The maximum revenue, to the nearest whole dollar, is $180,625 (option c).
To find the maximum revenue, we need to determine the vertex of the quadratic function represented by the revenue equation.
The revenue equation is given as:
R(p) = -4p² + 1700p
The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be found using the formula:
x = -b / (2a)
For the given revenue equation, a = -4 and b = 1700. Plugging these values into the formula, we have:
p = -1700 / (2 × -4)
p = -1700 / -8
p = 212.5
The maximum revenue occurs at p = 212.5.
To find the maximum revenue, we substitute this value back into the revenue equation:
R(p) = -4(212.5)² + 1700(212.5)
R(p) = -4(45256.25) + 361250
R(p) = -181025 + 361250
R(p) = 180625
Therefore, the maximum revenue, to the nearest whole dollar, is $180,625 (option c).
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Write an inequality with a variable on one side, a negative integer on the other side, and one of the inequality symbols in between. Give a value that is a solution of the inequality you wrote, AND a value that is not a solution of the inequality.
Answer: Let's try x = 1 as a potential solution:
Substituting x = 1 into the inequality:
3(1) - 7 ≥ -10
3 - 7 ≥ -10
-4 ≥ -10
Since -4 is greater than or equal to -10, x = 1 is a solution to the inequality.
Let's try x = -3 as a potential solution:
Substituting x = -3 into the inequality:
3(-3) - 7 ≥ -10
-9 - 7 ≥ -10
-16 ≥ -10
Since -16 is not greater than or equal to -10, x = -3 is not a solution to the inequality.
Therefore, x = 1 is a solution to the inequality 3x - 7 ≥ -10, while x = -3 is not a solution.
Step-by-step explanation:
A woman is selected at random from the population of the United States. Let event A represent "The woman is a professional basketball player" and event B represent "The woman is taller than 5 feet 4 inches."
Are these probabilities equal? If so, explain your reasoning. If not, explain which one is the greatest and why.
P(B) when you have no other information.
P(B) when you know A is true.
P(B) when you know A is false.
The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller.
To determine the probabilities in question, we need to consider the information provided and make some assumptions based on general knowledge about the population of the United States.
P(B) when you have no other information:
Without any other information, we cannot accurately determine the probability of event B, which represents "The woman is taller than 5 feet 4 inches." We would need additional data on the height distribution of women in the United States to calculate this probability.
P(B) when you know A is true:
If we know that event A is true, meaning "The woman is a professional basketball player," we can make some assumptions based on the nature of professional basketball players.
Generally, professional basketball players tend to be taller than the average population due to the physical requirements of the sport. Therefore, the probability of event B, "The woman is taller than 5 feet 4 inches," would likely be greater when we know event A is true.
P(B) when you know A is false:
If event A is false, meaning "The woman is not a professional basketball player," we cannot make any definitive conclusions about the probability of event B, "The woman is taller than 5 feet 4 inches." The height of an individual is not solely determined by their profession, so without further information, we cannot determine if event B is more or less likely when event A is false.
In summary, based on the given information, we can conclude that the probabilities of event B are not equal under different scenarios. The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller. However, without any other information, we cannot determine the probability of event B or make comparisons when event A is false.
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you Solve for x:
10
8
12
The value of x in the given figure is 15.
In the given figure
The length of section of chords are given
We have to find the value of x
In order to find the value of x
Apply the intersecting chord theorem,
The intersecting chords theorem, often known as the chord theorem, is a basic geometry statement that defines a relationship between the four line segments formed by two intersecting chords within a circle. It asserts that the products of the line segment lengths on each chord are equal.
Therefore,
From figure we get,
⇒ 10/x = 8/12
⇒ x = 120/8
⇒ x = 15
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Can anyone help me with this question?
Answer:
-7/4
Step-by-step explanation:
we can see that every time the value of x increases by 4, the value of y decreases by 7.
let's pick two sets of coordinates (first two will be fine).
that is (-4, 6) and (0, -1)
Slope = (change in y values) / (change in x values)
= (-1 - 6) / (0 - -4)
= -7 / (0 + 4)
= -7/4.
so our slope (gradient) is -7/4
100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answer:
[tex]x^2+2y=1[/tex]
Step-by-step explanation:
The equation of a parabola with a vertical axis of symmetry,
focus (h, k+p), and directrix x=h-p is given by:
[tex](x - h)^2 = 4p(y - k)[/tex]
In this case, the focus is (0, 1) and the directrix is x =3.
Comparing this to the general equation,
we have
h = 0, k = 1, and x = h - p = 3.
From x = h - p, we can solve for p:
3 = 0 - p
p = -3
Substituting the values of h, k, and p into the equation, we get:
[tex](x - 0)^2 = 4(-3)(y - 1)[/tex]
Simplifying further:
[tex]x^2 = -12(y - 1)[/tex]
[tex]x^2=-12y+1[/tex]
[tex]x^2+12y=1[/tex]
Therefore, the parabola equation is [tex]x^2+2y=1[/tex]
find the volume of cylinder 8in r 2in h
Answer:
402.12 or just 402
Calculate the weight of a bed if its mass is 120 kg and gravitational acceleration is 20m/s2. Use weight equation.
Answer:
2400 N (Newtons)
Step-by-step explanation:
The weight of an object can be calculated using the equation:
Weight = mass * gravitational acceleration
Given:
Mass of the bed (m) = 120 kg
Gravitational acceleration (g) = 20 m/s²
Using the weight equation:
Weight = mass * gravitational acceleration
Weight = 120 kg * 20 m/s²
Weight = 2400 kg·m/s²
The unit of the weight is kilogram-meter per second squared (kg·m/s²), which is equivalent to the unit of force called Newton (N).
Therefore, the weight of the bed is 2400 Newtons (N).
50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answer: B
Step-by-step explanation: 63.3 in^2
What is the meaning of "[tex] Y^{X}\subset P(X \times Y) [/tex]"?
It implies that the collection of all ordered pairs (x, y) formed by taking an element from the set x and an element from the set y is a subset of the set containing all possible subsets of the Cartesian product of sets X and Y.
The expression "y^x ⊂ p(X x Y)" represents a subset relationship between two sets.
Let's break it down:
"y^x" represents the set of all possible ordered pairs (x, y) where x is an element of the set x and y is an element of the set y. This set represents the Cartesian product of the sets x and y.
"⊂" denotes a subset relationship. If we have two sets A and B, A ⊂ B means that every element in A is also an element of B. In other words, A is a subset of B.
"p(X x Y)" represents the power set of the Cartesian product of sets X and Y. The power set of a set is the set of all possible subsets of that set.
Therefore, "y^x ⊂ p(X x Y)" means that the set of all possible ordered pairs (x, y) where x is an element of the set x and y is an element of the set y is a subset of the power set of the Cartesian product of sets X and Y.
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Two hundred eighty-two people attended a recent performance of Cinderella. Adult tickets sold for $5 and children’s tickets sold for $3 each. Find the number of adults and the number of children that attended the play if the total revenue was $1046.
Part A: Write a system of equations in standard form (Ax + By = C) that can be solved to find the number of adults and children who attended the performance. Define the variables used in the equations. (4 points)
Part B: How many adults attended the performance? How many children attended the performance? Show your work and steps of how you found your answer using elimination.
A. A system of equations in standard form that can be solved to find the number of adults and children who attended the performance is:
x + y = 282
3x + 5y = 1046
B. The number of adults who attended the performance is 182 adults.
The number of children who attended the performance is 100 children.
How to determine the number of each type of tickets sold?In order to write a system of linear equations to describe this situation, we would assign variables to the number of adult tickets sold and number of children tickets sold, and then translate the word problem into an algebraic equation as follows:
Let the variable x represent the number of adult tickets sold.Let the variable y represent the number of children tickets sold.Since 282 people attended the recent performance by Cinderella, a linear equation that models the situation is given by:
x + y = 282 ....equation 1.
Additionally, adult tickets sold for $5 while children tickets sold for $3 each with a total revenue was $1046, a linear equation that models the situation is given by:
3x + 5y = 1046 .......equation 2.
Part B.
By multiplying equation 1 by 3, we have:
3x + 3y = 846 .......equation 3.
By subtracting equation 3 from equation 2, we have:
2y = 200
y = 100 children.
For the x-value, we have:
x = 282 - y
x = 282 - 100
x = 182 adults.
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Solve for e.
38
Ө
27
Answer:
θ = 35.39°-----------------
Given a right triangle with two legs known.
Find the missing angle using tangent function:
tangent = opposite leg / adjacent legSubstitute values to get:
tan θ = 27/38θ = arctan (27/38)θ = 35.39° (rounded)Find the degrees of freedom, alpha or significance level, and the t-critical value using the t-table n = 27 ;CL=98\%
The degrees of freedom for a sample size of 27 is 26, the alpha or significance level for a 98% confidence level is 0.02, and the t-critical value can be found using the t-table corresponding to a 98% confidence level and 26 degrees of freedom.
To find the degrees of freedom, alpha (significance level), and the t-critical value using the t-table, we need to consider the given information:
n = 27: This represents the sample size, which is 27 in this case.
CL = 98%: CL stands for the confidence level.
It indicates the level of confidence we want to have in our interval estimate.
In this case, the confidence level is 98%.
Degrees of Freedom (df): For a t-distribution, the degrees of freedom depend on the sample size.
Since we have a sample size of 27, the degrees of freedom would be n - 1.
Therefore, the degrees of freedom would be 27 - 1 = 26.
Alpha (α): Alpha, or the significance level, represents the probability of making a Type I error, which is rejecting a true null hypothesis.
The alpha level is determined by subtracting the confidence level from 1. In this case, the alpha level would be 1 - 0.98 = 0.02.
T-Critical Value: To find the t-critical value corresponding to a specific confidence level and degrees of freedom, we can consult the t-table. Since we have a confidence level of 98% and degrees of freedom of 26, we need to find the value that corresponds to the area of 0.01 (half of 1 - 0.98) in the t-distribution table.
The t-critical value would be the value from the table that matches these criteria.
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This number pattern -1:5 ;x; 35 ; ...
Is a quadratic number pattern.
a) Calculate x
b) Hence, or otherwise, determine the nth term of the sequence.
This sequence 4;9; x; 37; .... is a quadratic sequence.
a) Calculate x
b) Hence, or otherwise, determine the nth term of the sequence.
Answer:
x = 17; an = 3n² -3n -1x = 20; an = 3n² -4n +5Step-by-step explanation:
Given the following quadratic sequences, you want the value of x and the expression for the n-th term.
-1, 5, x, 354, 9, x, 37DifferencesOne way to determine x is to look at the differences between terms. The "second difference" is constant for a quadratic sequence, and the third difference is zero.
N-th termThe quadratic equation for the n-th term can be found by solving for its coefficients. The three known values of the sequence can give rise to three linear equations in the three unknown coefficients. These can be solved by your favorite method. We use this approach in the following.
1. -1, 5, x, 35First differences are the differences between each term and the one before:
{6, x-5, 35-x}
Second differences are the differences of these:
{x -11, 40 -2x}
Third differences are zero:
51 -3x = 0 ⇒ x = 17
The value of x is 17.
The expression for the n-th term of the sequence can be written as ...
an = a·n² +b·n +c
We are given values of a1, a2, and a4. This lets us write 3 equations for a, b, and c. The solution of those is shown on the first line of the first attachment. (The second line shows the evaluation of this quadratic equation for n=3. It gives 17, which we already knew.)
an = 3n² -3n -1
2. 4, 9, x, 37The last line of the first attachment shows us the expression for the third differences. The value of that is zero, so ...
-3x +60 = 0 ⇒ x = 20
The value of x is 20.
As in the above problem, the matrix of equations for the quadratic coefficients can be reduced to give the coefficient values. That tells us the n-th term of this sequence is ...
an = 3n² -4n +5
The last line in the second attachment tells us this expression for the n-th term properly computes the 3rd term (x), as above.
__
Additional comments
You can also use quadratic regression to find the coefficients of the formula for the quadratic sequence. This is shown in the 3rd attachment.
If you're trying to avoid using a calculator, you can write the equations out and solve them in an ad hoc way. In case you cannot tell, the equations for the coefficients of an = a·n² +b·n +c for the first problem are ...
1·a +1·b +1·c = -14·a +2·b +1·c = 516·a +4·b +c = 35You can also use the first values of the sequence (p), first difference (q), second difference (r) to write the quadratic:
an = p +(n -1)(q +(n -2)/2(r))
For (p, q, r) = (-1, 6, 6), this is an = -1 +(n -1)(3n) . . . . . . for the first sequence.
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Use the image to answer the question.
Which line of reflection would make rectangle A'B'C'D' the image of rectangle ABCD?
2
B
0
D'
B3
OA. line 1
OB. line 2
OC. line 3
1
✓
OD. line 4
The line of reflection that would make A'B'C'D' the image of ABCD is line 3
How to determine the line of reflection that would make A'B'C'D' the image of ABCD?From the question, we have the following parameters that can be used in our computation:
Rectangles ABCD and A'B'C'D'
Also, we can see that
Both rectangles are in opposite quadrants
This means that the line of reflection must be slant line in the adjacent quadrants
In this case, the line is line 3
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Pls help me in math!!!!!!!!!!
In the given triangle value of b is,
⇒ b = 16
We have to given that,
A triangle with three angles (b + 20), (b + 32) and 6b.
Now, WE know that;
Sum of all interior angles of triangle is 180 degree.
Hence., We can formulate;
⇒ (b + 20) + (b + 32) + 6b = 180
Solve for b,
⇒ 8b + 52 = 180
⇒ 8b = 180 - 52
⇒ 8b = 128
⇒ b = 128/8
⇒ b = 16
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100 Points! Multiple choice Geometry question. Photo attached. Thank you!
Answer:
2. C. 50.3 ft²
3. A. 75.4 ft²
Step-by-step explanation:
The lateral surface area of a cylinder is the area of the curved surface of the cylinder. It is calculated by multiplying the circumference of the base by the height of the cylinder. The formula for the lateral surface area of a cylinder is:
Lateral Surface Area = 2πrh
Where:
r is the radius of the baseh is the height of the cylinderThe total surface area of a cylinder is the area of the lateral surface plus the area of the two circular bases. The formula for the total surface area of a cylinder is:
Total Surface Area = 2πrh + 2πr^2
Where:
r is the radius of the baseh is the height of the cylinder2.
r=2 ft
h=4 ft
Lateral Surface Area = 2πrh=2*22/7*2*4=50.3 ft²
3.
Total Surface Area = 2πrh + 2πr^2=2*22/7*2*4+2*22/7*2
=50.3+25.1=75.4 ft²
Write in slope intercept form 6y+y=5
The equation 6y + y = 5 can be written in a slope-intercept form as y = 5/7.
We have,
To write the equation 6y + y = 5 in slope-intercept form (y = mx + b), we need to simplify the equation and isolate the y variable on one side.
Starting with the equation 6y + y = 5:
Combining the like terms on the left side gives us:
7y = 5
To isolate the y variable, we divide both sides of the equation by 7:
y = 5/7
Now the equation is in the form y = mx + b, where m represents the slope and b represents the y-intercept.
In this case, since the equation only contains the variable y and no x, the slope (m) is not present, and the y-intercept (b) is 5/7.
Therefore,
The equation 6y + y = 5 can be written in a slope-intercept form as y = 5/7.
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x^2-2y=5 and 4y+z=7 write z in terms of x
The equation is written as z = 7 + (20 -4x²/2)
How to make the subject
From the information given, we have that the equations as;
x²-2y=5 ( 1)
4y+z=7 (2)
From equation (1), make y the subject of formula, we have;
-2y= 5 - x²
Divide both sides by the coefficient of the variables, we have;
y = 5 - x²/-2
y = -5 + x²/2
Now, substitute the value of y in (2), we have;
4 (-5 + x²/2) + z = 7
expand the bracket
-20 + 4x²/2 + z = 7
collect the like terms, we have;
z = 7 + (20 -4x²/2)
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Find the equation of the axis of symmetry of the following parabola algebraically. y=−3x^2−42x−159
The equation of the axis of symmetry is x = -7.
Given is an equation of a parabola, y = -3x² - 42x - 159, we need to find the equation of the axis of the symmetry.
To find the equation of the axis of symmetry of a parabola in the form of y = ax² + bx + c, you can use the formula x = -b / (2a).
In this case, the given equation is y = -3x² - 42x - 159.
Comparing it to the general form, we have a = -3 and b = -42.
Applying the formula, we can calculate the x-coordinate of the vertex (the axis of symmetry):
x = -b / (2a)
x = -(-42) / (2(-3))
x = 42 / (-6)
x = -7
Therefore, the x-coordinate of the vertex is -7.
To find the equation of the axis of symmetry, we use the value of x in the form x = h, where (h, k) is the vertex.
Hence, the equation of the axis of symmetry is x = -7.
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100 Points! Geometry question. Photo attached. Name the angle of depression and the angle of elevation in each figure. Thank you!
Answer:
carnival one
elevation = SWT
depression = RTW
deer one
elevation = ABC
depression = DCB
Step-by-step explanation:
Show your work show me how you got the answer HELP DUE TOMORROW!!
im on the final exam for edmentum
What is the difference 5 2/6 -2 4/6
Answer:
Therefore, the expression becomes:
7/1 - 2/6
Now, we need to find a common denominator for the fractions, which is 6:
7/1 - 2/6 = (76)/(16) - 2/6
= 42/6 - 2/6
= (42 - 2)/6
= 40/6
Finally, we can simplify the fraction:
40/6 = 20/3
So, the difference between 5 2/6 and -2 4/6 is 20/3.
Step-by-step explanation:
First, let's subtract the whole numbers: 5 - (-2) = 5 + 2 = 7.
Next, let's subtract the fractions: 2/6 - 4/6 = (2 - 4)/6 = -2/6.
Combining the whole number and fraction results, we have:
7 - 2/6
Now, to simplify this result further, we can express 7 as a fraction with a common denominator of 6:
7 = 7/1
Use the function f(x) to answer the questions:
f(x) = 2x2 − 5x + 3
Part A: What are the x-intercepts of the graph of f(x)? Show your work.
Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work.
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.
The x-intercepts of the graph of f(x) are x = 3/2 and x = 1,the Vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point, The vertex is at (5/4, 3/8). This is the minimum point of the graph.
Part A: To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.
2x^2 - 5x + 3 = 0
To factor this quadratic equation, we look for two numbers that multiply to give 3 (the coefficient of the constant term) and add up to -5 (the coefficient of the linear term). These numbers are -3 and -1.
2x^2 - 3x - 2x + 3 = 0
x(2x - 3) - 1(2x - 3) = 0
(2x - 3)(x - 1) = 0
Setting each factor equal to zero, we get:
2x - 3 = 0 --> x = 3/2
x - 1 = 0 --> x = 1
Therefore, the x-intercepts of the graph of f(x) are x = 3/2 and x = 1.
Part B: To determine whether the vertex of the graph of f(x) is a maximum or a minimum, we look at the coefficient of the x^2 term, which is positive (2 in this case). A positive coefficient indicates that the parabola opens upwards, so the vertex will be a minimum.
To find the coordinates of the vertex, we can use the formula x = -b/2a. In the equation f(x) = 2x^2 - 5x + 3, the coefficient of the x term is -5, and the coefficient of the x^2 term is 2.
x = -(-5) / (2*2) = 5/4
Substituting this value of x back into the equation, we can find the y-coordinate:
f(5/4) = 2(5/4)^2 - 5(5/4) + 3 = 25/8 - 25/4 + 3 = 3/8
Therefore, the vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point.
Part C: To graph f(x), we can use the information obtained in Part A and Part B.
- The x-intercepts are x = 3/2 and x = 1. These are the points where the graph intersects the x-axis.
- The vertex is at (5/4, 3/8). This is the minimum point of the graph.
We can plot these points on a coordinate plane and draw a smooth curve passing through the x-intercepts and the vertex. Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the graph will be concave up.
Additionally, we can consider the symmetry of the graph. Since the coefficient of the linear term is -5, the line of symmetry is given by x = -(-5) / (2*2) = 5/4, which is the x-coordinate of the vertex. The graph will be symmetric with respect to this line.
By connecting the plotted points and sketching the curve smoothly, we can accurately graph the function f(x).
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PLEASE HELP There are 30 people waiting outside in line to enter the auditorium. There are 8 times as many people already inside the auditorium. How many people are inside the auditorium?
Answer: There are 240 people inside the auditorium.
Step-by-step explanation:
30 x 8 = 240
50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answer:
C. 50 cm²
Step-by-step explanation:
The volume of a triangular pyramid is calculated using the following formula:
Volume = (1/3) * Area of the base * Height
Area of the base is the area of the triangular base of the pyramid.Height is the distance from the apex of the pyramid to the plane of the base.For Question:
length: 5cm
Breadth:5cm
Height : 6 cm
Now,
Volume = ⅓* Area of the base * Height
Volume = ⅓*length*breadth*height
Volume= ⅓*5*5*6=50 cm³
Which two points could be removed to make this relation a function?
A. Points R and S
B. Points Q and T
C. Points P and Q
D. Points Q and R
Option (D) D. Points Q and R is used to removed to make this relation a function.
To determine which two points could be removed to make the relation a function, we need to check if there are any repeated x-values (inputs) in the given set of points. In a function, each input should have a unique output.
Let's analyze the given options:
A. Points R and S: (R, 3), (S, 5)
B. Points Q and T: (Q, 2), (T, 5)
C. Points P and Q: (P, 1), (Q, 2)
D. Points Q and R: (Q, 2), (R, 3)
In this case, if we remove points Q and R, we eliminate the repeated x-value of 2, which ensures that each input has a unique output. Therefore, the answer is:
D. Points Q and R.
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Pleaseseee help
Two one-step equations
Two equations that contains fractions
One equation with distributive property
One equation with decimals
One real-world problem that is solved by an equation
Remember that each equation must include at least one variable
The correct equations are:
[tex]3x + 2 = 11 \\\[5y - 7 = 18\][/tex][tex]\frac{2}{3}x - \frac{1}{4} = \frac{5}{6}\\\frac{3}{5}y + \frac{2}{7} = \frac{1}{3}[/tex][tex]\[2(4x - 3) = 10\][/tex][tex]\[0.5x + 0.25 = 1.75\][/tex]1. Two one-step equations:
[tex]\[3x + 2 = 11\]\[5y - 7 = 18\][/tex]
2. Two equations that contain fractions:
[tex]\[\frac{2}{3}x - \frac{1}{4} = \frac{5}{6}\]\[\frac{3}{5}y + \frac{2}{7} = \frac{1}{3}\][/tex]
3. One equation with distributive property:
[tex]\[2(4x - 3) = 10\][/tex]
4. One equation with decimals:
[tex]\[0.5x + 0.25 = 1.75\][/tex]
5. Real-world problem solved by an equation:
A bakery sells cakes for $[tex]15[/tex] each. Let's say the total cost of cakes sold in a day is $[tex]180[/tex]. We can use the equation [tex]\(15x = 180\)[/tex] to find the number of cakes sold, represented by the variable [tex]x[/tex]. Solving the equation, we find [tex](x = 12\)[/tex]), indicating that the bakery sold [tex]12[/tex] cakes that day.
Here's a basic explanation for the real-world problem:
Imagine there is a bakery that sells cakes for $[tex]15[/tex]each. We want to find out how many cakes the bakery sold in a day if the total revenue from cake sales is $[tex]180[/tex]. To solve this problem, we can use an equation. Let's represent the number of cakes sold as [tex]x[/tex].
The equation [tex]\(15x = 180\)[/tex] is used to express that the total cost of the cakes sold [tex](\$15\ per \ cake)[/tex] is equal to $[tex]180[/tex]. To solve the equation, we divide both sides by [tex]15[/tex] to isolate the variable [tex]x[/tex]. The equation simplifies to [tex]\(x = 12\),[/tex] which means that the bakery sold [tex]12[/tex] cakes that day.
By using the equation, we can determine the number of cakes sold based on the given information and calculate the desired result.
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Verify:
sin(x)/1-cos(x) - sin(x) cos(x)/1+cos(x) = csc (x) (1 + cos² (x))
Using trigonometric identities sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)),
What are trigonometric identities?Trigonometric identities are equations that contain trigonometric ratios.
To verify the trigonometric identity
sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)), we need to show that Left Hand Side, L.H.S equals Right Hand Side R.H.S. We proceed as follows.
L.H.S = sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)]
Taking the L.C.M, we have that
{sin(x)[1 + cos(x)] - sin(x)cos(x)[1 - cos(x)]}/[1 - cos(x)][1 + cos(x)]
Expanding the brackets, we have that
{sin(x) + sin(x)cos(x)] - sin(x)cos(x) + sin(x)cos²(x)]}/[1 - cos(x)][1 + cos(x)]
Simplifying, we have that
= {sin(x) + 0 + sin(x)cos²(x)]}/[1 - cos²(x)] Since ([1 - cos(x)][1 + cos(x)] = [1 - cos²(x)]
= {sin(x) + sin(x)cos²(x)]}/sin²(x) [since sin²(x) = 1 - cos²(x)]
Factorizing out sinx in the equation, we have that
= {sin(x)(1 + cos²(x)]}/sin²(x)
= (1 + cos²(x)]}/sin(x)
= cosec(x)(1 + cos²(x)]} (since cosec(x) = 1/sin(x))
= R.H.S
Since L.H.S = R.H.S, we have that
sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x))
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Rectangle CDEF has vertices C (-10, 10),D (5, 10), E (5, 5), and F (-10, 5). It is dilated 5 by a scale factor of centered at (0, 0) to
produce rectangle C'D'E'F'. What is the perimeter in units of rectangle C'D'E'F?
The perimeter of the dilated rectangle C'D'E'F' is 200 units.
To find the perimeter of the dilated rectangle C'D'E'F', we need to determine the new coordinates of its vertices after the dilation.
Given that the scale factor is 5 and the dilation is centered at (0, 0), each coordinate of the original rectangle CDEF will be multiplied by 5 to obtain the corresponding coordinate of the dilated rectangle C'D'E'F'.
The original coordinates of CDEF are:
C (-10, 10)
D (5, 10)
E (5, 5)
F (-10, 5)
To find the coordinates of the dilated rectangle C'D'E'F', we multiply each coordinate by 5:
C' = (-10 × 5, 10 × 5) = (-50, 50)
D' = (5 × 5, 10 × 5) = (25, 50)
E' = (5 × 5, 5 × 5) = (25, 25)
F' = (-10 × 5, 5 × 5) = (-50, 25)
Now, we can calculate the perimeter of the dilated rectangle C'D'E'F' by summing the lengths of its sides.
Length of side C'D':
√[(-50 - 25)² + (50 - 50)²] = √[(-75)² + 0²] = √[5625] = 75
Length of side D'E':
√[(25 - 25)² + (50 - 25)²] = √[0² + 625] = √[625] = 25
Length of side E'F':
√[(25 - (-50))² + (25 - 25)²] = √[75² + 0²] = √[5625] = 75
Length of side F'C':
√[(-50 - (-50))² + (25 - 50)²] = √[0² + 625] = √[625] = 25
Now, we add up the lengths of all four sides to find the perimeter:
Perimeter = C'D' + D'E' + E'F' + F'C'
= 75 + 25 + 75 + 25
= 200
Therefore, the perimeter of the dilated rectangle C'D'E'F' is 200 units.
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