find the area of the region that lies inside both r=sin() and r=cos(). hint: the final example on the final video lecture goes through a similar problem.

Okay, let's solve this step-by-step:1) The equations for the two curves are: r = sin() and r = cos()2) We need to find the intersection points of these two curves. This is done by setting them equal and solving for :sin() = cos() => = /4 3) The intersection points are (1, /4) and (1, 3/4). The region lies between = /4 and = 3/4.4) To find the area, we use the formula: A = =3/4 =/4 2r d 5) Substitute r = sin() or r = cos(): A = =3/4 =/4 2sin() d = 2 =3/4 =/4 sin() d 6) Integrate: A = 2(cos() - sin() )|/4 to 3/4 = 2(0 - 1) = 2 7) Therefore, the area of the region is 2 square units.Let me know if you have any other questions!

Mathematics High School

find the area of the region that lies inside both r=sin(θ) and r=cos(θ). hint: the final example on the final video lecture goes through a similar problem.

Answers

Answer 1

Okay, let's solve this step-by-step:

1) The equations for the two curves are:

r = sin(θ) and r = cos(θ)

2) We need to find the intersection points of these two curves. This is done by setting them equal and solving for θ:

sin(θ) = cos(θ)

=> θ = π/4

3) The intersection points are (1, π/4) and (1, 3π/4). The region lies between θ = π/4 and θ = 3π/4.

4) To find the area, we use the formula:

A = ∫θ=3π/4 θ=π/4 2πr dθ

5) Substitute r = sin(θ) or r = cos(θ):

A = ∫θ=3π/4 θ=π/4 2πsin(θ) dθ

= 2π ∫θ=3π/4 θ=π/4 sin(θ) dθ

6) Integrate:

A = 2π(cos(θ) - sin(θ) )|π/4 to 3π/4

= 2π(0 - 1) = 2π

7) Therefore, the area of the region is 2π square units.

Let me know if you have any other questions!

Related Questions

Express tan R as a fraction in simplest terms.

Answers

Answer:

RS = 24, so tan R = 18/24 = 3/4

lizs algebra 1 class is taking a field trip to the cryptology museum. one of the geometry classes is going too. this table shows how many tickets each class bought for the field trip. algebra1: 164$ geometry: 120$. student tickets: 59. adult tickets: 11

Answers

Each Algebra ticket cost $5.125 for kids and $20.50 for adults, while each Geometry ticket cost $4.46 for pupils and $30.125 for adults.

For Algebra I:

Cost per student ticket = Total cost of student tickets / Number of student tickets

Cost per student ticket = $164.00 / 32

Cost per student ticket = $5.125

For Geometry:

Cost per student ticket = Total cost of student tickets / Number of student tickets

Cost per student ticket = $120.50 / 27

Cost per student ticket = $4.46 (rounded to two decimal places)

Next, we can find the cost of one adult ticket for each class using the same method.

For Algebra I:

Cost per adult ticket = Total cost of adult tickets / Number of adult tickets

Cost per adult ticket = $164.00 / 8

Cost per adult ticket = $20.50

For Geometry:

Cost per adult ticket = Total cost of adult tickets / Number of adult tickets

Cost per adult ticket = $120.50 / 4

Cost per adult ticket = $30.125

Therefore, the price of each ticket for Algebra I was $5.125 for students and $20.50 for adults, while the price of each ticket for Geometry was $4.46 for students and $30.125 for adults.

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arrange the following in ascending order 16 upon 22, - 5 upon 18,2 upon - 21 ,- 7 upon 12

Answers

We can convert all the fractions to decimals and then arrange them in ascending order:

- 16/22 ≈ 0.727

- -5/18 ≈ -0.278

- 2/-21 ≈ -0.095

- -7/12 ≈ -0.583

Therefore, the ascending order would be:

2/-21 ≈ -0.095 < -5/18 ≈ -0.278 < -7/12 ≈ -0.583 < 16/22 ≈ 0.727

So the final arrangement in ascending order is:

2/-21, -5/18, -7/12, 16/22

Find a non-zero vector w∈R4 which is orthogonal to all of the following vectors:Enter the vector w in the form [c1,c2,c3,c4]:

Answers

To write the vector w in the form [c1,c2,c3,c4], we can simply list the components of w in order. So our final answer is w = [c1, c2, c3, c4] (where c1, c2, c3, and c4 are the components of the vector w that we found using the cross product).

To find a non-zero vector w∈R4 which is orthogonal to all of the given vectors, we can use the cross product. Let's call the given vectors u1, u2, u3, and u4. Then we can find w as:

w = u1 x u2 x u3 x u4

where "x" represents the cross product. This means we take the cross product of u1 and u2, then take the cross product of that result with u3, and so on until we have taken the cross product of all four vectors.

The resulting vector w will be orthogonal to all of the given vectors, since the cross product of two vectors is always orthogonal to both of the original vectors.

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A sphere has a volume of 65.5 cubic inches. What is the diameter of the
sphere, to the nearest tenth of an inch?

Answers

Answer:

5.0 inches

Step-by-step explanation:

The formula for the volume of a sphere is:

[tex]\boxed{V=\dfrac{4}{3}\pi r^3}[/tex]

where r is the radius of the sphere.

Given a sphere has a volume of 65.5 cubic inches, substitute V = 65.5 into the formula and solve for the radius, r:

[tex]\begin{aligned}\implies \dfrac{4}{3}\pi r^3&=65.5\\\\3 \cdot \dfrac{4}{3}\pi r^3&=3 \cdot 65.5\\\\4\pi r^3&=196.5\\\\\dfrac{4\pi r^3}{4 \pi}&=\dfrac{196.5}{4 \pi}\\\\r^3&=15.636973...\\\\\sqrt[3]{r^3}&=\sqrt[3]{15.636973...}\\\\r&=2.50063840...\; \sf in\end{aligned}[/tex]

The diameter of a sphere is twice its radius.

Therefore, if the radius is 2.50063840... inches, then the diameter is:

[tex]\begin{aligned}\implies d&=2r\\&=2 \cdot 2.50063840...\\&=5.00127681...\\&=5.0\; \sf in\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, the diameter of a sphere with a volume of 65.5 cubic inches is 5.0 inches, to the nearest tenth of an inch.

Answer:

5 cm

Step-by-step explanation:

The formula to find the volume of a sphere is:

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

Here,

V ⇒ volume ⇒ 65.5 cm³

r ⇒ radius

Let us find the value of r.

[tex]\sf V =\frac{4}{3} \pi r^3\\\\65.5=\frac{4}{3} \pi r^3\\\\65.5*3=4 \pi r^3\\\\196.5=4 \pi r^3\\\\\frac{196.5}{4} =\pi r ^3\\\\49.125=\pi r^3\\\\\frac{49.125}{\pi} = r^3\\\\15.63=r^3\\\\\sqrt[3]{15.63} =r\\\\2.5=r[/tex]

Let us find the diameter now.

d = 2r

d = 2 × 2.5

d = 5 cm

Which of these does NOT represent the distance a car travels when going 55 miles per hour?
A d=55c, where d represents distance in miles and t represents time in hours
B
D
Car Travel
Time
(hours)
1
1.5
2
2.5
Distance
(miles)
C In 3 hours a car will travel a distance of 160 miles.
200
150
100
50
0
Distance
(miles)
55
82.5
Your answer
110
137.5
1
Car Travel
2
3
Time
(hours)
4

Answers

The statement which does not represent the distance is

C) In 3 hours car will travel the distance of 160 miles and D).

What is proportion?

A percentage is created when two ratios are equal to one another. We write proportions to construct equivalent ratios and to resolve unclear values.

Here the car can travel 55 miles in one hour.

Then in 1.5 hour distance traveled by car is x.

Using proportion,

=> x = 55*1.5 = 82.5 miles

Now in 2 hours distance traveled by car = 55*2=110 miles

In 2.5 hours distance traveled by car = 55*2.5 = 137.5 miles

In 3 hours distance traveled by car = 55*3 = 165 miles.

Then distance = 55t . where t = time

Hence the statement which does not represent the distance is

C) In 3 hours car will travel the distance of 160 miles and D).

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find the exact length of the curve x = 6 + 3t^2 ,y = 6 + 2t^3 for 0 ≤ t ≤ 4

Answers

To find the length of the curve, we need to use the formula:

length = ∫[a,b] √[dx/dt)^2 + (dy/dt)^2] dt

In this case, a=0, b=4, and:

dx/dt = 6t
dy/dt = 6t^2

So, we can plug these values into the formula and integrate:

length = ∫[0,4] √[(6t)^2 + (6t^2)^2] dt
length = ∫[0,4] √[36t^2 + 36t^4] dt
length = ∫[0,4] 6t√(1 + t^2) dt

This integral is not easy to solve analytically, so we'll use numerical methods to approximate the answer. Using a numerical integration method such as Simpson's Rule or the Trapezoidal Rule, we can get:

length ≈ 244.36

So the exact length of the curve x = 6 + 3t^2, y = 6 + 2t^3 for 0 ≤ t ≤ 4 is approximately 244.36 units.
To find the exact length of the curve x = 6 + 3t^2, y = 6 + 2t^3 for 0 ≤ t ≤ 4, you can use the arc length formula:

Length = ∫[√(dx/dt)^2 + (dy/dt)^2] dt from t=0 to t=4

First, find the derivatives dx/dt and dy/dt:
dx/dt = 6t
dy/dt = 6t^2

Now, square the derivatives and find their sum:
(6t)^2 + (6t^2)^2 = 36t^2 + 36t^4

Take the square root of the sum:
√(36t^2 + 36t^4)

Now, integrate the expression with respect to t from 0 to 4:
Length = ∫[√(36t^2 + 36t^4)] dt from t=0 to t=4

This integral is not easy to evaluate directly, and numerical methods are usually required. To obtain an approximate value, you can use an appropriate numerical integration technique, like Simpson's Rule or a computer algebra system.

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Explanation needed aswell please

Answers

The image is plotted and attached

Description of the plot

The rectangle started with ABCD. Then following the reflection along line AC. point B and point D swapped so we have B' replacing D and D' replacing B.

180 degrees rotation through C, resulted to B'' D'' and A'. Point C maintains it's position since the rotation is about point C.

A' replacing AB'' replacing B'D'' replacing D'

Enlargement by a factor of 2 results to C' B''' D''' A'' and this is the final image.

While the reflection and rotation preserves the geometry, the enlargement affects the geometry, producing a rectangle with a bigger size twice the initial size

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Bella qualifies for $9,750 in scholarships and grants per year, and she will earn $3,100
through the work-study program.

Answers

a. Bella should estimate her cost per year to be $17,150.

b. It looks like Bella's family is contributing more than her estimated cost per year, so she may not need to contribute anything.

What is subtraction?

The act of deleting items from a collection is represented by subtraction. Subtraction is denoted by the minus sign. For instance, suppose there are nine oranges stacked. If four oranges are then transferred to a basket, there will now be nine oranges left in the stack (9 – 4).

a. To estimate Bella's cost per year, we need to subtract her financial aid from the total cost of attendance. Let's assume the total cost of attendance is $30,000 per year. Then, Bella's estimated cost per year would be:

Total cost of attendance - Financial aid = Estimated cost per year

$30,000 - $9,750 - $3,100 = $17,150

Therefore, Bella should estimate her cost per year to be $17,150.

b. If Bella's family is contributing $20,000 towards her expenses each year, she needs to contribute the remaining amount. To calculate how much she needs to contribute each year, we can subtract her family's contribution from her estimated cost per year:

Estimated cost per year - Family contribution = Bella's contribution

$17,150 - $20,000 = -$2,850

It looks like Bella's family is contributing more than her estimated cost per year, so she may not need to contribute anything. However, it's important to keep in mind that these are just estimates and actual costs may vary.

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is a continuous uniform (−9,9) random variable. define the event ={||≤7}. (a) what is the conditional pdf?

Answers

(a) The conditional PDF is f_X(x|A) = 1/14 for -7 ≤ x ≤ 7 and 0 otherwise.

A continuous uniform random variable defined on the interval (-9, 9) and the event A = {X: |X| ≤ 7}.

The event A implies that the random variable X lies in the interval [-7, 7]. In this interval, X is still uniformly distributed.

Since the length of the interval is 14, the PDF is 1/14, indicating a constant probability density within the given interval. Outside of this interval, the probability density is 0, as the event A does not cover those values of X.

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Find the exact value of the expression:
sin ( cos^-1 (1/2) + tan^-1 (1) )

Answers

Answer:

Step-by-step explanation:

The solution to the trigonometric expression is:

sin ( cos^-1 (1/2) + tan^-1 (1) )

= sin (60° + 45°) (since cos^-1 (1/2) = 60° and tan^-1 (1) = 45°)

= sin 105°

= 0.966

pls see the question in attachment and solve it

Answers

Answer:

42°

Step-by-step explanation:

Measured with potractor

Write the example of non polyhydrons the length of a fields and a rectangular ,cone,sphere,semi- circle

Answers

The example of non polyhydrons the length of a fields are equals to the cone and sphere. So, option (b) and (c) is rigth choices for answering this problem.

The solid objects which have faces (flat faces) are called polyhedra (singular is polyhedron) and the solid objects which have curved faces are called non-polyhedra. Some examples of non-polyhedra are sphere, cylinder, cone . A sphere is not a polyhedron because it is not composed of flat faces connected at straight edges, thus it does not form a shape. Cone is not a polyhedron because it has a curved surface. Rectangle is a polyhedron because it has a curved shape.

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Complete question:

Write the example of non polyhydrons the length of a fields and

a) a rectangular

b) cone

c) sphere

d) semi- circle

determine the value of 'c' for which the following system of equations have infinite number of solutions. 3x y 4z = 'c' 2x 3z = 2.7 2y - z = 12.0

Answers

X can be any real number, there are infinite values of c that will make the system consistent and dependent.

To have an infinite number of solutions, the system must be consistent and dependent. Thus, we need to find the value of 'c' for which the third equation is a linear combination of the first two equations.

Multiplying the second equation by 3 and adding it to the third equation, we get:

2y - z + 3(2x + 3z) = 12.0 + 3(2.7)

Simplifying, we get:

2y - z + 6x + 9z = 20.1

6x + y + 13z = 20.1

Now we have a system of two equations with three variables. To have an infinite number of solutions, one of the variables must be a free variable. Let's solve for z:

z = (20.1 - 6x - y) / 13

Now we can substitute this expression for z into the first two equations:

3x + y + 4[(20.1 - 6x - y) / 13] = c

2x + 3[(20.1 - 6x - y) / 13] = 2.7

Simplifying, we get:

39x + 13y = 52c - 321.6

39x - 6y = 41.7

To have an infinite number of solutions, the two equations must be linearly dependent. We can multiply the second equation by 13 and add it to the first equation to eliminate y:

754x = 52c - 525.9

Solving for c, we get:

c = (754x + 525.9) / 52

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Please help to find the answer to a b and c please

Answers

Answer:

a) 5p (or 5q)

b) 5p ( or 5q)

c) 10p (or 10q)

Step-by-step explanation:

We will use the following information for solving and using the given figure

In a regular hexagon all six sides are equal
Therefore
[tex]\overrightarrow{AB} = \overrightarrow{BC} = \overrightarrow{CD} = \overrightarrow{DE} = \overrightarrow{EF} = \overrightarrow{FA}[/tex]
The circumradius R which is the length of the segment from O to the center of the hexagon to any of the vertices A, B, C, D, E, F is the exact same. So in the figure,
[tex]\overrightarrow{OA} = \overrightarrow{OB} = \overrightarrow{OC} = \overrightarrow{OD} = \overrightarrow{OE} = \overrightarrow{OF}[/tex]
The distance between a pair of opposite vertices is twice the circumradius and is the same for all
[tex]\overrightarrow{EB} = 2\overrightarrow{OB}[/tex] and similarly for all other pairs

Preliminary computation

We have [tex]\overrightarrow{AB} = \overrightarrow{BC}[/tex]

Therefore 4p + q = 5p

5p = 4p + q

5p-4p = q

p = q

[tex]\overrightarrow{AB} = 4p + q = 4p + p = 5p = 5q\\[/tex]

So each side is 5p in length which is also equal to 5q since p = q

Part a

[tex]\overrightarrow{AO} = \overrightarrow{OA} = \overrightarrow{AB} = 5p[/tex] (same as 5q)

Part b

[tex]\overrightarrow{OB} = \overrightarrow{OA} = 5p[/tex] (same as 5q)

Part c
[tex]\overrightarrow{EB} = 2 \cdot \overrightarrow{OB} = 2 \cdot 5p = 10p[/tex] (also 10q)

A baker has 20 eggs and 18 cups of flour.

One batch of chocolate chip cookies requires 4 eggs and 3 cups of flour.
One batch of oatmeal raisin cookies requires 2 eggs and 3 cups of flour.
The baker makes $5 profit for each batch of chocolate chip cookies and $3 profit for each batch of oatmeal raisin cookies.

How many batches of each type of cookie should she make to maximize profit?

Answers

Answer:

Step-by-step explanation:

Note that maximum profit is $26. This point is obtained when the baker has made 4 batches of chocolate chip cookies and 2 bactches of oatmean raisons cookies.

How did we arrive at the above?

Lets define x as the number of batches of chocolate chip cokies

We want to maximize profit, which is given by:

P = 5x + 3y

subject to the constraints:

4x + 2y ≤ 20 (egg constraint)

3x + 3y ≤ 18 (flour constraint)

x, y ≥ 0 (non-negativity constraint)

We can rewrite the constraints as:

2x + y ≤ 10

x + y ≤ 6

Graphing these constraints on a coordinate plane, we see that the feasible region is a triangle with vertices at (0,0), (0,6), and (4,2)

See agraph attached.

We want to find the point (x,y) within this region that maximizes P.

One way to do this is to calculate P at each vertex of the feasible region:

P( 0,0) = 0

P (0, 6) = 3(6) = 18

P (4,2) =
5(4) + 3(2) =

So the point of profit maximization is at $ 26.

Thica can happen when the baker is baking 4 batches of chocolate chip cookies and 2 batches of oatmeal raisin cookies.

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find the orthogonal trajectories of the family of curves. (use c for any needed constant.) y = 17 k x

Answers

To find the orthogonal trajectories of the family of curves y = 17kx, we need to follow some steps including differentiation or reciprocal etc.

Steps are:
Step 1: Differentiate the given equation with respect to x.
Differentiating both sides with respect to x, we have:
dy/dx = 17k

Step 2: Find the negative reciprocal of dy/dx.
The negative reciprocal of dy/dx is the slope of the curve orthogonal to the given family of curves: -(dx/dy) = -1/(17k)

Step 3: Solve the new differential equation.
Now, we need to solve the new differential equation: dx/dy = 1/(17k)

Notice that k is a constant, so let's rewrite the equation as: dx/dy = C, where C = 1/(17k)

Step 4: Integrate both sides of the equation.
Integrate dx/dy with respect to y:
∫(1) dx = ∫(C) dy

x = Cy + D, where D is the constant of integration.

Step 5: Express the equation in terms of y and x.
Now, substitute the original equation y = 17kx back into our orthogonal trajectory equation:

y = (1/C)(x - D)

This is the equation for the orthogonal trajectories of the family of curves y = 17kx, where C and D are constants related to k.

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1. for x = 1001 1010 0011 1101, show the result of the following operations. a) shl(x) b) shr(x) c) cil(x) d) cir(x) e) ashl(x) f) ashr(x) g) dshl(x) h) dshr(x)

Answers

The following parts can be answered by the concept of Operations.

a) shl(x): The result of left shifting x by 1 bit is 0010 0100 0111 1010.

b) shr(x): The result of right shifting x by 1 bit is 0100 1100 0111 1011.

c) cil(x): The result of circularly left shifting x by 1 bit is 0010 0100 0111 1010.

d) cir(x): The result of circularly right shifting x by 1 bit is 1100 1101 1001 1110.

e) ashl(x): The result of arithmetic left shifting x by 1 bit is 0010 0100 0111 1010.

f) ashr(x): The result of arithmetic right shifting x by 1 bit is 1100 1101 1001 1110.

g) dshl(x): The result of double precision left shifting x by 1 bit is 0010 0100 0111 1010 0000.

h) dshr(x): The result of double precision right shifting x by 1 bit is 0000 1001 0100 1000 1111.

a) shl(x): Left shifting x by 1 bit means shifting all the bits in x to the left by 1 position. The leftmost bit is lost, and a 0 is shifted in from the right. Therefore, the result is 0010 0100 0111 1010.

b) shr(x): Right shifting x by 1 bit means shifting all the bits in x to the right by 1 position. The rightmost bit is lost, and a 0 is shifted in from the left. Therefore, the result is 0100 1100 0111 1011.

c) cil(x): Circularly left shifting x by 1 bit means shifting all the bits in x to the left by 1 position, and the leftmost bit is shifted to the rightmost position. Therefore, the result is 0010 0100 0111 1010.

d) cir(x): Circularly right shifting x by 1 bit means shifting all the bits in x to the right by 1 position, and the rightmost bit is shifted to the leftmost position. Therefore, the result is 1100 1101 1001 1110.

e) ashl(x): Arithmetic left shifting x by 1 bit is similar to logical left shifting, except that the sign bit (the leftmost bit) is preserved. Therefore, the result is 0010 0100 0111 1010.

f) ashr(x): Arithmetic right shifting x by 1 bit is similar to logical right shifting, except that the sign bit (the leftmost bit) is preserved. Therefore, the result is 1100 1101 1001 1110.

g) dshl(x): Double precision left shifting x by 1 bit means shifting all the bits in x, including the sign bit, to the left by 1 position. The leftmost bit is lost, and a 0 is shifted in from the right. Therefore, the result is 0010 0100 0111 1010 0000.

h) dshr(x): Double precision right shifting x by 1 bit means shifting all the bits in x, including the sign bit, to the right by 1 position. The rightmost bit is lost, and the sign bit is duplicated to fill the leftmost bit positions. Therefore, the result is 0000 1001 0100 1000 1111

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