

Using the multiplication law of indices, the two numbers are in the same base therefore add their powers

8⁷⁺³⁺² = 8¹²

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