π/16 is the area enclosed by the curve r= sin(4θ)
The given curve is polar curve and hence the area of the polar curve is given by:
Let A be the area of the curve so,
A = [tex]\int\limits^b_a {\frac{1}{2}r^2 } \, d\theta[/tex]
where a and b is the boundary at which r=0
so after equation r=0
sin(4θ) =0
=> sin(4θ) =0
=> 4θ = 0,π
=> θ = 0, π/4
so a=0 , b= π/4
now
A = [tex]\int\limits^b_a {\frac{1}{2}r^2 } \, d\theta[/tex] ------(i)
so [tex]r^2[/tex] = (sin(4θ))^2
=> [tex]sin^2[/tex]( 4θ )
ans we know that
cos(2α) = 1 - [tex]2sin^2[/tex]2 α
so [tex]sin^2[/tex]= (1- cos(8θ) )/2
putting the value of r in the equation (i) we get :-
A = [tex]\int\limits^b_a {\frac{1}{4} *(1-cos8\theta) } \, d\theta[/tex]
=> 1/4* [tex]\int\limits^b_a {(1-cos8\theta) } \, d\theta[/tex]
here a=0 and b=π/4
after putting the value and solving the integral
A = π/16
so A is the area enclosed by r=sin(4θ) is π/16
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given 12 points on a plane, no three of which are collinear, what is the number of lines that can run through any two of these points?
The number of lines that can run through any two of these points is 66
What are Collinear Points?
Points that are on the same straight line are referred to as collinear points. They must lie on the same straight line, even if they don't have to be co-planar. The Latin terms "col" and "linear," where "col" means "together" and "linear" means "in the same line," are the source of the English word "colinear." Collinearity is the state of being shared by two points.
A group of three or more points that are located along the same straight line is called a collinear point. On separate planes, collinear points may exist, but not on different lines. Collinearity is the quality of two points being parallel to one another. Any three or more points are only considered to be collinear if they are situated along the same straight line.
12 points are given, not three of which are collinear.
∴ Number of lines formed = ¹² C ₂
= 12!/ 10!⋅2!
=66
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Find the lengths of the hypotenuses (x) of the triangle whose legs are given
1) 5cm,12cm
a = 5 cm
b = 12 cm
c = ?
c = √{ a² + b }
c = √{ 5² + 12² }
c = √{ 25 + 144 }
c = 13 cm
____
x
y
4
2
2
4
4
2
2
4
y=1x2
a = 1
As |a| increases, the parabola becomes
Answer:
narrower
Step-by-step explanation:
As [tex]|a|[/tex] increases, the parabola becomes narrower.
As [tex]|a|[/tex] decreases, the parabola becomes wider.
Consider the functions f(x) = 23x 5 and g(x) = 32x 7. what do the slopes indicate about the lines?
The lines of the functions f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7 have slopes that are negative reciprocals and are said to be perpendicular lines.
How to interpret the slope of a linear equation?The general formula for the equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept
We are given the linear equations as; f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7
Now, when two lines are parallel, their slopes will be the same and equal. However, when two lines are perpendicular, their slopes will be a negative reciprocal of each other.
In our given functions, we see that the slope of g(x) is a negative reciprocal of the slope of f(x). Thus, both lines are concluded to be perpendicular to each other.
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The complete question is;
Consider the functions f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7. what do the slopes indicate about the lines?
Statistics problem.
These are independent events. Suppose that P(A)=0.2 and P(B)=0.4 and P(B|A)= 0.8.
P(A and B) is
a) 0.40
b) 0.16
c) 0.48
d) 0.32
e) 0.60
The probability P(A and B) has a value of 0.08
How to determine the probability?From the question, we have the following parameters that can be used in our computation:
P(A) = 0.2
P(B) = 0.4
P(B|A) = 0.8
From the question, we have
Independent events
This means that
P(A and B) = P(A) * P(B)
So, we have
P(A and B) = 0.2 * 0.4
Evaluate
P(A and B) = 0.08
Hence, value is 0.08
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At wrestling competitions, the wrestling mat is placed on a square that has an area of 1,444 square feet.
What is the length of one side of the square?
Joe i building a lide for hi kid. If the ladder i 8 feet tall and he want the bottom of the lide to be 6 feet from the ladder, how long doe the lide need to be?
The length of the slide is 10 feet.
From the provided information, the height of the slide ladder is 8 feet and the distance between the bottom of the slide and the ladder is 6 feet. The three sides form a right-angle triangle.
Pythagoras theorem postulates that in a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides which are perpendicular and base. Hence,
Hypothenuse^2 = Base^2 + Perpendicular^2
The length of the slide using Pythagoras theorem is where length of the slide is the hypothenuse of the triangle,
Length^2 = 8^2 + 6^2 = 100
Length = √100 = 10 feet
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In a triangle, the lengths of the sides are 3, 7, and 8. If the perimeter of a similar
triangle is 54, what is the length of the longest side of the larger triangle?
Answer:
Answered Below :)
Step-by-step explanation:
To solve this problem, we can use the concept of similar triangles. Two triangles are similar if they have the same shape, but possibly different sizes. In other words, the angles in the two triangles are the same, and the lengths of their sides are in proportion to each other. In this case, we are given that the lengths of the sides of the first triangle are 3, 7, and 8, and the perimeter of the second triangle is 54. We are asked to find the length of the longest side of the second triangle.
To solve this problem, we first need to find the perimeter of the first triangle. Since the perimeter of a triangle is the sum of the lengths of its sides, the perimeter of the first triangle is 3 + 7 + 8 = 18. Now that we know the perimeter of the first triangle, we can use this information to find the scale factor between the two triangles. The scale factor is the ratio of the lengths of corresponding sides in the two triangles. Since the perimeter of the second triangle is 3 times the perimeter of the first triangle, we can say that the scale factor between the two triangles is 3. In other words, the lengths of the sides of the second triangle are 3 times the lengths of the corresponding sides of the first triangle.
Now that we know the scale factor between the two triangles, we can use it to find the length of the longest side of the second triangle. The longest side of the first triangle has length 8, so the longest side of the second triangle has length 8 * 3 = 24. Therefore, the length of the longest side of the second triangle is 24.
Note: It is important to realize that the scale factor between two similar triangles is always the same for all sides of the triangles. In other words, if one side of a triangle is x times the corresponding side of a similar triangle, then all sides of the second triangle are x times the corresponding sides of the first triangle. This means that if we know the scale factor between two triangles for any one side, we can use that scale factor to find the lengths of all sides of the second triangle.
help me with my math
Answer:
y = 2/3 x + 6
Step-by-step explanation:
I do not see a point C, so I guess you get to make your own point C. I will make it at (06,)
The equation of the line seen is
y = 4/6x + 5 If you start at A and count up, you will get 4 unit and then count right you will be 6 units. This is the slope 4/6. the 5 is where the line crosses the x axis (0,5)
We can simplify 4/6 to 2/3
y = 2/3 x + 5
When lines are parallel, they have the same slope, but different y intercept.
Since I made C (0,6) The equation of this parallel line would be
y = 2/3 x + 6
Draw a line parallel to AB but make sure that it goes through the point (0,6)
If x = 3, y = 6, then cosO is equal to:
3√5.
√5/5
9.
15.
Answer the questions below. Be sure to mark all answers that apply.
(a) Which numbers are divisible by 5?
(b) Which numbers are divisible by 10?
(c) Which numbers are divisible by 2?
435
859 850
0
None of these
The diagram shows a rhombus inside a regular hexagon.
Work out the size of angle x.
X
Answer: 60 degrees
Step-by-step explanation:
Make the hexagon into a bunch of equilateral triangles, and split the rhombus into two equilateral triangles.
Answer:
60°----------------------------
Sides of the given rhombus are parallel to sides of the hexagon.
If you extend sides of the rhombus they will make diagonals connecting opposite vertices..
These diagonals are angle bisectors and hence the angle x if half the interior angle of the hexagon:
x = 120°/2 = 60°someone answer this correctly please, I’ll give you brainliest and your getting 50 points someone answer this no fraud answers don’t be weird.
The proof of the congruent sides RU and TS is given below.
What is congruent triangles?
In any orientation, two triangles are said to be congruent if their three sides and three angles are equal.
Therefore, if two triangles have the same number of sides on each side, they are said to be congruent. Additionally, we can determine that they are congruent if we have a side, an angle between the sides, and then another side that is congruent, in that order: side, angle, side.
Let the given figure is a rectangle RSTU.
Since the all four angles of rectangle are of 90 degree.
So, each angle of rectangle RSTU is 90 degree.
Let, US divides the angle RUT and angle RST and form two new triangles, ΔSRU and ΔSTU.
Both the triangles having same measure of angles and sides.
⇒ ΔSRU ≅ ΔSTU
As the triangles are congruent so their sides are congruent.
Hence, RU is congruent to TS.
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State if the two triangles are congruent based on the given information. If they are, state
how you know (SSS, SAS, ASA, or HL). (2pts each)
The triangle LMO and LNO are, if LM ≅ ON, ∠ M ≅ ∠ N concurrent by SAS congruence.
What is triangle?Three interconnected vertices make up one of the basic geometric shapes denoted by the symbol Δ.
Given:
LM ≅ ON
∠ M ≅ ∠ N
As you can see, the side LO is common in both the triangle,
So In the concurrency SAS, the two sides of a triangle are two to be equal, and an angle is to be equal.
Therefore, the triangle LMO and LNO are, if LM ≅ ON, ∠ M ≅ ∠ N concurrent by SAS congruence.
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(Marks: 9) If your current age is x years, and 9 years later your age will be 4 less than double of your current age, then an equation representing the statement is:
X + 9 = 2x - 4
X-9 = 2x - 4
X + 9 = 2x + 4
X-9 = 2x + 4
Answer:
x+9=2x-4
Step-by-step explanation:
it just has to be
age=x
9 years later =+9
x+9
will be is =
x+9=
4 less = -4
double of current age =2x
=x+9=2x-4
Answer:
The current age is X and 9 years later the age will be X+9
Step-by-step explanation:
And it was stated that in 9 years time the age will be 4 less than double of the current age
And the current age is X, double of the current age is 2×X= 2x and it was stated that the double of the age will be 4 less than the person age in 9 years time so the equation statement is
X + 9 = 2x + 4
20 The distance between two towns is 600 km, correct to the nearest 10 km.
A car takes 8 hours 40 minutes, correct to the nearest 10 minutes, to travel this distance.
Calculate the lower bound for the average speed of the car in km/h.
The average speed should be almost 69 - 70 km/h.
What is a average speed and an illustration?
To calculate average speed, divide the total distance traveled by the passing time. For instance, if a person is moving at a speed of one mile per minute, or 40 miles per hour, they would be traveling 20 miles north and then 20 miles south (60 mph).
The mean value of a body's speed over a period of time is its average speed. Since a moving body's speed is not constant over time and fluctuates, the average speed formula is required.
Time = distance / speed
Speed = Distance / Time
Speed = 600/8.66
Speed = 69.28 km/h
So the average speed should be almost 69 - 70 km/h
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Value of x is and length of AB is
7 is the value of x in the given figure.
What is Triangle?A triangle is a three-sided polygon that consists of three edges and three vertices.
A line segment that connects two midpoints of the sides of a triangle is called a midsegment.
DE is the midsegment between AC and CB.
DE=1/2(AB)...*
From the figure, DE=2x+13 and AB=13x-37.
Now plug in the value of DE and AB in Equation (*).
2x+13=1/2(13x-37)
Apply distributive property on RHS.
2x+13=13/2x-37/2
2x-13/2x=-37/2-13
4x-13x/2=-37-26/2
-9x/2=-63/2
9x=63
Divide both sides by 9
x=7
Hence, the value of x in the figure is 7.
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-will give brainliest, thx
Find all real values of t that satisfy the equation t^2 = 324
the Answer: t=18, - 18
A student walks 3 blocks south, 4 blocks west, and 3 blocks north. What is the displacement of the student?.
The displacement of the student after taking mentioned path is 4 blocks.
Lets understand the displacement before heading to calculations. So, displacement is the shortest distance between initial and final points. The distance is the total path covered. In this problem, we see that 3 blocks covered in South direction were equated by 3 blocks covered in North direction.
So, the displacement is the distance covered in West direction. Based on this, the displacement of the student is 4 blocks in treading mentioned path.
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if Johnny has 123 apples falling from a feet of 500 how many seconds will it take for it to fall
It will take approximately 12.3 seconds for the apples to fall to the ground.
What is gravity?A strong, invisible force called gravity pulls everything on Earth down. Gravity keeps you and all the things that are around you from floating up into the sky. An example of gravity in action is when you jump from the ground. Gravity is the force that pulls you back down.
To find the time taken to fall to the ground
Using the formula;
[tex]H = \frac{1}{2} gt^{2}[/tex]
where H = 500 feet
Converting feet to meters
H = 152.4 meters
152.4 = [tex]\frac{1}{2} (9.8)t^{2}[/tex]
152.4 = [tex]4.9t^{2}[/tex]
[tex]t^{2} = 152.4[/tex]
[tex]t = \sqrt{152.4}[/tex]
t = 12.3seconds
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Hi, it would be much appreciated if someone could help me with this question. All screenshots are attached. Thanks so much :)
Answer:
Step-by-step explanation:
25x6
kenny owned so many beseball cards that he sold 82 of them.after selling the cards, he still had 159 lefft.write and solve an equation to find the number of beseball cards b kenny had originally
By applying the simple concept of algebra, the equation showing the number of Kenny's baseball cards (b) is b = 82 + 159. So it can be concluded that the number of Kenny's baseball cards is 241 cards.
The use of algebra in everyday life.
Algebra is a branch of mathematics that uses numbers, symbols, or letters to solve math problems. In everyday life, algebra is often used, starting from simple mathematical operations, to financial planning, calculating time/mileage, and so on.
In the above questions, we can take some notes as follows:
Baseball cards sold (x) = 82 cards.
Remaining baseball cards (y) = 159 cards.
The number of baseball cards owned by Kenny (b):
b = x + y
= 159 + 82
= 241 cards
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The base of a triangle is twich the height of the triangle. If the area is 16 square inches, what is the height?.
The height of a triangle when the area is 16 inches h=4
The area of the triangle(A) is given by:
A=1/2bh---(1)
where,
b is the base
h is the height
As per the statement:
The area of a triangular block is 16 square inches.
A=16 in
It is given: If the base of the triangle is twice the height
⇒ b=2h---(2)
Substitute these in [1] we have;
A=1/2bh
16=1/2(2h)h
h^2=16
h=4in
Substitute h =4 in [2] we have;
b=2h
b=2(4)
b=8in.
Therefore, the base and height of the triangle are 8 inches and 4 inches.
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The graph represents the piecewise function
The graph represents the piecewise function is [tex]y=\frac{5 x}{4}+\frac{15}{4}[/tex].
What is graph?
A graph is a structure that resembles a set of items in discrete mathematics, more especially in graph theory, in which certain pairs of the objects are conceptually "connected." The items are represented by mathematical abstractions known as vertices, and each pair of connected vertices is referred to as an edge.
Given,
Graph representing a piecewise function:
A piecewise function is a function that is defined by different formulas or functions for each given interval. It’s also in the name: piece. The function is defined by pieces of functions for each part of the domain.
So,
Using the formula:
[tex]y=\frac{5 x}{4}+\frac{15}{4}[/tex]
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What is the average of 127 and 46?
Answer:
86.5
We add them together, which is 173, and since there are two terms, divide by 2
173/2=86.5
Answer:
86.5
127 + 46 / 2 = 86.5
I could be wrong but I believe this is correct.
Keith buy 3 yard of material to make a blanket. He trim off a total of 1/6 yard before he begin ewing. How much material remain for the blanket?
The material remained for the blanket after trimming off by Keith is 2.5 yards.
Simple subtraction and multiplication can provide the result. Beginning will be by the multiplication. Firstly finding the exact amount of material trimmed = 3×1/6
Performing multiplication and division on Right Hand Side of the equation
Trimmed material = 1/2 or 0.5
Now, we need to subtract the trimmed material from overall amount of material
Remaining material = 3 - 0.5
Performing subtraction on Right Hand Side of the equation
Remaining material = 2.5 yards
Hence, 2.5 yards of material remained.
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What is the average rate of change of the function y = 3 sin(2x) on the interval [0,π4] ?
The average rate of change of the trigonometric function f(x) = 3 · sin 2x on the interval [0, π / 4] is 12 / π. (Correct choice: D)
How to find the average rate of change of a trigonometric function in a given interval
In this question we need to determine the average rate of change of a trigonometric function on the interval [0, π / 4]. Graphically speaking, the average rate of change is represented by a line secant to the curve of the function, whose definition is shown below:
m = [f(π / 4) - f(0)] / (π / 4 - 0)
If we know that f(x) = 3 · sin 2x, then the average rate of change is:
m = [3 · sin (π / 2) - 3 · sin 0] / (π / 4 - 0)
m = (3 - 0) / (π / 4)
m = 12 / π
The average rate of change of the function is 12 / π.
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Answer question with steps get brainliest! Please answer ASAP with steps. Use picture above
An air traffic controller Is tracking two planes. To start, Plane A is at an altitude of 5000 feet and Plane B is at an altitude of 4207 feet. Plane A is gaining altitude at 45.25 feet per second and Plane B is gaining altitude at 60.5 feet per second.
seconds:
feet:
The planes will both meet in 53 seconds based on the information.
How to illustrate the equation?From the information, Plane A is at an altitude of 5000 feet and Plane B is at an altitude of 4207 feet. Plane A is gaining altitude at 45.25 feet per second and Plane B is gaining altitude at 60.5 feet per second.
The equation will be:
5000 + 45.25f = 4207 + 60.5f
Collect the like terms
60.5f - 45.25f = 5000 - 4207
15.25f = 793
Divide
f = 793 / 15.25
f = 53
They'll meet in 53 seconds.
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An air traffic controller Is tracking two planes. To start, Plane A is at an altitude of 5000 feet and Plane B is at an altitude of 4207 feet. Plane A is gaining altitude at 45.25 feet per second and Plane B is gaining altitude at 60.5 feet per second. When will that meet
seconds:
feet: