1) A contractor needs to know the height of a building to estimate the cost of a job. From a point 96 feet away from the base of the building the angle of elevation to the top of the building is found to be 46 . Find the height of the building. Round your answer to the hundredths place.

If the [tex]f(x)[/tex] and [tex]t(x)[/tex] are even function then [tex]fo\ t\ (x)[/tex] is an even function, if [tex]f(x)[/tex] and [tex]t(x)[/tex] are odd function then the function [tex]fo\ t\ (x)[/tex] is an odd function and if [tex]f(x)[/tex] is even and [tex]t(x)[/tex] is odd then the function [tex]fo\ t\ (x)[/tex] is an even function.

Further explanation:

An even functrion satisfies the property as shown below:

[tex]\boxed{f(-x)=f(x)}[/tex]

An odd functrion satisfies the property as shown below:

[tex]\boxed{f(-x)=-f(x)}[/tex]

Consider the given composite function as follows:

[tex]\boxed{fo\ t\ (x)=f\left(t(x))\right}[/tex]

If both the function [tex]f(x)[/tex] and [tex]t(x)[/tex] are even function.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(t(x))\right\\&=fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an even function.

If both the function [tex]f(x)[/tex] and [tex]t(x)[/tex] are odd function.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(-t(x))\right\\&=-fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=-fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an odd function.

If the function [tex]f(x)[/tex] is even and [tex]t(x)[/tex] is odd.

[tex]\begin{aligned}fo\ t\ (-x)&=f\left(t(-x))\right\\&=f\left(-t(x))\right\\&=fo\ t\ (x)\end{aligned}[/tex]

From the above calculation it is concluded that,

[tex]\boxed{fo\ t\ (-x)=fo\ t\ (x)}[/tex]

This implies that the composite function [tex]fo\ t\ (x)[/tex] is an even function.

Answer:

C

Step-by-step explanation:

Credit Score is a numerical expression which analyzes a person's credit level by looking at this financial conditions. Will he/she be worthy of loan or not.

payment history comprises 35% of a person's credit score. This is a huge factor. If you consistently make your payments on time, your credit score increases.

length of credit history tells how secure you will be to lenders. Usually 7 years+ is a great length of credit history. This pretty much affects credit score.

marital status doesn't affect credit score. Lenders assess a person based on their financial condition and past activity, NOT whether or not he/she is married or not. That's personal agenda.

debt ratio is the ratio of total debt to total assets. If this is high, it means a person owes money to banks/individuals and is more likely to be not given credit. It affects credit score highly.

THus, the correct answer is C

Answer:

C

Step-by-step explanation:

Answer:

Hi, my friend! The answer to this questions are "0.125" and "12.5%"

Step-by-step explanation:

First, if you divide 1 by 8 using a calculator, you will see that the result will be 0.125. Also, the option 12,5% is correct because de symbol "%" means "divided by 100". If you check in the calculator, 12.5 divided by 100 will also be 0.125.

Hoping to be clear! Good luck!

Final answer:

Only 0.125 and 12.5% are equivalent to the fraction 1/8. This is because they both reflect a '125-like' number pattern which is crucial when identifying representations of 1/8 in decimal or percentage form.

Explanation:

The fraction 1/8 is equivalent to a few different representations in decimal and percentage forms.

To identify which options among 0.125, 0.18, 12.5%, and 1.25% are equivalent to 1/8, we should look for a number pattern resembling '125' as it indicates a connection to the fraction 1/8 or its reciprocal nature.

So, 0.125 is exactly 1/8 as a decimal, and 12.5% represents 1/8 in percentage form because 12.5% is 12.5 parts per 100, which can be equated to 125 parts per 1000, resembling the multiplication of 1000 by the reciprocal of 8, that is 0.125.

Therefore, the options 0.125 and 12.5% are equivalent to 1/8.

Answer:

The value of the car in very nice condition will be $35500.

Step-by-step explanation:

Let the value of car in a very nice condition be = x

The value $62125 is [tex]1\frac{3}{4}[/tex] or [tex]\frac{7}{4}[/tex] or 1.75 times of x.

Now, we can calculate for x:

[tex]1.75x=62125[/tex]

[tex]x=\frac{62125}{1.75}[/tex]

x = 35500

Hence, the value of the car in very nice condition will be $35500.

The value of the car in very nice condition is $35,500. This is determined by dividing the value of the car in excellent condition by 1.75.

To determine the value of the car in very nice condition, you need to use the information that a car in excellent condition is worth 1 3/4 times (or 1.75 times) as much as a car in very nice condition.

Given the value of the car in excellent condition is $62,125, you can set up the equation:

Substitute the known value:

To find the Value in Very Nice Condition, divide both sides by 1.75:

Hence, the value of the car in very nice condition is $35,500.

Answer:

5r + 2p + 6

Step-by-step explanation:

4r+r = 5r

9-3 = 6

Your first two tests add up to:

85 + 89 = 174

To average 87 for 3 tests, the tests need to add up to : 87 x 3 = 261

To average 91, the 3 tests need to equal: 91 x 3 = 273

So to average 87, the third test needs to be: 261 - 174 = 87

To average 91, the third test needs to be: 273 - 174 = 99

Your score needs to be between 87 and 99

Answer:

P ( 500<X<650 ) = 0.8577

Step-by-step explanation:

Since μ=572 and σ=51 we have:

P ( 500<X<650 ) = P ( 500−572< X−μ<650−572 )

[tex]\RightarrowP ( \frac{500-572}{51} < \frac{x-\mu}{\sigma} < \frac{650-572}{51})[/tex]

⇒ P ( 500<X<650 ) = P ( −1.41<Z<1.53 )

Now, Using the standard normal table to conclude that:

P ( −1.41< Z <1.53 ) = 0.8577

Final answer:

Approximately 85.77% of Indiana tenth-grade students scored between 500 and 650 on the English/language arts ISTEP exam, as calculated by converting the given scores to z-scores and finding the proportion within the standard normal distribution.

Explanation:

To find the proportion of Indiana tenth-grade students who scored between 500 and 650 on the English/language arts exam, given that the ISTEP scores are approximately normal with a mean (μ) of 572 and a standard deviation (σ) of 51, we can use the standard normal distribution.

First, we'll convert the scores to z-scores using the formula z = (X - μ) / σ where X is the score.

Then we'll look up these z-scores in a standard normal distribution table or use a calculator with normal distribution functions to find the proportion of students whose scores fall between these two z-scores. The table or calculator will provide us with the areas under the curve to the left of each z-score.

Let's assume the area to the left of z=-1.41 is approximately 0.0793 (7.93%) and to the left of z=1.53 is approximately 0.937 (93.7%). To find the proportion between 500 and 650, we'll subtract the smaller area from the larger one:

Proportion = Area to the left of z=1.53 - Area to the left of z=-1.41 = 0.937 - 0.0793 = 0.8577 or 85.77%

Therefore, approximately 85.77% of the students scored between 500 and 650 on the ISTEP English/language arts exam.

After 22 s, the car has velocity

[tex]v=\left(1.7\dfrac{\rm m}{\mathrm s^2}\right)(22\,\mathrm s)=36.4\dfrac{\rm m}{\rm s}[/tex]

In this time, it will have traveled a distance of

[tex]\dfrac12\left(1.7\dfrac{\rm m}{\mathrm s^2}\right)(22\,\mathrm s)^2=411.4\,\mathrm m[/tex]

Over the next 29 s, the car moves at a constant velocity of 36.4 m/s, so that it covers a distance of

[tex]\left(36.4\dfrac{\rm m}{\rm s}\right)(29\,\mathrm s)=1055.6\,\mathrm m[/tex]

so that after the first 51 s, the car will have moved 1467 m.

After the 29 s interval of constant speed, the car's negative acceleration kicks in, so that its velocity at time [tex]t[/tex] is

[tex]v(t)=36.4\dfrac{\rm m}{\rm s}+\left(-5.8\dfrac{\rm m}{\mathrm s^2}\right)t[/tex]

The car comes to rest when [tex]v(t)=0[/tex]:

[tex]36.4-5.8t=0\implies t=6.3[/tex]

That is, it comes to rest about 6.3 s after the first 51 s. In this interval, it will have traveled

[tex]\left(36.4\dfrac{\rm m}{\rm s}\right)(6.3\,\mathrm s)+\dfrac12\left(-5.8\dfrac{\rm m}{\mathrm s^2}\right)(6.3\,\mathrm s)^2=114.2\,\mathrm m[/tex]

so that after 57.3 s, the total distance traveled by the car is 1581.2 m, or about 1.6 km.

Answer:

Step-by-step explanation:

Easy. its [tex]x^{2} \int\limits^a_b {x} \, dx \\ \\ \left \{ {{y=2} \atop {x=2}} \right. \\ \neq \frac{x}{y} \beta \al\left \{ {{y=2} \atop {x=2}} \right. pha \neq \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \geq \\ \left \{ {{y=2} \atop {x=2}} \right. \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]

The inequality that represents the graph is:

                        [tex]y\geq x+3[/tex]

By looking at the graph we observe that the graph is a solid line which passes through the point (-1,0) and (1,2).This means that the inequality is a inequality with a equality sign(i.e. not strict)

This means that the equation of line is:

[tex]y-2=\dfrac{2-0}{1-(-1)}\times (x-(-1))\\\\y-2=\dfrac{2}{2}\times (x+1)\\\\y-2=x+1\\\\y=x+1+2\\\\y=x+3[/tex]

The shaded region is away from the origin.

This means that the inequality does not pass the zero test.

Hence, the inequality is:

               [tex]y\geq x+3[/tex]

Answer:

The mammoth died 17.190 years ago.

Step-by-step explanation:

Given information:

For every 5700 years, the elephant's bones losses a half-life of C-14. And supposing that the living mammoth bones had the same percentage of C-14.

If the mammoth when living has 40%, then:

Answer:

[tex]\large \boxed{\text{17 190 yr ago}}[/tex]

Step-by-step explanation

The amount of C-14 has declined from 40 % to 5 % of the original, that is, to ⅛ of the original amount.

The half-life of C-14 (5730 yr) is the time it takes for half of the isotope to decay.

We can make a table of the amount remaining after each successive half-life.

[tex]\begin{array}{ccc}\textbf{Number of} && \textbf{Fraction} \\\textbf{half-lives} & \textbf{Years} & \textbf{remaining} \\0 &0 & 1 \\1 &5730 &\frac{1}{2 } \\\\2 &11460 &\frac{1}{4 } \\\\3 &17190 &\frac{1}{8} \\\\4 & 22920& \frac{1}{16} \\\end{array}\\\text{We see that the fraction of C-14 is reduced to $\frac{1}{8}$ after three half-lives.}\\\text{The mammoth died $\large \boxed{\textbf{17 190 yr ago}}$}[/tex]

Answer:

  Domain of set B: {2, 4, -4, 0}

Step-by-step explanation:

The domain of the function whose ordered pairs are listed in set B is the set of first numbers of those pairs: {2, 4, -4, 0}.

_____

Comment on the question

A "set" does not have a domain. A "function" has a domain. To make any sense of this question, we have to interpret the question to mean the function described by the ordered pairs in the set.

Answer:

20

Step-by-step explanation:

8/2 = 4

4 * 5 = 20

Answer:

(a) What is the research objectives?

B. To determine the number of adults in the country who believe the federal government wastes 51 cents or more of every dollar.

(b) What is the population?

C. Adults in the country aged 18 years or older.

(c) What is the sample?

D. The 1019 adults in the country that were surveyed.

D. The 1019 adults in the country that were surveyed. The research objective is to determine the number of adults in the country who believe the federal government wastes 51 cents or more of every dollar. The population is adults in the country aged 18 years or older. The sample is the 1019 adults in the country that were surveyed.

The research objective of the survey conducted by the news service is B. To determine the number of adults in the country who believe the federal government wastes 51 cents or more of every dollar.

The population in this study is C. Adults in the country aged 18 years or older.

The sample in this study is D. The 1019 adults in the country that were surveyed.

Answer:

The intersection is 10

Step-by-step explanation:

You just have to look at them and see what number is in all the sets.

Answer:

the answer is 10

Answer:

  Any of 7,111,777 or 7,333,555 or 7,555,333 or 7,777,111.

Step-by-step explanation:

The first digit must be 7. The sum of the unknown digits is then 31-7 = 24. Since this represents 3 pairs of digits (a thousands period digit and a units period digit), each pair must total 24/3 = 8.

There are two ways that odd numbers can total 8: 1+7 = 3+5 = 8. Since there is no restriction on the digits other than they must be the same in any period and they must be odd, there are 4 ways the 2 pairs of digits can be arranged into a number:

Answer:

[tex]x=(c-60)/85[/tex]

Step-by-step explanation:

we have

[tex]c=85x+60[/tex]

Solve for x

That means -----> isolate the variable x

subtract 60 both sides

[tex]c-60=85x+60-60\\c-60=85x[/tex]

Divide by 85 both sides

[tex](c-60)/85=85x/85\\(c-60)/85=x[/tex]

Rewrite

[tex]x=(c-60)/85[/tex]

Answer:

For solar: [tex]S(x)=6240+75x[/tex]

For traditional: [tex]T(y)=850+320y[/tex]

Step-by-step explanation:

Cost to install solar water heater = $6240

The average annual cost to run the solar water heater is about = $75

Total cost to run x years : [tex]S(x)=6240+75x[/tex]

A traditional gas water heater costs about = $850

The average annual cost to run = $320

Total cost to run y years : [tex]T(y)=850+320y[/tex]

1. The total cost of a solar water heater, including installation and annual running costs, can be calculated as $6240 + ($75 * years).

2. The total cost of a traditional gas water heater, including installation and annual running costs, can be calculated as $850 + ($320 * years).

Given that,

Solar water heater:

Installation cost: $6240 (after rebates)

Annual running cost: $75

Traditional gas water heater:

Installation cost: $850

Annual running cost: $320

Let's create the total cost functions for each water heater as a function of the number of years.

For the solar water heater, the total cost (TC) can be calculated as:

TC_solar(years) = installation cost + (annual running cost * years)

Given the installation cost of $6240 and annual running cost of $75,

The total cost function for the solar water heater becomes:

TC_solar(years) = $6240 + ($75 * years)

For the traditional gas water heater, the total cost (TC) can be calculated as:

TC_gas(years) = installation cost + (annual running cost * years)

Given the installation cost of $850 and annual running cost of $320,

So, The total cost function for the gas water heater becomes:

TC_gas(years) = $850 + ($320 * years)

These equations allow you to determine the total cost of each water heater based on the number of years they are used.

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Answer:

The answer is 54.5

Step-by-step explanation:

The median of a set of observations is the value in the middle in case there is an uneven number of observations or the average between the two values in the middle.

First we arrange the times from less time to longer times.

50 - 52 - 57 - 230 .

The values in the middle are  52 - 57

And the average between this two values is 54.5. Thus that is the median legth of time.

Answer:

The median length of time required to review an application is 54.5 minutes.

Step-by-step explanation:

The times (in minutes) that several underwriters took to review applications for similar insurance coverage are 50, 230, 52, and 57.

Arranging in ascending order we get

50,52,57,230

Median length of time = [tex]\frac{52+57}{2}=54.5[/tex]

Hence, the median length of time required to review an application is 54.5 minutes.

Answer:

A triangle with three sides that are each equal in length to those of another triangle, for example, are congruent. This statement can be abbreviated as SSS. Two triangles that feature two equal sides and one equal angle between them, SAS, are also congruent.

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

First, when fractions get raised to an exponent, both the numerator and the denominator get raised to the power.

When exponents get raised to a power, the powers multiply.

E.g. [tex](x^2)^3=(x^2)(x^2)(x^2)=(x*x)(x*x)(x*x)=x^6[/tex]

In this case,

[tex](\frac{4^3}{5^-^2})^5=\frac{4^(^3^*^5^)}{5^(^-^2^*^5^)}=\frac{4^1^5}{5^-^1^0}\\[/tex]

Note that [tex]x^-^n=\frac{1}{x^n}[/tex]

So, 1/5^(-10) = 5^10

So, our answer is C

let's firstly convert both fractions with the same denominator, by simply multiplying one by the denominator of the other, let's proceed,

[tex]\bf \cfrac{1}{5}\cdot \cfrac{3}{3}\implies \cfrac{3}{15}~\hspace{7em}\cfrac{1}{3}\cdot \cfrac{5}{5}\implies \cfrac{5}{15} \\\\[-0.35em] ~\dotfill\\\\ \boxed{\cfrac{3}{15}}\rule[0.35em]{10em}{0.25pt}~~\cfrac{4}{15}~~\rule[0.35em]{10em}{0.25pt}\boxed{\cfrac{5}{15}}[/tex]

well, low and behold, 4/15 doesn't simplify further and is right between those two, and its denominator is not 4.

Answer:

See explanation

Step-by-step explanation:

7.8 9.1 9.5 10.0 10.2 10.5 11.1 11.5 11.7 11.8

12.2 12.2 12.5 13.1 13.5 13.7 13.7 14.0 14.4 14.5

14.6 15.2 15.5 16.0 16.0 16.1 16.5 17.2 17.8 18.2

19.0 19.1 19.3 19.8 20.0 20.2 20.3 20.5 20.9 21.1

21.4 21.8 22.0 22.0 22.4 22.5 22.5 22.8 22.8 23.1

23.1 23.2 23.7 23.8 23.8 23.8 23.8 24.0 24.1 24.1

24.5 24.5 24.9 25.1 25.2 25.5 26.1 26.4 26.5 26.7

27.1 29.5

A. The five-number summary is

B. The interquartile range is

[tex]Q_3-Q_1=23.8-14.2=9.6[/tex]

C. See attached diagram

D. The distribution is not symmetric, the left half shows more data than the right part

Answer:

  8x + 3(20 - x) = 5(20)

Step-by-step explanation:

If x represents the number of pounds of $8 chocolates, then (20-x) is the number of pounds of $3 chocolates. The cost of the mix attributable to the $8 chocolates will be 8x, and the cost associated with the $3 chocolates will be 3(20-x). The sum of these costs is the cost of the 20 pounds of mix: $5×20.

In equation form, this is ...

  8x + 3(20-x) = 5(20)

_____

The solution is x=8, so 8 lb of $8 chocolates should be mixed with 12 lb of $3 chocolates to make a mix worth $5 per pound.

Final answer:

The correct equation to determine the pounds of each type of chocolate for the mixture is 8x + 3(20 - x) = 5(20), where x represents the pounds of the $8 chocolate.

Explanation:

To solve for how many pounds of each type of chocolate should be used in the mixture, let's designate x as the number of pounds of the $8 chocolate and 20-x as the number of pounds of the $3 chocolate. The total cost of the $8 chocolate will be 8x dollars, and the total cost of the $3 chocolate will be 3(20-x) dollars. The total sales price for the 20-pound mixture must be $5 per pound, which equals $100 (since 20 pounds × $5 per pound = $100). Therefore, the correct equation that represents the situation is 8x + 3(20 - x) = 5(20).

Let's break down this equation:

The $8 chocolate: 8x dollars

The $3 chocolate: 3(20 - x) dollars

Total cost: 8x + 3(20 - x)

The mixture sells for: $5 × 20

The equation balances the cost of both chocolates with the selling price of the mixture.

Answer:

The answer to your questions is: 25 new teachers

Step-by-step explanation:

Data

# of students = 2000

ratio = 3:80 teachers to students

New teachers = ?

Process

I suggest to use rule of three to solve this problem

                     3 teachers ----------------  80 students

                      x                ----------------  2000 students

                  x = (2000 x 3) / 80 = 75 teachers

               Number of initial teachers = 75

The ratio change to 1:20

                     1 teacher -------------------  20 students

                     x             -------------------   2000 students

                     x = (2000 x 1) / 20

                    x = 100 teachers

Number of new teachers = 100 - 75 = 25

Firstly we calculate the height(h)...,

by using trigonometry ratios

;tan 45° = h/5

;where.. h = 5 tan 45°

;h = 5

Then we calculate the side x...using the height h and the angle of 30°

by also using trigonometry ratios

; sin 30° = 5/x

;where.. x = 5/(sin 30°)

Hence...., x = 10

Answer: x = 10

Answer:

B

Step-by-step explanation:

Giving away money because individuals loan dollars to other entitles at a a cost

Answer:

  D.  lending money

Step-by-step explanation:

It depends on the institution. A "for profit" financial institution has the purpose of making money for its investors. To do that, it offers a variety of money-oriented services, some involving payments to customers (interest) and some involving charging customers for the service (fees or interest).

A service offered in the latter category is "lending money."

__

A "not for profit" financial institution has the purpose of providing financial services to its members. Again, some of these services may involve payments to customers, and others may involve collecting interest or fees from customers. These institutions, too, offer the service of "lending money."

__

Among the services a financial institution may offer that do not involve lending money are ...

Answer:

Production has increased 20/day

Step-by-step explanation:

In the first scenario production is 500/day and productivity 25 set/hour. After the changes, production is 600/day and 25 set/hour.

So productivity remains the same, nevertheless, as there are more productive hours per day, production raises, in this case the can be calculated as (New Production-Old Production)/Old Production=(600-500)/500=100/500=0.2=20%.

As productivity remains the same, you do not ge more sets/shift, as shifts are shorter (8 instead of hours, so you get 200/shift instead of 250/shift). The rest of the option is false as productivity remains constant

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

If you are given, say, the polynomial equation y = x2 + 5x + 6, you can factor the polynomial as y = (x + 3)(x + 2). Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y.

You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x = –2 and x = –3 are the zeroes of a quadratic, then you know that x + 2 = 0, so x + 2 is a factor, and x + 3 = 0, so x + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a(x + 3)(x + 2).

(The extra number "a" in that last sentence is in there because, when you are working backwards from the zeroes, you don't know toward which quadratic you're working. For any non-zero value of "a", your quadratic will still have the same zeroes. But the issue of the value of "a" is just a technical consideration; as long as you see the relationship between the zeroes and the factors, that's all you really need to know for this lesson.)

Anyway, the above is a long-winded way of saying that, if x – n is a factor, then x = n is a zero, and if x = n is a zero, then x – n is a factor. And this is the fact you use when you do synthetic division.

Let's look again at the quadratic from above: y = x2 + 5x + 6. From the Rational Roots Test, you know that ± 1, 2, 3, and 6 are possible zeroes of the quadratic. (And, from the factoring above, you know that the zeroes are, in fact, –3 and –2.) How would you use synthetic division to check the potential zeroes? Well, think about how long polynomial divison works. If we guess that x = 1 is a zero, then this means that x – 1 is a factor of the quadratic. And if it's a factor, then it will divide out evenly; that is, if we divide x2 + 5x + 6 by x – 1, we would get a zero remainder. Let's check:

As expected (since we know that x – 1 is not a factor), we got a non-zero remainder. What does this look like in synthetic division? Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

First, write the coefficients ONLY inside an upside-down division symbol:

Answer:

Step-by-step explanation:

[tex]A=\frac{1}{2} r^{2} \theta\\\frac{dA}{d\theta} =r \\\\when r=2\\rate of change at r=2 is  2[/tex]

We want to find the rate of change of a given function of two variables when we fix one of the two, and then we want to evaluate it at r = 2.

We will get that the rate of change is:

[tex]\frac{dA(r, \theta)}{d\theta} = 12*r^2[/tex]

And when r = 2, the rate is 48.

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

We define the rate of change of a function with respect to some variable as the differentiation of the function with respect to that variable.

Here we have the function:

[tex]A(r, \theta) = 12*r^2*\theta[/tex]

We need to differentiate it with respect to θ, we will get:

[tex]\frac{dA(r, \theta)}{d\theta} = 1*12*r^2*\theta^0 = 12*r^2[/tex]

Where I used the general rule to derive functions with exponents:

[tex]f(x) = x^n\\\\\frac{df(x)}{dx} = n*x^{n -1}[/tex]

Now that we know the rate of change, we want to evaluate it in r = 2, we will get:

[tex]\frac{dA(2, \theta)}{d\theta} = 12*2^2 = 48[/tex]

Notice that this does not depend on the value of θ.

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Explanation:

5(y+3)-11=-y-6 Given

5(y+3)=-y+5 . . . . addition property of equality (11 is added)

5y+15=y+5 . . . . . distributive property (5 is distributed)

5y=-y-10 . . . . . . . addition property of equality (-15 is added)

6y=-10 . . . . . . . . . addition property of equality (y is added)

y=-5/3 Division Property of Equality/Reduce