The area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
How to find the area of the region that lies inside both curves?Since the curves are
r² = 2sin(2θ) and r = 1.We find their point of intersection.
So, r² = r²
2sin(2θ) = 1²
2sin(2θ) = 1
sin(2θ) = 1/2
2θ = sin⁻¹(1/2)
2θ = π/6
θ = π/12
So, we integrate the area from θ = 0 to θ = π/12
Now the area A of the region between two curves between θ = α to θ = β is
[tex]A = \int\limits^{\beta }_\alpha ({r^{2} - r^{'2} }) \, d\theta[/tex]
So, the area betwwen the curves r² = 2sin(2θ), r = 1 between θ = 0 to θ = π/12 is
[tex]A = \int\limits^{\frac{\pi}{12} }_0 ({r^{2} - r^{'2} }) \, d\theta \\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1^{2} }) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 ({2sin2\theta - 1}) \, d\theta\\= \int\limits^{\frac{\pi}{12} }_0 {2sin2\theta}\, d\theta - \int\limits^{\frac{\pi}{12} }_0 {1} \, d\theta\\= [2(-cos2\theta)/2 - \theta]_{0}^{\frac{\pi}{12} } \\= [-cos2\theta - \theta]_{0}^{\frac{\pi}{12} } \\= -cos2\frac{\pi}{12} - \frac{\pi}{12} - ( -cos2(0) - 0)\\[/tex]
[tex]= -cos2(\frac{\pi}{12}) - \frac{\pi}{12} - ( -cos2(0) - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -cos0 - 0)\\= -cos\frac{\pi}{6}- \frac{\pi}{12} - ( -1)\\= -(\frac{\sqrt{3} }{2} ) - \frac{\pi}{12} + 1\\= -0.8660 - 0.2618 + 1\\= -1.1278 + 1\\= -0.1278[/tex]
So, the area of the region that lies inside both curves r² = 2sin(2θ), r = 1 is -0.1287 square units.
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If you randomly selected a British Columbia ski resort, what is the probability that it will have an annual snowfall greater than 170 inches?
The probability that it will have an annual snowfall greater than 170 inches is 0.4.
How to compute the probability?The information is incomplete and the complete question wasn't found. An overview will be given. It should be noted that a probability means the likelihood of the occurence of an event.
Probability is calculated as:
= Number of ways of achieving success/Total number of possible outcomes
Let's assume that the total number of resort is 10 and those that have more than 170 inches are 4.
In this case, the probability will be:
= 4/10
= 0.4
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Which choice is equivalent to the product below?
√14•√10
A. 2√35
B. 4√7
C. 35
D. 4√35
A school play sold 575 tickets total They were either adult or student tickets there were 75 fewer student tickets sold than adult tickets how many adult tickets were sold
Answer:
325 adult tickets
Step-by-step explanation:
Let adult tickets be x, student tickets are 75 less than adult tickets, so let student tickets be x-75
The total is 575
x+x-75 = 575
2x = 650
x = 325 adult tickets
The princess of the Kingdom of Abundance accidentally drops her magic salamander's feeding bowl into a fire. She orders a new feeding bowl to be made out of Dragon Alloy #18, which is 86% magic steel and the rest titanium. The
dwarfs have a lot of Dragon Alloy #33, which is 28% titanium. How much of
that alloy and how much magic steel should they combine to make 700 grams of Dragon alloy #18?
350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the amount of magic steel and y represent the amount of alloy #33, hence:
(0 * x) + (y * 0.28) = 700(1 - 0.86) (1)
Also:
x + (1 - 0.28)y = 0.86(700) (2)
From both equations:
x = 350, y = 350
350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.
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Kylie has a modern quarter with a mass of 5.623 g and an older silver quarter with a mass of 6.24 g What is the combined mass of the quarters?
Answer:
11.863 g
Step-by-step explanation:
5.623 g + 6.24 g = 11.863 g
I need help with this geometry question asap!
Answer:
A, D
Step-by-step explanation:
Lines are parallel if consecutive interior angles at a transversal are supplementary.
ApplicationIn the given figure, angles EAH and AED are consecutive interior angles associated with transversal AE. If they total 180°, then AC║EF.
Similarly, angles FCI and CFG are consecutive interior angles associated with transversal CF. If they total 180°, then AC║EF.
Either of these will prove AC║EF:
∠EAH +∠AED = 180°∠FCI +∠CFG = 180°Part 1:
A. Express the new length of the long side of the note card, once the two corners are removed.
B. Express the new width of the short side of the note card, once the two corners are removed.
Part 2:
Write a function A(x) that defines the area of the bottom of the box, once the corners are removed and the sides are folded up.
Part 3:
A. Suppose you want the bottom of your box to cover a total area of 16 in2. Set up an equation in standard form that will help you find the size (x) of the corner you need to cut in order for your box to have this area.
B. Solve this equation and take note of any extraneous solutions. Explain why the answer is extraneous, and clearly state the correct answer
The size of the corner you need to cut in order for your box to have this area is 0.5 inches
Part 1: A. Express the new length of the long side of the note card, once the two corners are removed.The base length is given as:
Length = 7
When the edges are removed, the new length becomes
New length = 7 - x - x
Evaluate
New length = 7 - 2x
Part 1: B. Express the new width of the short side of the note card, once the two corners are removed.The base width is given as:
Width = 4
When the edges are removed, the new length becomes
New width = 4 - x - x
Evaluate
New width = 4 - 2x
Part 2: Write a function A(x) that defines the area of the bottom of the box, once the corners are removed and the sides are folded up.The area of the bottom of the box is calculated as:
Area = New length * New width
This gives
Area = (7 - 2x) * (4 - 2x)
Rewrite as:
A(x) = (7 - 2x) * (4 - 2x)
Part 3: Set up an equation in standard form that will help you find the size (x)The area is given as:
Area = 16
So, we have:
(7 - 2x) * (4 - 2x) = 16
Expand
28 - 14x - 8x + 4x^2 = 16
Rewrite as
4x^2 - 14x - 8x + 28 - 16 = 0
Evaluate the like terms
4x^2 - 22x + 12= 0
Part 3: Solve this equation and take note of any extraneous solutionsWe have:
4x^2 - 22x + 12= 0
Expand the equation
4x^2 - 24x - 2x + 12 = 0
Factorize the equation
4x(x - 6) - 2(x - 6) = 0
Factor out x - 6
(4x - 2)(x - 6) = 0
Solve for x
x = 0.5 or x = 6
The value of x = 6 is too big for the dimensions of the box.
So, x = 6 is an extraneous solution for the equation
Hence, the size of the corner you need to cut in order for your box to have this area is 0.5 inches
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A(4,3) B(1/2, 1/5) C( 3/4, 7/4). Which point lies on the circumstance of the unit circle?
None of the three points (A, B, C) lies on the circumference of the unit circle.
What point is in the circumference of an unit circle?
Unit circles are circles centered at the origin and with a radius of 1. A point is on the circumference if and only if the distance of the point respect to the origin is equal to 1. The distance of each point is determine by Pythagorean theorem:
Point A
[tex]d = \sqrt{4^{2}+3^{2}}[/tex]
d = 5
Point B
[tex]d = \sqrt{\left(\frac{1}{2} \right)^{2}+\left(\frac{1}{5}\right)^{2} }[/tex]
d = √29 /10
d ≈ 0.539
Point C
[tex]d = \sqrt{\left(\frac{3}{4} \right)^{2}+\left(\frac{7}{4} \right)^{2}}[/tex]
d = √58 /4
d ≈ 1.904
None of the three points (A, B, C) lies on the circumference of the unit circle.
RemarkThe statement presents a typing mistake, correct form is shown below:
A(x, y) = (4, 3), B(x, y) = (1/2, 1/5), C(x, y) = (3/4, 7/4). Which point lies on the circumference of the unit circle?
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Find the vertex of the quadratic function.
ƒ(x) = 2(x−1)² +3
a. (1,3)
b. (2, -1)
c. (-1,3)
d. (2,3)
hi,
the function is given with it's canonic form.
canonic form is : f(x) = a ( x-α)² + β
α and β are the value of the coordonnates of the vertex.
so here we have : ƒ(x) = 2(x−1)² +3
with α = 1 and β = 3
so vertex is V (1;3)
So yes, answer is A.
Perform the indicated operation 7 1/8×5 2/3
Answer:
40 3/8
Step-by-step explanation:
You can just multiple it on calculator, which 1/8 is 0.125 so convert 40.375 to 40 3/8
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What is the solution to the inequality -6+l2p+3l>7?
Answer: A
Step-by-step explanation: add 6 on each side you get 2p+3>13 then subtract 3 you get
2p> 10 divide by 2 you get p>5
Find the eighth term of the
geometric sequence, given the
first term and common ratio.
a1=6 and r=-1/3
Answer:
7
Step-by-step explanation:
haha
for
for
haha
but
birthday
can
7 answer
Answer:
the answer is -2/729
Step-by-step explanation:
When you're given the first term and the common ratio, you just need to multiply the first term by the common ratio and then raise to the power of the nth term minus 1.
In this case, 6(-1/3)^7
Harry is building a decking area in his garden.
250 cm
150 cm
4 m
7m
150 cm
He needs to build a wooden frame in the shape of the decking.
Calculate the perimeter of the decking area.
The perimeter of the decking area is 800cm.
How to determine the perimeterIt is important to note that the decking area takes the shape of a rectangle
The formula for determining the perimeter of the decking area is given as;
Perimeter = 2 multiplied by the sum of the length and width of the rectangle
It is mathematically written as;
Perimeter = 2 ( length + width)
Where;
length = 250cmwidth = 150 cmNow, let's substitute the values into the formula
Perimeter = 2 ( 150 + 250)
Sum the bracket
Perimeter = 2 ( 400 )
Perimeter = 2 × 400
Perimeter = 800 cm
Thus, the perimeter of the decking area is 800cm
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The function h(x) = 2|x] is a transformation of the absolute value parent
function, f(x) = |xl. Which graph shows h(x)?
The graph is shown in the attached image.
Give the name (monomial,
binomial, trinomial etc.) and
degree of the polynomial.
√29x6
Name? [? ]
Degree? [
find the lcm of 48,64,72,96,108
Please help me fast i will give you50 pts
2|48,64,72,96,108
2|24,32,36,48,54
2|12,16,18,24,27
2|6,8,9,12,27
2|3,4,9,6,27
2|3,2,9,3,27
3|3,1,9,3,27
3|1,1,3,1,9
3|1,1,1,1,3
LCM
2⁶×3⁴64×815184Select the correct triangle. Which triangle represents a reflection of triangle Aacross line m?
3 + b/d
HELP !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Fractions are mathematical expressions that relate two numbers as a quotient. Thus the answer to this question is: [tex]\frac{3d + b}{d}[/tex]
Fractions are expressions in a form of a quotient that relates one number to another number. The two numbers can be classified as either the numerator or the denominator. The number above the division line is the numerator, while that below the line is the denominator. Some types of fractions are proper fractions, improper fractions, and mixed fractions.
A given fraction can undergo basic arithmetic operations when two or more are involved. Also, a fraction can be added to one another, or subtracted from a whole number.
The given expression in the question involves the addition of a whole number (3), to a fraction (b/d).
Given: 3 + b/d
The LCM is d, thus;
3 + b/d = [tex]\frac{3d + b}{d}[/tex]
Therefore, the required answer is: [tex]\frac{3d + b}{d}[/tex]
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give at least five problems solving about area of a sector of a circle.
five example of arc length
fixe example or arc of a circle
See below for the examples of sectors and arcs of a circle
Area of sector of a circleThe area of a sector is calculated as:
[tex]A = \frac{\theta}{360} * \pi r^2[/tex] ---- when the angle is in degrees
[tex]A = \frac{\theta}{2} *r^2[/tex] ---- when the angle is in radians
Take for instance, we have the following problems involving sector areas
Calculate the area of a sector where the radius of the circle is 7, and
The central angle is 30 degreesThe central angle is π/12 radThe central angle is 90 degreesThe central angle is π/4 radThe central angle is 180 degreesUsing the above formulas, the sector areas are:
1. [tex]A = \frac{30}{360}* \frac{22}{7} * 7^2 = 12.83[/tex]
2. [tex]A = \frac{\pi}{12} * 7^2 = 12.83[/tex]
3. [tex]A = \frac{90}{360}* \frac{22}{7} * 7^2 = 38.5[/tex]
4. [tex]A = \frac{\pi}{2} * 7^2 = 38.5[/tex]
5. [tex]A = \frac{180}{360}* \frac{22}{7} * 7^2 = 77[/tex]
Examples of arc lengthThe length of an arc is calculated as:
[tex]L= \frac{\theta}{360} * 2\pi r[/tex] ---- when the angle is in degrees
[tex]L = r\theta[/tex] ---- when the angle is in radians
Take for instance, we have the following problems involving arc lengths
Calculate the length of an arc where the radius of the circle is 7, and
The central angle is 30 degreesThe central angle is π/12 radThe central angle is 90 degreesThe central angle is π/4 radThe central angle is 180 degreesUsing the above formulas, the arc lengths are:
1. [tex]L = \frac{30}{360}* 2 * \frac{22}{7} * 7 = 3.7[/tex]
2. [tex]L = \frac{\pi}{12} * 7 = 3.7[/tex]
3. [tex]L = \frac{90}{360}*2 * \frac{22}{7} * 7 = 11.0[/tex]
4. [tex]L = \frac{\pi}{4} * 7 = 11[/tex]
5. [tex]L = \frac{180}{360}*2 * \frac{22}{7} * 7 = 22[/tex]
Examples of the arcs of a circleThe examples include:
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A committee of 6 members is to be selected from 8 senior and 10 junior. find how many ways can this be done if there is no restriction
Answer:
18564 ways
Step-by-step explanation:
The number of committees is equal to the number of combinations of 6 members that we can form out of 18 person :
then
This can be done in [tex]C^{6}_{18}[/tex] ways.
[tex]C^{6}_{18}=\frac{18!}{6!\left(18-6 \right)! } = 18564[/tex]
I'm have some issues with the problem shown in the screenshot and would love some help.
The dimensions and volume of the largest box formed by the 18 in. by 35 in. cardboard are;
Width ≈ 8.89 in., length ≈ 24.89 in., height ≈ 4.55 in.Maximum volume of the box is approximately 1048.6 in.³How can the dimensions and volume of the box be calculated?The given dimensions of the cardboard are;
Width = 18 inches
Length = 35 inches
Let x represent the side lengths of the cut squares, we have;
Width of the box formed = 18 - 2•x
Length of the box = 35 - 2•x
Height of the box = x
Volume, V, of the box is therefore;
V = (18 - 2•x) × (35 - 2•x) × x = 4•x³ - 106•x² + 630•x
By differentiation, at the extreme locations, we have;
[tex] \frac{d V }{dx} = \frac{d( 4 \cdot \: {x}^{3} - 106 \cdot \: {x}^{2} + 630\cdot \: {x} ) }{dx} = 0[/tex]
Which gives;
[tex] \frac{d V }{dx} =12\cdot \: {x}^{2} - 212\cdot \: {x} + 630 = 0[/tex]
6•x² - 106•x + 315 = 0
[tex]x = \frac{ - 6 \pm \sqrt{106 ^2 - 4 \times 6 \times 315} }{2 \times 6} [/tex]
Therefore;
x ≈ 4.55, or x ≈ -5.55
When x ≈ 4.55, we have;
V = 4•x³ - 106•x² + 630•x
Which gives;
V ≈ 1048.6
When x ≈ -5.55, we have;
V ≈ -7450.8
The dimensions of the box that gives the maximum volume are therefore;
Width ≈ 18 - 2×4.55 in. = 8.89 in. Length of the box ≈ 35 - 2×4.55 in. = 24.89 in. Height = x ≈ 4.55 in.The maximum volume of the box, V ≈ 1048.6 in.³Learn more about differentiation and integration here:
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16
A Ferris wheel with a radius of 25 m makes 2 rotations every minute.
a. State the period of the Ferris wheel.
b. Complete a tables of values showing the height above the ground for 2 rotations of
Ferris wheel by assigning the correct variables
c. Graph 2 rotations of the Ferris wheel. (Hint: Time Vs Height above the platform)
d. State the amplitude of the Ferris wheel.
The period of this Ferris wheel can be calculated as 30 seconds. The amplitude is 25
How to solve for the period.a. We have 60 seconds in a minute
In a minute it makes 2 rotations. The period = 60 secs / 2 rotations
= 30 seconds
b. We have h = 25 - 25cos [tex]\frac{25}{30} t[/tex]
25 - 25cos[tex]\frac{25}{15} t[/tex]
We would have the table in the attachment
d. Amplitude
= max - min / 2
= 50 - 0 / 2
= 25
can someone please help me with this
What is the volume of a solid with dimensions 5 inches, 3 inches, and 4 inches? 15 cubic inches 30 cubic inches 45 cubic inches 60 cubic inches
Answer:
60 cubic inches
Step-by-step explanation:
You multiply 3 and 4 to get 12, then 12 and 5 to get 60
3 x 4 x 5 = 60
The volume of a solid exists the measurement of how much area an object brings up. It exists measured by the number of unit cubes it brings to fill up the solid.
Volume of a solid = 5 [tex]*[/tex] 3 [tex]*[/tex] 4 = 60 cubic inches
Therefore, the correct answer is option d) 60 cubic inches.
What is the volume in dimension?Volume exists as a three-dimensional measurement; that exists, it measures the collective length, width, and height of a figure. Because planes, polygons, and other two-dimensional figures don't contain height, they contain no volume. Only solids and figures with dimensions more significant than or equivalent to three have volume.
What is the volume of solid?The volume of a solid exists as the measurement of how much space an object brings up. It exists estimated by the number of unit cubes it brings to fill up the solid.
Volume of a solid = 5 [tex]*[/tex] 3 [tex]*[/tex] 4 = 60 cubic inches
Therefore, the correct answer is option d) 60 cubic inches.
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half a number is 21 less than double the number find the number.
Answer:
14
Step-by-step explanation:
let the number be x
2x is larger than half of x by 21: 2x = 0.5+21
Solve:
2x = 0.5+21
1.5x = 21
x = 14
What are the exact values of a and b?
Right triangle A B C is shown with labeled sides and one labeled angle. Side A B is the hypotenuse and it is labeled 10. Side B C is the height and it is labeled a. Side C A is the base and it is labeled b. The angle opposite side B C is 30 degrees.
Answer:
a = 5b = 5√3Step-by-step explanation:
A right triangle with one acute angle measuring 30° is the 30°-60°-90° "special" triangle. The ratios of its side lengths are well known to be ...
1 : √3 : 2
Side lengthsIn the given triangle, the longest side is labeled 10, so is 5 times the measure of the longest side in the ratios given above. Multiplying those ratio values by 5, we have side lengths ...
a : b : c = 5 : 5√3 : 10
That is ...
a = 5
b = 5√3
__
Additional comment
If you want to figure the side lengths using trigonometry, you have ...
a = c·sin(30°) = 10·(1/2) = 5
b = c·cos(30°) = 10·(√3/2) = 5√3
Fill in the blank
Which form of compensation is generally paid on a weekly or biweekly basis?
______ are paid weekly or biweekly, while salaries are paid bimonthly or biweekly.
The blank space in the task content should be filled with; Wages.
What type of employee compensation is Wage?A wage by it's primitive definition is a fixed regular payment usually paid on a daily or weekly basis by the employer to an employee which could be skilled or unskilled.
Commission, and salary which is a monthly analogue of wages is another form of employee compensation.
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Use an outside source to search for a quadratic equation that models something from your daily life. (Be sure to cite the source you use.)
Solve the equation in two ways.
Discuss which method you liked better and why.
In your responses to peers, compare and contrast your preferences for how to solve quadratic equations.
A quadratic equation that models something from one's daily life is when calculating areas, and determining a product's profit.
What is a quadratic equation?It should be noted that a quadratic equation means an algebraic equation of the second degree in x.
The quadratic equation in its standard form is ax² + bx + c = 0. Here, a and b are the coefficients, x is the variable, while c is the constant term.
Quadratic equations are used in everyday life, as when calculating areas, and the determination of a product's profit and formulating the speed of an object.
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24g ____ 1,679mg?
Complete the inequality statement
Answer:
24g > 1,679mgYou are ordering shirts for a club at your school. The function f(x) = 8x + 12 represents the cost of
ordered a shirts. How much would it cost to buy 32 shirts?
The cost of buying 32 shirts according to the function given is $268.
Given that the function f(x) = 8x + 12 represents the cost of ordered a shirt.
We need to find the cost to buy 32 shirts.
To find out how much it would cost to buy 32 shirts, you can use the given function f(x) = 8x + 12, where 'x' represents the number of shirts and 'f(x)' represents the cost of ordering 'x' shirts.
Let's plug in x = 32 into the function:
f(32) = 8 × 32 + 12
f(32) = 256 + 12
f(32) = 268
So, it would cost $268 to buy 32 shirts.
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