precalculus Question 11.2
a) show that the distance between the two points.
b) describe the positions of the points relative to eachother. simplify the distance formula for this case.
c) simplify the distance formula.
d) choose two points on the polar coordinate system and find the distance between them. then choose different polar representations of the same two points and apply the distance formula again.
I will attach the answer, this work did not come from me but I figured I'd upload it anyways.
Answers
A can also be represented as A(2,-11π/6) and B can be represented as B(3,-π/6). Using the distance formula again, we have: distance = sqrt(2^2 + 3^2 - 223*cos(-11π/6 - (-π/6))) = sqrt(13)
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
a) To find the distance between two points in polar coordinates, we can use the Pythagorean theorem to calculate the distance in the radial direction and then use the law of cosines to take into account the angular separation. Thus,
distance = sqrt((r2-r1)^2 + (r1^2 + r2^2 - 2r1r2*cos(θ1 - θ2)))
Simplifying this expression, we get:
distance = sqrt(r1^2 + r2^2 - 2r1r2*cos(θ1 - θ2))
Using the identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we can rewrite the expression as:
distance = sqrt(r1^2 + r2^2 - 2r1r2cos(θ1)cos(θ2) - 2r1r2sin(θ1)sin(θ2))
Using the identity cos^2(x) + sin^2(x) = 1, we can simplify the expression further:
distance = sqrt(r1^2 + r2^2 - 2r1r2*cos(θ1-θ2))
b) If θ1 = θ2, then the points lie on the same radial line and are separated only by their radial distances from the origin. In this case, the distance formula simplifies to:
distance = sqrt((r2-r1)^2) = abs(r2-r1)
This is what we would expect, as the distance between two points on the same radial line is simply the absolute difference in their radial distances from the origin.
c) If θ1 - θ2 = 90 degrees, then the points are separated by the maximum angular distance and lie on perpendicular radial lines. In this case, the distance formula simplifies to:
distance = sqrt(r1^2 + r2^2)
This is also what we would expect, as the distance between two points on perpendicular radial lines is simply the Pythagorean sum of their radial distances from the origin.
d) Let's choose two points in polar coordinates: A(2,π/6) and B(3,5π/6). Using the distance formula derived in part (a), we have:
distance = sqrt(2^2 + 3^2 - 223*cos(5π/6 - π/6)) = sqrt(13)
Now let's express these same points in different polar representations.
Therefore, a can also be represented as A(2,-11π/6) and B can be represented as B(3,-π/6). Using the distance formula again, we have:
distance = sqrt(2^2 + 3^2 - 223*cos(-11π/6 - (-π/6))) = sqrt(13)
As expected, the distance between the two points is the same regardless of their polar representations.
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Related Questions
A hotel has 320 rooms. Some are singles, and some are doubles. The single room cost 40$ and the double rooms cost 65$. Because of and event in town all of the hotel rooms are occupied. The sales tonight are 15,600$. How many of each type of room does the hotel have?
Answers
As a result, the hotel has 208 single rooms and 112 double rooms. We can now use these two equations to find the values of x and y.
what is equation ?An equation in mathematics is a claim that two mathematical expressions are equivalent. In an equation, the two expressions on either side are separated by the equals sign (=). As in the following equation: 3x + 2 = 11 declares that the formula 11 is identical to the formula 3x + 2. Finding the value of x that causes the equation to be true is the first step towards solving this equation. By first removing 2 from both sides of the equation and then dividing by 3, we may find x in this situation: 3x + 2 - 2 = 11 - 2 (subtracting 2 from both sides) (subtracting 2 from both sides) .3x = 9 (simplifying the right side) (simplifying the right side) .
given
Assume that there are x single rooms and y double rooms at the hotel.
We may create an equation since we are aware that there are 320 rooms in the hotel overall:
x + y = 320 (Equation 1) (Equation 1)
We can create another equation based on sales since we also know the prices for each type of accommodation and the overall sales:
40x + 65y = 15,600 (Equation 2) (Equation 2)
We can now solve for x and y using these two equations. To accomplish this, one method is to solve for x in Equation 1 and replace it in Equation 2:
x + y = 320
x = 320 - y
Equation 2 using x = 320 - y as a replacement:
12,800 - 40y + 65y = 15,600
25y = 2,800 y = 112
40x + 65y = 15,600
40(320 - y) + 65y
= 15,600
There are 112 double rooms in the hotel. In order to determine the number of single rooms, we can reenter this value into Equation 1 as follows:
x + y = 320
x + 112 = 320
x = 208
As a result, the hotel has 208 single rooms and 112 double rooms. We can now use these two equations to find the values of x and y.
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What type of function is a circle?
Answers
is a pineapple
Step-by-step explanation:this is becuase you should work it out youself you silly goose
Answer:
circle is not a function.
Step-by-step explanation:
there are certain rules to define a mapping as function, one of them is function should not have more than 1 value at a given x.
for illustration purpose consider the below diagram,
as you can see in the given circle at x = 2, there are two different values of y, therefore equation of circle doesn't qualify as a function. half circle can be defined as a function (by restricting the range to only positive or negative values).
Hopefully you found the answer helpful
i need answer for 29 please!!!
Answers
Answer:
D. -3, -7
Step-by-step explanation:
F^-4 (x) : f(x) = 4 (3x+2)
Answers
Step-by-step explanation:
To find F^-1(x), we need to solve for x in terms of f(x) using the given equation for f(x):
f(x) = 4(3x+2)
First, we can simplify the expression inside the parentheses:
f(x) = 12x + 8
Next, we can solve for x by isolating it on one side of the equation:
f(x) - 8 = 12x
(x is isolated)
x = (f(x) - 8) / 12
Therefore, F^-1(x) = (x - 8) / 12.
how do you find surface area on the net of a rectangular prism
Answers
Answer: The area of the rectangular prism would be S.A. = 66m²
Step-by-step explanation: :)
What is the surface area of this rectangular prism?
A rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces. The surface area of this rectangular prism = l x b
We have given
4 rectangles with area
A₁ = 3m × 4m = 12 m²
2 rectangles with area
A₂ = 3m × 3m = 9 m²
Therefore the Surface Area of the rectangular prism is:
S.A. = 4A₁ + 2A₂
S.A. = 4(12) + 2(9)
= 48 + 18
= 66
Hence, the net surface area of the rectangular prism would be 66 meters sq.
Please I need help
Manuel makes bird food by mixing sunflower seeds and
corn kernels. He bought 2.5 pounds of sunflower seeds. He
also bought some corn kernels. Sunflower seeds and corn
kernels each cost $2.20 per pound. The total amount
Manuel spent on the sunflower seeds and corn kernels was
$9.35.
Enter the number of pounds of corn kernels Manuel
purchased.
Answers
Manuel purchased 1.75 pounds of corn kernels.
What is equation?An equation is a mathematical statement that shows the equality between two expressions. The expressions on either side of the equals sign are called the left-hand side and the right-hand side of the equation. An equation can be written in many different forms, but the most common form is: Left-hand side = Right-hand side
In the given question,
Let's use x to represent the number of pounds of corn kernels Manuel purchased.
We know that Manuel bought 2.5 pounds of sunflower seeds, and that both the sunflower seeds and corn kernels cost $2.20 per pound.
The total amount Manuel spent on the sunflower seeds and corn kernels was $9.35, so we can write the equation: 2.2(2.5) + 2.2x = 9.35
Simplifying the left side of the equation, we get: 5.5 + 2.2x = 9.35
Subtracting 5.5 from both sides, we get: 2.2x = 3.85
Dividing both sides by 2.2, we get: x = 1.75
Therefore, Manuel purchased 1.75 pounds of corn kernels.
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Harold ran a total of 4,464 miles the last 9 months. How many miles did he ride each month?
Answers
Answer:
I think 496
Step-by-step explanation:
Divide
Please help me!! This has to be turned in by the end of the day
Answers
The solution to the system of equations is x = 31/7 and y = 1/7.
What is equation?An equation is a mathematical statement that shows the equality between two expressions. It consists of two parts, the left-hand side and the right-hand side, separated by an equal sign (=). The left-hand side and right-hand side may contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
Here,
We can solve this system of equations using the elimination method. We will eliminate one of the variables by adding or subtracting the equations to get an equation with just one variable.
Multiply the first equation by 4 and the second equation by 3 to get:
16x + 12y = 76
9x + 12y = 45
Subtract the second equation from the first to eliminate y:
16x + 12y - (9x + 12y) = 76 - 45
7x = 31
Solve for x by dividing both sides by 7:
x = 31/7
Substitute the value of x into either equation to solve for y. Let's use the first equation:
4x + 3y = 19
4(31/7) + 3y = 19
3y = 19 - 124/7
3y = 3/7
y = 1/7
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Complete question:
Solve the equation for x and y:
4x+3y=19
3x+4y=15
What is the square root of 92 rounded to the nearest tenth
Answers
Answer:
answer is 2 into 2 into twenty three
Step-by-step explanation:
2×2×23
√2^2×23
2√23
Answer: The square root of 92 is approximately 9.59166304663. Rounding this to the nearest tenth gives 9.6. Therefore, the square root of 92 rounded to the nearest tenth is 9.6.
Step-by-step explanation:
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(D) What is the surface area of this rectangular pyramid?
4 yd
Submit
444
3 yd
4 yd
square yards
3 1.8
9
Answers
Check the picture below.
so if we cut the pyramid like a pie with a knife from the top and open it up, it'd look more or less like that picture, so its area is just the area of four triangles with a base of 4 and a height of 3 and a 4x4 square.
[tex]\stackrel{ \textit{\LARGE Areas} }{\stackrel{\textit{four triangles}}{4\left[ \cfrac{1}{2}(4)(3) \right]}~~ + ~~\stackrel{square}{(4)(4)}}\implies 24+16\implies \text{\LARGE 40}~yd^2[/tex]
Findthe
y -intercept
oftheparabola y = x2 + 2x + 10
Answers
The y -intercept of the parabola y = x^2 + 2x + 10 is (0, 10)
the y-intercept represents the value of y when x=0. In the equation y = x^2 + 2x + 10, the x^2 and 2x terms create a parabola that opens upwards. As x approaches infinity or negative infinity, the value of y also increases towards infinity.
The y-intercept of a graph is the point at which the graph crosses the y-axis, which is where x=0. To find the y-intercept of the parabola y = x^2 + 2x + 10, we substitute x=0 into the equation, which gives
y = 0^2 + 2(0) + 10
y = 10
Therefore, the y-intercept of the parabola is (0, 10), which means the graph crosses the y-axis at the point (0, 10).
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The given question is incomplete, the complete question is:
Find the y -intercept of the parabola y = x^2 + 2x + 10
At how many points does the line with equation y = -3/4x + 25/4 intersect the circle shown?
A. 0
B. 1
C. 2
D. There is not enough information to determine the number of points of intersection.
Answers
Answer:
We need to find the number of points of intersection between the line with equation y = -3/4x + 25/4 and the circle. The equation of the circle is not given, so we cannot directly solve for the intersection points. However, we can use the fact that the intersection points must satisfy both the equation of the line and the equation of the circle.
Let (x, y) be a point on the circle. Then the coordinates satisfy the equation of the circle, which we will assume is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center of the circle and r is the radius. Substituting y = -3/4x + 25/4, we get:
(x - a)^2 + (-3/4x + 25/4 - b)^2 = r^2
This is a quadratic equation in x, which we can expand and simplify:
(x^2 - 2ax + a^2) + (-3/2)x^2 + (15/2)x - 3bx + (625/16) + b^2 - (25/2)b + (625/16) - r^2 = 0
Simplifying further, we get:
(-5/2)x^2 + (-2a - 3b + 15/2)x + (2a^2 - 25/2b + 125/8 - r^2) = 0
This is a quadratic equation in x, with coefficients that depend on the center and radius of the circle. We can use the quadratic formula to solve for x, and then substitute into y = -3/4x + 25/4 to get the corresponding value of y.
The discriminant of the quadratic equation is:
(-2a - 3b + 15/2)^2 - 4(-5/2)(2a^2 - 25/2b + 125/8 - r^2)
Simplifying and factoring, we get:
(4a + 6b - 15)^2 - 25(4a^2 - 50b + 250/8 - 2r^2)
This is a quadratic expression in b, which tells us whether the equation has real solutions (i.e., whether the line intersects the circle). If the discriminant is positive, then the equation has two real solutions, which means the line intersects the circle at two points. If the discriminant is zero, then the equation has one real solution, which means the line is tangent to the circle. If the discriminant is negative, then the equation has no real solutions, which means the line does not intersect the circle.
Without knowing the equation of the circle, we cannot determine the discriminant and therefore the number of intersection points. Therefore, the answer is D: "There is not enough information to determine the number of points of intersection."
Step-by-step explanation:
Find the value of x that makes AABC~ ARTS.
Answers
The value of x that will make ∆ABC similar to ∆ RST is 12
What are similar triangles?Similar triangles are triangles with similar shapes but their sizes are not necessarily thesame. The ratio of the corresponding sides of similar triangles are equal.
This means ,
2x+6/36 = 40/48 = 25/x+18
therefore, 2x+6/36 = 40/48
48( 2x+6) = 36× 40
96x + 288 = 1140
96x = 1140-288
96x = 1152
divide both sides by 96
x = 1152/96
x = 12
therefore the value of x that will make the two triangles similar is 12
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factor. If the trinomial will not factor write "prime"
a²-13a-30
Answers
Answer:
( a + 2 ) ( a - 15 )
Step-by-step explanation:
a² - 13a - 30
a² + 2a - 15a - 30
( a² + 2a ) - ( 15a - 30 )
a ( a + 2 ) - 15( a + 2 )
( a + 2 ) ( a - 15 )
You draw two marbles without replacement from a bag containing three green marbles and three black marbles. The number of possible outcomes in the sample space
Answers
Answer: There are 7 marbles in the bag and in the first attempt, all 7 marbles are there. After one marble is taken away, then we have 6 marbles to choose from. Therefore we will multiply 7*6 = 42.
Hence, there are 42 possible outcomes.
pls it wud be really helpful
Answers
The value of 2 (x²+2/x²) is D) 1/4.the calculation is given below.
What is value?Value in math is a measure of the amount of a quantity, such as size, cost, or weight. It is used to compare different quantities or compare different elements in a set. Value can be measured in a variety of ways, such as absolute value, relative value, or fractional value. Ultimately, value helps to explain how different elements interact and relate to one another.
Solution:
The given equation is:
sin θ = 2x
cos θ = 2
0°< θ < 90°
Now we can solve for x using the first equation to get
x = sinθ/2
Now we can substitute this value of x in the second equation and solve for θ
2 = cosθ
θ = cos-1 (2)
θ = 60°
Thus, x = sin60°/2
x = √3/2
Now we can substitute this value of x in the given equation and solve for the value of 2 (x²+2/x²)
2 (x²+2/x²) = 2 ((√3/2)²+ 2/ (√3/2)²)
2 (x²+2/x²) = 2 (3/2 + 2/3)
2 (x²+2/x²) = 2 (5/3)
2 (x²+2/x²) = 10/3
Therefore, the value of 2 (x²+2/x²) is D) 1/4.
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An expression equivalent to B +B +B +B + B that is a product of a coefficient and a variable
Answers
Answer:
One expression equivalent to B + B + B + B + B that is a product of a coefficient and a variable is 5B, where 5 is the coefficient and B is the variable
Step-by-step explanation:
Find the median, mode, and range
of the data.
Birth weight (pounds) 6, 9, 8, 6, 9, 6
Answers
Answer:
answer: 7
Step-by-step explanation:
u have to find the median by finding the middle of all of those numbers which will me seven
Find all values of x that are not in the domain of h. If there is more than one value, separate them with commas
Answers
Answer:
x = - 1, 6
Step-by-step explanation:
the denominator of h(x) cannot be zero as this would make h(x) undefined
equating the denominator to zero and solving gives the values of x that cannot be in the domain of h(x)
x² - 5x - 6 = 0
(x + 1)(x - 6) = 0
equate each factor to zero and solve for x
x + 1 = 0 ⇒ x = - 1
x - 6 = 0 ⇒ x = 6
then x = - 1, 6 are values not in the domain of h(x)
Are these triangles similar, congruent or neither? What theorem supports your answer?
options:
Similar: SAS
Similar: AA
Similar: SSS
Congruent: ASA
Congruent: SSS
Congruent: SAS
Neither
Answers
Both of the two triangles are similar by SSS similarity postulate.
How to Identify Similar Triangles?Similar triangles are defined as triangles that have the same shape, but their sizes may vary. This means that, if two triangles are similar, then we can say that their corresponding angles are congruent and corresponding sides are in equal proportion.
Looking at the 2 given triangles, we can see the ratio if corresponding sides as:
39/13 = 3
39/13 = 3
21/7 = 3
Since the ratios of the corresponding sides of both triangles are the same, then they are both Similar by SSS similarity postulate.
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1. Find the zeroes of the following quadratic polynomials and verify the relationship between
the zeroes and the coefficients.
(i) x²-2x-8
Answers
Answer:
The zeroes are -2 and 4.
Step-by-step explanation:
x^2 - 2x - 8 = 0
(x + 2)(x - 4) = 0
x + 2 = 0 or x - 4 = 0
x = -2, 4
The general form of the equation is ax^2 + bx + c = 0.
The sum of the zeroes should be equal to -b/a:-
-2 + 4 = 2
and -(-2)/1 = 2
Verified
The product of the zeroes should be c/a = -8:-
-2 * 4 = -8.
Verified.
Which of the following is the surface area of the pyramid?
36 m2
72 m2
88 m2
144 m2
please answer quick, give quick explanation please:) thanks!! 60 points!
Answers
The Surface area of Pyramid is equal to Option (B) [tex]72 m^2[/tex] when base is square and their are 4 triangles.
The surface area of a pyramid is the total area of all its faces. The formula to calculate the surface area of a pyramid depends on the shape of its base.
To find the surface area of this pyramid we should add area of all 4 triangles and 1 square base.
Area of Base = 4m * 4m
= [tex]16m^2[/tex]
Area of Triangle = 1/2 *4* 7
= [tex]14m^2[/tex]
Surface Area of Pyramid = Area of Base + 4 * (Area of Triangle)
= [tex]16m^2 + 4(14m^2)[/tex]
= [tex]72 m^2[/tex]
Therefore, Surface area of Pyramid is equal to [tex]72 m^2[/tex].
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Can someone help me with this the image is below use a trigonometric ratio to solve for x. round to two decimal places as necessary some person keeps giving me the wrong answer
Answers
Therefore, the length of the perpendicular side x is approximately 12.64 feet, rounded to two decimal places.
What is trigonometric ratio?A trigonometric ratio is a mathematical function that relates the sides of a right triangle to its angles. There are three basic trigonometric ratios: sine, cosine, and tangent. Each of these ratios is defined as the ratio of two sides of the triangle. Sine (abbreviated as sin) is the ratio of the length of the side opposite an acute angle to the length of the hypotenuse. That is, sin(theta) = opposite/hypotenuse. Cosine (abbreviated as cos) is the ratio of the length of the adjacent side to the length of the hypotenuse. That is, cos(theta) = adjacent/hypotenuse. Tangent (abbreviated as tan) is the ratio of the length of the opposite side to the length of the adjacent side. That is, tan(theta) = opposite/adjacent.
Here,
We can use the trigonometric ratio of sine to solve for x in this problem.
Sine is defined as the ratio of the length of the opposite side (in this case, x) to the length of the hypotenuse (21 feet):
sin(37°) = x/21
We can solve for x by multiplying both sides of the equation by 21:
x = 21 sin(37°)
Using a calculator to evaluate sin(37°), we get:
sin(37°) ≈ 0.6018
Substituting this value into the equation, we get:
x ≈ 21(0.6018)
x ≈ 12.64
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Hector has a collection of nickels, dimes, and quarters totaling 122 coins. The number of dimes he has is 3 more than four times the number of nickels, and the number of quarters he has is 19 less than the number of dimes. How many coins of each kind does he have?
Answers
Answer: 15 nickels, 63 dimes, 44 quarters
Step-by-step explanation:
x=number of nickels
4x+3=number of dimes
4x+3-19=number of quarters
Solve the following equation:
x+4x+3+4x+3-19=122
9x-13=122
x=15
Plug x back into the expressions:
Number of nickels=15
Number of dimes:
4(15)+3
63
Number of quarters:
4(15)+3-19
44
if you have an 88, a 95, a 93, and an 80, what do you need to make on the next quiz to have a mean quiz score of 90
Answers
You would need to make a 94 on the next quiz to have a mean quiz score of 90.
If 90 is the desired mean, then you have to multiply it by the number of quizzes
5 x 90 = 450 is the total in the 5 quizzes.
Now, add the scores of the first four quizzes.
88 + 95 + 93 + 80 = 356
Then, take the amount of the 5 quizzes and subtract it from the amount of the total 4 quizzes.
450 - 356 = 94
PLEASEEE HELP I HAVE IT DUE TODAY
cather added the ages of a crowd of 18 year olds and 22 year olds and got a sum of 290 if there were 10 18 year olds how many 22 year olds were in the crowd?
PLEASE HELP
Answers
Answer: 5
Step-by-step explanation:
If Cather added the ages of a crowd of both 18-year-olds and 22-year-olds and got 290, we can write that as 18x+22y=290. Knowing that there were 10 18-year-olds, we can plug 10 in for x.
18 times 10 + 22y = 290
So, first, we should multiply.
18 times 10 is 180, leaving us with 180+22y=290
Next, we should subtract both sides by 180.
So, 22y=110
Finally, we need to divide both sides by 22.
Leaving us with the answer of 5.
Answer:
5
Step-by-step explanation:
10×18=180
290-180=110
110/22=5
There Are 450 students in a school & 74% of them are boys.
i)Find the number of boys
ii)Find the numver of girls
Answers
Answer:
333 boys117 girls
Step-by-step explanation:
There Are 450 students in a school & 74% of them are boys.
1)Find the number of boys
2)Find the numver of girls
first we find 74% of 450 and we have the number of boys, we subtract the result from the total number and we have the number of girls
Find boys450 : 100 x 74 =
4.5 x 74 =
333
Find girls
450 - 333 =
117
On my homework I am suck because I do not know how to multiply decimals 25.5 x 10 x 10
Answers
Answer:
2550
Step-by-step explanation:
Step-by-step explanation:
To multiply decimals, we can simply multiply the numbers as if they were whole numbers and then count the total number of decimal places in the factors. In this case, we have:
25.5 x 10 x 10 = 255 x 10
Now, we multiply 255 by 10 as if they were whole numbers:
255 x 10 = 2550
Finally, we count the total number of decimal places in the factors, which is two (one in 25.5 and one in 10). Therefore, the answer should have two decimal places, so we can add a decimal point to the result and place two zeros after it:
25.5 x 10 x 10 = 2550.00
Therefore, 25.5 x 10 x 10 = 2550.00.
what is spearman's rank?
Answers
1.5 + [ (-1) x (1/2+1/4) ] / 1/2
A) 0
B) 1
C) 3
D) -1.5
Answers
Answer:
A) 0
Step-by-step explanation:
1.5+(-1 x (.5 + .25)) / .5
1.5+(-1 x .75) / .5
1.5 + -0.75 / .5 Because of order of operations division comes first
1.5 + (-1.5)
1.5 - 1.5
0