ind the general solution of the given differential equation. dr / d theta + r sec theta = cos theta
r = ___

Answers

Answer 1

r = (sin theta + C)^(-1) where C is an arbitrary constant of integration.

Any equation with at least one ordinary or partial derivative of an unknown function is referred to as a differential equation. Assuming that a function's rate of change with regard to x is inversely proportional to y, we may write it down as dy/dx = k/y.

r = (cos(theta) - d/d(theta) of sec(theta)) / sec(theta)

The differential equation assists us in presenting a relationship between the changing quantity with respect to the change in another variable. The derivative represents nothing more than a rate of change. Be a function with the form: y=f(x), where x is an independent variable, f is an unknown function.

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Related Questions

Shaheen ha many triangular piece, each of area 1 q. Unit and many rectangular piece, each of area 2 q. Unit

Answers

If Shaheen has many triangular piece, each of area 1 sq. Unit and many rectangular piece, each of area 2 sq. Unit  she arranges these to form the given shape then the area of shape is 14 Square Units .

The Area of one triangular piece is given as = 1 Sq . Unit ;

The Area of one rectangular piece is = 2 Sq. Unit ;

From the shape given below ,

we can see that the shape has 4 triangular pieces and 5 rectangular pieces .

So , total area of given shape is ⇒[tex]4\times Area Of Triangle + 5\times Area Of Rectangle ;\\[/tex]

Substituting the values of Area ,

we get ;

[tex]Total Area = 4\times 1 +5\times 2[/tex]

= [tex]4+10[/tex]

= 14 Sq . Units .

Therefore , the area of the given shape is 14 Sq. Units .

The given question is incomplete , the complete question is

Shaheen has many triangular piece, each of area 1 sq. Unit and many rectangular piece, each of area 2 sq. Unit .

If she arranges these to form a shape as shown below, what would be the area of this shape ?

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Points D, B, and E are collinear. Find the value of a so that points A, B, and C
are collinear.

Answers

Answer:

Since A, O and B are collinear,

i.e.,

A

O

B

=

180

or,

A

O

D

+

D

O

C

+

B

O

C

=

180

(

x

10

)

+

(

4

x

25

)

+

(

x

+

5

)

=

180

Or,

x

+

4

x

+

x

10

25

+

5

=

180

Or,

6

x

30

=

180

Or,

6

x

=

180

+

30

=

210

Or,

x

=

210

6

=

35

B

O

C

=

x

+

5

=

35

+

5

=

40

How to graph 4x-6y=36

What’s the x intercept

What’s the y intercept

Answers

Therefore . the solution of the given problem of equation comes out to be x-intercept(s): (9,0) and y-intercept(s): (0,−6).

Clarify the equation

A depiction of two equal variables, one on either side of a "equals" sign, is what constitutes an equation in mathematics. To solve common issues, one can use equations. We frequently look for help with pre algebra to overcome obstacles in real life. Basic mathematical concepts are covered in pre-algebra lessons.

Here,

the x-intercepts,

In order to determine the x-intercept(s), put in for y and calculate 4x6y=36.

Make the equation work.

x=9 point form x-intercepts.

x-intercept(s): \s(9,0) (9,0)

the y-intercepts, please.

In order to determine the y-intercept(s), replace with and solve for y=.4(0-)6y=36.

Make the equation work.

To continue, tap.

Point shape y=6 y-intercept(s).

The y-intercept is (0, 6).

Write down the junctions.

x-intercept(s): (9,0)

The y-intercept is (0, 6).

Therefore . the solution of the given problem of equation comes out to be x-intercept(s): (9,0) and y-intercept(s): (0,−6).

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Anybody know the answer to this ?

Answers

On solving the provided question, we can say that - the data from graphs represents a curvilinear and straight graph.

What is graphs?

Graphs are visual representations or charts used in mathematics to methodically express data or values. A relationship between two or more objects is frequently represented by a point on a graph. A non-linear data structure called a graph is made up of nodes, or vertices, and edges. Connect the nodes, also known as vertices. This graph comprises a set of vertices V= 1, 2, 3, 5, and a set of edges E= 1, 2, 1, 3, 2, 4, and (2.5), (3.5), (4.5). Statistics graphs (bar charts, pie charts, line charts, etc.) Exponential diagrams. triangle graph, a logarithmic graph

Here,

in graphs,

as per data we can say that there two types

one is curvilinear and other is straight graph.

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Aisha has 50 chocolates she fills 3 boxes and has 2 left over How many chocolates does she have left? Write it as an equation

Answers

Aisha has 2 chocolates left. You can represent the problem as an equation as follows:

x = 3b + 2

Where x is the total number of chocolates, b is the number of boxes filled and 2 is the remaining chocolates left.

Since we know that Aisha has 50 chocolates, we can substitute this value for x in the equation:

50 = 3b + 2

Solving for b, we find that b = 16. So, Aisha filled 16 boxes and had 2 chocolates left over. The equation is true, and Aisha has 2 chocolates left.

It's worth noting that this is a simple algebraic equation, we can use the same equation to find the number of chocolates given the number of boxes and remaining chocolates, or to find the number of boxes given the number of chocolates and remaining chocolates or to find the remaining chocolates given the number of boxes and the number of chocolates.

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Solve the system of equations.
y = 3x
y=x²-4
A. (-1, -3) and (4, 12)
O B. (-4, 12) and (1, -3)
C. (-1, 3) and (4, -12)
OD. (-4,-12) and (1, 3)

Answers

Answer:

A

Step-by-step explanation:

Se them equal to each other

3x = [tex]x^{2}[/tex] - 4

[tex]x^{2}[/tex] -3x -4 = 0

(x + 1) ( x -4)

x + 1 = 0

x = -1

y = 3x

y = 3(-1)

y = -3

(-1,-3)

x - 4 = 0

x = 4

y = 3x

y = 3(4)

y = 12

(4, 12)

Need the answer plsss!

Answers

By solving, we see that the product of the polynomials 4 - a and a3 + 7a - 18 is a3 + 3a2 + 46a + 72.

what is polynomial ?

The location on the y-axis where the slope of the line passes is known as the intersection point in mathematics. a point on a line or curve where the y-axis crosses. The equation for the straight line is given as Y = mx+c, where m denotes the slope and c the y-intercept. In the intercept form of the equation, the line's slope (m) and y-intercept (b) are highlighted. The slope and y-intercept of an equation with the intercept form (y=mx+b) are m and b, respectively. There are several equations that can be rewritten to look like slope intercepts. The slope and y-intercept are both modified to 1 if y=x is rewritten as y=1x+0, for example.

given

(4 - a) * ( a3 + 7a - 18 )

= 4a3 + 28a - 72 - a4 + 7a2 + 18 a

= a3 + 3a2 + 46a + 72

By solving, we see that the product of the polynomials 4 - a and a3 + 7a - 18 is a3 + 3a2 + 46a + 72.

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can somebody please help me solve this question:

you are going to climb the roof of your two-story house using a 15-foot ladder. If the safety directions say that you can put the ladder only 2 feet from the house, how far up will you be able to reach on your house?


thanks

Answers

Using Pythagoras theorem to solve the right angle triangle, the height of the building is 14.87 feet

What is Pythagoras Theorem

Pythagoras Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as an equation:

x² = y² + z²

where x is the hypothenuse and y, z are the legs of the right angle triangle.

To determine the height of the building in which the ladder will reach, we have to use Pythagoras theorem for that.

x² = y² + z²

15² = 2² + z²

z² = 15² - 2²

z² = 221

z = √221

z = 14.87 feet

The height is 14.87 feet

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Using the following equation, find the center and radius of the circle. You must show and explain all work and calculations to receive credit. Be sure to leave your answer in exact form.
x^2+y^2+8x-6y-15=0

Answers

The answers for Center of the circle is (-4,3) and radius is 2√10 units.

How to find the Center of the circle using Algebraic Expressions?

(x - h)2 + (y - k)2 = r2 is the equation of the center of the circle. The radius (r) is specified as 5 units, while the center's (h, k) coordinates are (0, 0). (x - 0)2 + (y - 0)2 = 52 is the result of substituting the values of h, k, and r in the equation.

How to simplify Algebraic expressions?

Finding the simplified term of the given expression is the goal of simplifying the algebraic statement. We must first understand how to join like terms, factor a number, and the order of operations before we can factorize or simplify the statement. When like words are combined, the constant terms are divided for the purpose of simplicity and the variables with the same degree are grouped together.

Examples of  Algebraic Expressions

2x2+3xy+4x+7

Given,

[tex]x^2+y^2+8x-6y-15=0[/tex]

⇒ x².+8x+16–16+y²-6x+.9–9–15=.0

(x+4)²+(y-3)²-16–9–15=0

I.e. (x+4)²+(y-3)²=40

(x+4)²+(y-3)²={2√10}²

Hence center is (-4,3) and radius is 2√10 units.

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GIVING BRAINLIEST AND 50 POINTS

Answers

Answer:

2x + 2

Step-by-step explanation:

Combine all the like terms:

-5x + 7x -12 + 14 = 2x + 2

Answer:

2x + 2

Step-by-step explanation:

(- 5x - 12) + (14 + 7x) ← remove parenthesis

= - 5x - 12 + 14 + 7x ← collect like terms

= (- 5x + 7x) + (- 12 + 14)

= 2x + 2

Is Avogadro's number equal to one mole?

Answers

One mole consists of 6.02214076×10²³ units which is Avogadro's number. Thus one mole is known as Avogadro's number.

What is one mole?

A mole is 6.02214076×10²³ of any chemical unit, including atoms, molecules, ions, and others. Due to the large number of atoms, molecules, or other components that make up any substance, the mole is a useful measure to utilize. The mole was initially defined as the quantity of atoms contained in 12 grammes of carbon-12, but the General Conference on Weights and Measures declared in 2018 that the mole will only contain 6.02214076×10²³ of some chemical unit as of May 20, 2019.

In chemistry, a mole, sometimes spelled mol, is a common scientific measurement unit for significant amounts of very small objects like atoms, molecules, or other predetermined particles.

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Is 0 Infinity closed interval?

Answers

Thus, 0 itself is the sole position at which [0,∞] and (0,∞] are separated. Since it is in [0,∞] the set is closed. Given that it is not in (0,∞), the set is open interval.

A (real) interval in mathematics is a collection of real numbers that includes all real numbers that fall within any two of the collection's numbers. For instance, the interval containing 0, 1, and all integers in between is the set of values x satisfying 0 x 1. The collection of numbers such that 0 x 1 and the collection of all real numbers are other examples of intervals.

the empty set, any singleton, the set of positive real numbers, the set of nonnegative real numbers, and (set of one element).

Because they are the most basic sets whose "length" (or "measure" or "size") is straightforward to define, real intervals are crucial in the theory of integration. The Borel measure and finally the Lebesgue measure result from extending the notion of measure to more intricate sets of real numbers.

Interval arithmetic is a broad numerical computing technique that automatically generates assured enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff. Intervals are at the centre of this technique.

Therefore, 0 itself is the sole place at which [0,∞] and (0,∞] have a boundary. Due to its location in [0,∞] the set is closed. Since it is not in (0,∞), the set is unclosed.

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Find whether x^n+y^h is disable by x-y(y not equal o)or not

Answers

Yes, it is equal x-y disable

Makayla and Addison are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Makayla is 290 miles away from the stadium and Addison is 345 miles away from the stadium. Makayla is driving along the highway at a speed of 40 miles per hour and Addison is driving at speed of 51 miles per hour. Let MM represent Makayla's distance, in miles, away from the stadium tt hours after noon. Let AA represent Addison's distance, in miles, away from the stadium tt hours after noon. Write an equation for each situation, in terms of t,t, and determine the interval of hours, t,t, for which Makayla is closer to the stadium than Addison.

Answers

The linear functions for each situation are given as follows:

Makayla: 290 - 40x.Addision: 345 - 51x.

Makayla is closer than Addision when:

x < 5.

How to define the linear functions?

The linear functions in slope-intercept format for this problem are given as follows:

y = mx + b.

In which:

m is the slope, representing the velocity, as a negative as they are getting closer to the stadium with time.b is the intercept, representing the initial distance.

Hence the functions are defined as follows:

Makayla: 290 - 40x.Addision: 345 - 51x.

Makayla is closer when:

290 - 40x < 345 - 51x

11x < 55

x < 5.

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Write a polynomial function in standard form
-6,0,0,2

Answers

The polynomial function in standard form would be:

f(x) = 2x^3 - 6

This function has a degree of 3 (the highest exponent on the x term), and the leading coefficient (the coefficient of the x^3 term) is 2. The constant term (the term with no x) is -6.

when journalizing an accounts payable account what account is credited
a. it depends on the transaction
b. accounts payable
c. cash
d. accounts receivable

Answers

Answer:

cash

Step-by-step explanation:

debit - account payable

credit - cash




Finn leaned a 100-foot ladder against the side of a building. The base
of the ladder is 14 feet from the base of the building. The top of the
ladder is 11 feet from the top of the building. How tall is the building?
Provide an answer accurate to the nearest tenth of a foot.

Answers

The height of the Building is 110.01 feet.

What is the Pythagorean theorem?

The square of the hypotenuse side in a right-angled triangle is equal to the sum of the squares of the other two sides, according to Pythagoras' Theorem.

Given, Finn leaned a 100-foot ladder against the side of a building. The base of the ladder is 14 feet from the base of the building. The top of the ladder is 11 feet from the top of the building.

For the triangle made by ladder

hypotenuse = 100

base = 14

thus, from the Pythagorean theorem

hypotenuse² = base² + height²

100² = 14² + height²

height² = 10000 - 196

height = √9804

height = 99.01

total height of the wall = height of triangle + remaining height of the wall

Thus the total height of the wall  = 99.01 + 11

the total height of the wall  = 110.01

therefore, the building is 110.01 feet tall.

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what is the answer thank ya mate

Answers

Answer: The ship is going to another planet.

Step-by-step explanation:

I hate i-ready,

Your welcome mate

The answer is B going to another planet

What is the value of 8 power 8?

Answers

Answer:16,777,216

Step-by-step explanation:

The coordinates of a triangle are: R(-4,4) S(-2,-2) T(2, 3) Brian determined A RST was a scalene triangle, but he wanted to the find the lengths of the three sides. Complete the table to determine the three side lengths of ARST. 37 38 140 141 length of RS length of ST length of TR​

Answers

The length of the sides RS, ST and TR are √72, √41 and √37 in the scalene triangle RST.

The distance between any two points A(a,b) and B(c,d) in the plane is given by,

AB = √((c-a)²+(d-b)²)

Here, it is given that the points of the triangle are R(-4,4) S(-2,-2) T(2, 3).

The triangle is scalene so length of every side is different.

Now, Using the above formula,

RS = √((2+4)²+(-2-4)²)

RS = √(36+36)

RS = √72

ST = √((2+2)²+(3+2)²)

ST = √(16+25)

ST = √41

TR = √((2+4)²+(3-4)²)

TR = √(36+1)

TR = √37

So, Brian is correct, the triangle is a scalene triangle.

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How do you solve basic equations in algebra 1

Answers

Really depends on what you mean by “basic” and “algebraic”. If I take “algebraic” to mean polynomial, and “basic” to mean having one variable and a solution that you can find with standard techniques, then it’s “simple” — for a suitably generous definition of the word.

First, get all your stuff on one side so that you have P(x)=0
P
(
x
)
=
0
for some polynomial P
P
. This is done by subtracting both sides by everything on the right hand side, and combining terms that have the same power of x
x
. For convenience, order your terms so that the biggest powers of x
x
come first, and they get smaller as you go, so that the last term is one without an x
x
, if you have one. If this isn’t a polynomial, then this equations isn’t algebraic, so stop following this guide.

Possible mid-step, if you don’t have a term without an x
x
, then x=0
x
=
0
is a solution, and you can divide by the smallest power of x
x
so that now you should have a term without an x
x
. If there’s no smallest power of x
x
, you started with 0=0
0
=
0
which is always true. If the term without an x
x
is the only one, your equation looks like a=0
a
=
0
for some nonzero a
a
, so you’re out of solutions and can stop here.

Next, figure out the degree of P
P
. This is super easy if you did everything correct up until now, because it will be the power of x
x
on the very first term, which should be the biggest power of x
x
you have, and should be at least 1
1
.

Another possible mid-step, if your degree is more than 4
4
, check if the least common multiple of all the powers of x
x
is greater than 1
1
, and check that the degree divided by the least common multiple is 4
4
or less. If this is the case, make the substitution z=xm
z
=
x
m
where m
m
is the least common multiple. Then continue on to solve the equation for z
z
, and then turn those into solutions for x
x
with x=z√m
x
=
z
m
. If you don’t have a degree 4
4
equation, then you might be out of luck. At this point, I think we can no longer call the equation “basic”, as it might not even have a closed-form solution.

Lastly, at this step you should have a polynomial of degree between 1
1
and 4
4
, in which case you will use the corresponding formula: linear, quadratic, cubic, or quartic. The term “linear formula” isn’t really in wide use, but it simply states that the equation
. The other three can be found easily via the internet. The quadratic formula is fairly manageable, the cubic formula is a bit unwieldy but not too bad, and the quartic formula is a bit of a nightmare but if you’re at this point it’s probably the only way.

So there you go, a more or less complete tutorial on how to solve any “basic” “algebraic” equation (for the definitions given above). There are a couple of heuristic techniques you can also try before giving up, such as factoring by grouping, the rational root theorem, and Descartes rule of signs, but none of these techniques are guaranteed to be of any help. Most polynomials of degree 5 or greater don’t have closed-form solutions, so don’t be discouraged if you can’t find them. At that point, if you really need those solutions, you can try numerical methods, but I’ll leave that out of this answer.

Use the distributive property to simplify each expression.

-2(x - 2) + 5x

please help

Answers

Answer:

3x + 4

Step-by-step explanation:

Using the distributive property simply means that when there is a number in front of a pair of parentheses, every value inside the parentheses is multiplied by the co-efficient to expand the parentheses.

Here, the coefficient is -2, therefore we multiply both terms inside the parentheses (x and -2) by -2 to expand them:

-2(x - 2) + 5x

= (-2)(x) + (-2)(-2) + 5x

Now we simply evaluate and combine like terms to simplify:

= -2x +4 +5x

= 3x + 4

A toaster has 4 slots for bread. Once the toaster is warmed up, it takes 35 seconds to make 4 slices of toast, 70 seconds to make 8 slices, and 105 seconds to make 10 slices. Explain how to predict the time it takes to toast any number of slices of toast.

Answers

It will take the toaster about 175 seconds to make 20 slices.

What is the unitary method?

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.

Given that The toaster takes 35 seconds to make 4 slices of toast.

It takes 70 seconds to make 8 slices of toast, It also takes 105 seconds to make 12 slices.

We are required to calculate how long it may probably take the toaster to make 20 slices.

To calculate the time it takes the toaster to make 1 slice:

4 slices ---------- 35 seconds

1 slice ----------- ? seconds

1/4 × 35 = 35/4 seconds

Now, if 1 slice will be made in 35/4 seconds,

20 slices will be made in:

1 slice --------- 35/4 seconds

20 slices ------ ? seconds

(20/1) × 35/4 = 175 seconds

Hence It will take the toaster about 175 seconds to make 20 slices.

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A tank contains 60 kg of salt and 1000 L of water. A solution of a concentration 0. 03 kg of salt per liter enters a tank at the rate 9 L/min. The solution is mixed and drains from the tank at the same rate. A. ) What is the concentration of our solution in the tank initially?concentration = (kg/L)b. ) Find the amount of salt in the tank after 2. 5 hours. Amount = (kg)c. ) Find the concentration of salt in the solution in the tank as time approaches infinity. Concentration = (kg/L)

Answers

(a) The concentration of our solution in the tank initially is 0.060 kg/L. (b) the amount of salt in the tank after 2. 5 hours is 44.45 kg. (c) The concentration of salt in the solution in the tank as time approaches infinity is 0.030 kg/L.

How do you find the concentration of a salt solution?

Divide the solute's gram weight by the total weight of the solution, then multiply the result by 100 to get the mass/mass percent of the solution. The amount of solute (measured in grams) per milliliter of solution is known as the mass/volume percent.

(A) Tank contains 60 kg of salt and 1000 L of water. So the concentration of solution in the tank initially = 60/1000 kg/l

The concentration of solution in the tank initially = 0.060 kg/L

(B) Let y(t) denote the amount of salt in the tank at any time t.

Rate In

A solution of concentration 0.03 kg of salt per liter enters a tank at the rate 9 L/min.

[tex]R_{i}[/tex]  =(concentration of salt in inflow)(input rate of solution)

= (0.03kg/L)*(9L/min)

= 0.27kg/min

Rate Out

The solution is mixed and drains from the tank at the same rate.

Concentration, C(t) = Amount/Volume = y(t)/1000

[tex]R_{out}[/tex] =(concentration of salt in outflow)(output rate of solution)

= {y(t)/1000}kg/L*9L/min

= 0.009y(t) kg/min

Therefore, the differential equation for the amount of Salt in the Tank  at any time t:

dy/dt = 0.27 - 0.009y(t)

So the amount of salt in the tank after 2. 5 hours (2.5*60 min = 150 min) is  44.45 kg.

(c) The concentration of salt in the solution in the tank as time approaches infinity

As t -> -∞, e^-∞ = 0

so, the concentration of salt in the solution in the tank as time approaches infinity is 0.030 kg/L.

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Jamila deposits $800 in an account that earns yearly simple interest at a rate of 2.65%. How much money is in the account after 3 years and 9 months?

Answers

Answer:

$821.2

Step-by-step explanation:

cause 2.65 percentage of 800 dollars is 800/100=8*2.65=21.2dollars.+800 =$821.2

How many times smaller is 2.7 x 10^3 than 5.481 x 10^5?

Answers

Function: This means that 2.7 x 10^3 is about 201.593 times smaller than 5.481 x 10^5. This can also be expressed as 0.049 or 4.9%.

What is function?

A function is a mathematical relation between two sets of values, usually between an input and an output. It can be thought of as a machine that takes a certain input and produces a certain output.

2.7 x 10^3 is about 0.049 times smaller than 5.481 x 10^5.
To determine this, we can divide the two numbers. 5.481 x 10^5 divided by 2.7 x 10^3 equals about 201.593.
This means that 2.7 x 10^3 is about 201.593 times smaller than 5.481 x 10^5. This can also be expressed as 0.049 or 4.9%.

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. A window is in the form of a rectangle surmounted by a semicir- cle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thick- ness of the frame.

Answers

The area of the window that lets in the most light will be the one that causes the window to transmit the most light overall.

To find the proportions of the window that will admit the lightest, we need to maximize the total area of the window while keeping the perimeter fixed. The total area of the window is the sum of the area of the rectangle and the area of the semicircle. The area of the rectangle is the product of its length and width, and the area of the semicircle is half the product of its radius and the area of a full circle with the same radius.

Since the perimeter is fixed, the length and width of the rectangle are also fixed. The radius of the semicircle can be found using the perimeter and the length and width of the rectangle.

Once we have the radius of the semicircle, we can find the area of the rectangle, the area of the semicircle, and the total area of the window. Since the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does, we will have to multiply the area of the semicircle by 0.5 to find the total amount of light transmitted by it.

Finally, we will have to compare the total amount of light transmitted by the window for different values of the length and width of the rectangle. The proportion of the window that admits the most light will be the one that results in the maximum total amount of light transmitted by the window.

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25PTS 25PTS don’t have to explain

Answers

Answer: y=3x-2

Hope that helps!

Answer:

y = 3x - 2

Step-by-step explanation:

The function f is defined by f(x) = sqrt 25 - x2 for -5\leq x\leq 5. a) Determine the average rate of change of f over the interval --4\leq x\leq 3 b) Find f (prime) (x). c) Write an equation for the line tangent to the graph of f at x= -3. d) Let g be the function defined by g(x) = f(x) for -5\leq x\leq -3 and x+7 for-3\leq x\leq 5. Is g continuous at x= -3? Use the defintion of continuity to explain your answer.

Answers

a) The average rate of change of a function over an interval is the change in the function's output (f(b) - f(a)) divided by the change in the function's input (b - a). So, to find the average rate of change of f over the interval -4 <= x <= 3, we can use the formula:

(f(3) - f(-4)) / (3 - (-4)) = (sqrt(25 - 9) - sqrt(25 - 16)) / (3 - (-4)) = (-2sqrt(7) + 4sqrt(3)) / 7

b) To find the derivative of f(x), we need to use the power rule and the chain rule. The power rule states that the derivative of x^n is nx^(n-1), and the chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).

So, the derivative of f(x) = sqrt 25 - x^2 is:

-2x

c) To find the equation of the line tangent to the graph of f at x = -3, we need to use the point-slope form of a line, which is:

y - y1 = m(x - x1)

where m is the slope of the line (which is f'(-3) = -6), (x1, y1) is the point on the graph where the line touches (which is (-3, f(-3)) = (-3, sqrt(25 - 9) = 2sqrt(7))

so we have: y - 2sqrt(7) = -6(x + 3)

d) To check if g(x) is continuous at x = -3, we need to check if the limit of g(x) as x approaches -3 exists and is equal to g(-3).

Since g(x) = f(x) for -5 <= x <= -3 and g(x) = x + 7 for -3 <= x <= 5, we have:

g(-3) = f(-3) = sqrt(25 - 9) = 2sqrt(7)

and the limit of g(x) as x approaches -3 is:

lim x->-3 g(x) = lim x->-3 (x + 7) = -3 + 7 = 4

Since g(-3) = 2sqrt(7) and lim x->-3 g(x) = 4, g(x) is not continuous at x = -3.

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Select the correct answer.
What is the slope of the line that goes through the points (-1,4) and (14,-2)?

A. -15/6
B. -6/13
C. -5/2
D. -6/15

Answers

Slope = (-2 - 4)/(14 - (-1)) = -6/15

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