1. To raise prices, the best alternative is to increase the price of 200-level seats to $60 or higher. To lower prices, the best alternative is to reduce the price of 300-midcourt seats to $35 or lower.
2. Administration could raise 300 mid-level seat prices to make them as attractive as the next best alternative. To calculate the optimal level, the cost of the next best alternative (200-level seats) needs to be compared with the cost of the 300-midcourt seats. If the cost of the 300-midcourt seats is lower than the cost of the 200-level seats, administration should raise the price of the 300-midcourt seats to match the cost of the 200-level seats. The attractiveness of the two options should be assessed to determine if the 300-midcourt seats should be priced at the calculated level.
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Find the equation of a line, in Slope Intercept Form, that has a slope of 3 and passes through the point (-4, 2).
[tex](\stackrel{x_1}{-4}~,~\stackrel{y_1}{2})\hspace{10em} \stackrel{slope}{m} ~=~ 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{ 3}(x-\stackrel{x_1}{(-4)}) \implies y -2= 3 (x +4) \\\\\\ y-2=3x+12\implies {\Large \begin{array}{llll} y=3x+14 \end{array}}[/tex]
Answer: y = 3x + 14
Step-by-step explanation:
The slope-intercept form is y=mx+b
We will plug in -4 for the x, 2 for the y, and 3 for the m, and leave b alone to solve for it.
2 = 3(-4) + b
2 = -12 + b
b = 14
The final equation is y = 3x + 14.
Hope this helps!
Postcard stamps are 20¢ each, while letter stamps are 33¢ each. If you have 50 stamps worth $12.60, how many of each type do you have?
Answer: 50 total stamps = 30 Postcard Stamps + 20 Letter Stamps
Step-by-step explanation:
Postcard Stamp = Ps
Letter Stamp = Ls
Ps = 20c
Ls = 33c
50 total stamps = $12.60
50 total stamps = X Ps + Y Ls
12.60 = 30 Ps + 20 Ls
$12.60 = 30(20c) + 20(33c)
50 total stamps = 30 Postcard Stamps + 20 Letter Stamps
Answer:
You have 30 postcard stamps and 20 letter stamps.
Step-by-step explanation:
To solve this you'll need to set up a system of equations. Let's use P for postcard stamps and L for letter stamps
Remember, since the price is in cents, it'll be 0.2 and 0.33.
The equations can be in any order, this is just the order I chose.
0.2P + 0.33L = 12.60
P + L = 50
Your next step is to cancel out one of the variables to solve for the other. Let's cancel out P and solve for L. (You can switch these if you want, you'll still get the same answsrs.) Remember to multiply by a negative so the variable cancels out.
Here's what your work will look like:
0.2P + 0.33L = 12.60
-0.2(P + L = 50)
Here's what your new equations will look like after distributing:
0.2P + 0.33L = 12.60
-0.2P - 0.2L = -10
Now add these two equations together. When you do so, the P cancels out, and you can now solve for L.
Here's what your new equation will look like after adding the two equations:
0.13L = 2.6
Now, divide both sides by 0.13 to get what L equals. After doing so, you should get L = 20. This means that you have 20 letter stamps.
Your last step is to plug the value of L (which is 20) into either of your original equations to solve for how many postcard stamps you have. Let's use our second equation, P + L = 50. (You can use either original equation and get the same answer, but this one is more simpler to use.)
Here's what your work will look like:
P + 20 = 50
Subtract 20 from both sides:
P = 30. This means you have 30 postcard stamps.
Hope this helps!
Use the equation to complete an algebraic proof that proves the answer is x = 7/6. Write your proof in your journal and upload your answer. You will be awarded 5 points for the statements and 5 points for the reasons.
2x+6/5= 4x-3
Proofs
The given statement [tex]x = \frac76[/tex] can be proved by using the given equation.
What is Algebra?Algebra, a discipline of mathematics where abstract symbols rather than concrete numbers are subjected to arithmetic operations and formal transformations. The idea that there is such a separate branch of mathematics, as well as the term algebra to identify it, came about gradually through time. tries to find solutions to equations that contain one or more unknown quantities. Although early mathematicians would not have been familiar with such a concept, the word "equation" is frequently employed to conveniently represent these operations when discussing the early history of algebra.
As per the given data:
[tex]\frac{2x+6}{5} = 4x -3[/tex]
Simplifying to evaluate the expression:
[tex]\frac{2x+6}{5} = 4x -3[/tex]
[tex]= 2x+6 = 5(4x -3)\\= 2x + 6 = 20x - 15\\= 18x = 21\\= x = \frac{21}{18}\\\\= x = \frac76[/tex]
Hence, the given statement [tex]x = \frac76[/tex] can be proved by using the given equation.
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Since (84/85, -13/85) is on the unit circle, with the center of
the origin, the point (a, b) in QII is also on
the circle.
(a,b) = ( , )
The point (a, b) in QII that is also on the unit circle is (-84/85, 13/85).
So, (a,b) = (-84/85, 13/85).
Since the point (84/85, -13/85) is on the unit circle with the center at the origin, the point (a, b) in QII is also on the circle if it satisfies the equation of the unit circle centered at the origin, which is x^2 + y^2 = 1.
In QII, the x-coordinate is negative and the y-coordinate is positive. So, we can write (a, b) as (-a, b), where a > 0 and b > 0.
Substituting (-a, b) into the equation of the unit circle, we get:
(-a)^2 + b^2 = 1
a^2 + b^2 = 1
Since (84/85, -13/85) is on the unit circle, we can use the Pythagorean theorem to find the value of a and b:
(84/85)^2 + (-13/85)^2 = 1
a^2 = (84/85)^2 = 7056/7225
b^2 = (-13/85)^2 = 169/7225
Taking the square root of both sides, we get:
a = sqrt(7056/7225) = 84/85
b = sqrt(169/7225) = 13/85
Therefore, the point (a, b) in QII that is also on the unit circle is (-84/85, 13/85).
So, (a,b) = (-84/85, 13/85).
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find tan a if necessary write answer as fraction
2 A DVD rental company charges $10 per month plus $0.75 per rental. Andy wants to spend no more than $25.00 per month on DVD rentals.
Select the inequality that represents how many DVDs Andy can rent in one month that satisfies this condition. The number of rentals in a month is represented by n.
The inequality that represents how many DVDs Andy can rent in one month that satisfies this condition is 10 + .75n ≤ 25
The correct answer choice is option B
Which inequality represents how many DVDs Andy can rent in one month?Cost of rental per month = $10
Additional cost = $0.75
Number of DVD's rented = n
Total amount Andy want to spend ≤ $25
The inequality:
10 + 0.75n ≤ 25
subtract 10 from both sides
0.75n ≤ 25 - 10
0.75n ≤ 15
divide both sides by 0.75
n ≤ 15 / 0.75
n ≤ 20
Ultimately, Andy can rent no more than 20 DVD's in a month.
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If h(z)=2z^(4)-10z^(3)+20z^(2)-25z+3, use synthetic division to find h(2). Submit
Using synthetic division, h(2) = -9.
To find h(2) using synthetic division, we will divide the polynomial h(z) by (z-2). The steps are as follows:
1. Write the coefficients of the polynomial in a row: 2 -10 20 -25 3
2. Write the value of z we are dividing by in the upper left corner: 2 | 2 -10 20 -25 3
3. Bring down the first coefficient: 2 | 2 -10 20 -25 3
-------------------
2
4. Multiply the first coefficient by the value of z and write the result below the second coefficient: 2 | 2 -10 20 -25 3
-------------------
2 4
5. Add the second coefficient and the result from step 4: 2 | 2 -10 20 -25 3
-------------------
2 -6
6. Repeat steps 4 and 5 for the remaining coefficients: 2 | 2 -10 20 -25 3
-------------------
2 -6 8 -6
7. The last number in the bottom row is the remainder: 2 | 2 -10 20 -25 3
-------------------
2 -6 8 -6 -9
8. The value of h(2) is the remainder, so h(2) = -9.
Therefore, the answer is h(2) = -9.
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Find the missing number to create a perfect-square binomial
___ y2-36y+81
Answer:
To create a perfect-square binomial of the form (y - k)^2, we need to find the value of k such that:
the first term of the binomial is y^2 (which is already the case)
the second term of the binomial is -2ky (which corresponds to -36y in the given expression)
the third term of the binomial is k^2 (which corresponds to 81 in the given expression)
To find k, we can use the formula:
k = (1/2)*(-b/a)
where a is the coefficient of y^2, b is the coefficient of y, and we are looking for the value of k that makes the expression a perfect square.
In this case, a = 1 and b = -36, so:
k = (1/2)(-b/a) = (1/2)(-(-36)/1) = 18
Therefore, the missing number to create a perfect-square binomial is 18:
(y - 18)^2 = y^2 - 36y + 324
One year, an estimated 6,955 sandhill cranes migrated in March. The next March, an estimated 3,480 sandhill cranes migrated. To the nearest percent, what is the percent change in the number of migrating cranes from the first March to the next? Decreases should be negative numbers and increases should be positive numbers. Do not include the percent in your answer
Rounding to the nearest percent, we get that the percent change in the number of migrating cranes from the first March to the next is approximately -50%.
How is a percentage calculated?To calculate the percentage, we must first divide the amount by the total value and then multiply the result by 100.
To find the percent change in the number of migrating cranes from the first March to the next, we can use the formula:
percent change = ((new value - old value) / old value) × 100%
where the old value is the number of migrating cranes in the first March and the new value is the number of migrating cranes in the next March.
Substituting the given values, we get:
percent change = ((3480 - 6955) / 6955) × 100%
Simplifying, we get:
\percent change = (-3475 / 6955) × 100%
percent change ≈ -50%
Rounding to the nearest percent, we get that the percent change in the number of migrating cranes from the first March to the next is approximately -50%.
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Ayudaaaa porfa ):porfavor
The vertices of the dilated image are given.
What is Dilation?Dilation is a type of transformation where the figure is enlarged or made smaller such that it preserves the shape but not size.
Every dilated image are similar figures to the original figure.
The point (x, y) with the scale factor of k will dilate to the vertex (kx, ky) if the center of dilation is origin.
1) Given for ΔQRS, Q(-1, 0), R(-1, 2) and S(-2, 1).
Scale factor = 2
Q'(2×-1, 2×0) = Q'(-2, 0)
R'(2×-1, 2×2) = R'(-2, 4)
S'(2×-2, 2×1) = S'(-4, 2)
2) Given for ΔTRK, T(-1, -2), R(1, 0) AND K(0, 1).
Scale factor = 3
T'(-3, -6), R'(3, 0) and K'(0, 3)
3) Given for ΔXYZ, X(-4, 0), Y(-4, 4) and Z(-2, -2).
Scale factor = 1/2
X'(-2, 0), Y'(-2, 2) and Z'(-1, -1)
4) If the center of dilation is not origin, then,
P'(x, y) = O(x, y) + k [P(x, y) - O(x, y)]
where, P'(x, y) is the dilated point, O(x, y) is the center of dilation, k is the scale factor and P(x, y) is the original point.
Given for ΔHAT, H(-1, -1), A(1, 0) and T(-1, 2).
Scale factor = 2
Center of dilation = (-1, 2)
H'(x, y) = (-1, 2) + 2 [(-1, -1) - (-1, 2)]
= (-1, 2) + 2 (-1+1, -1-2)
= (-1, 2) + 2 (0, -3)
= (-1, 2) + (0, -6)
= (-1, -4)
A'(x, y) = (-1, 2) + 2 [(1, 0) - (-1, 2)]
= (-1, 2) + 2 (2, -2)
= (3, -2)
T'(x, y) = (-1, 2) + 2 [(-1, 2) - (-1, 2)]
= (-1, 2) + 2 (0, 0)
= (-1, 2)
Hence the vertices of the dilated image are found.
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heeeellllllpppppp i havent done decimals and shaded parts in a whileeeee
Answer:
0.40 or 40% is shaded
Step-by-step explanation:
There are 40 shaded out of the 100 squares. Therefor, .40 is the answer
Zational Equations (Level 1) eb 22, 10:01:35 PM Watch help video Solve for all values of x : -6-(4x)/(x+2)=(8)/(x+2)
There is no solution for the equation -6-(4x)/(x+2)=(8)/(x+2).
To solve for all values of x in the equation -6-(4x)/(x+2)=(8)/(x+2), we need to use the following steps:
Step 1: Multiply both sides of the equation by (x+2) to eliminate the denominators. This gives us:
-6(x+2) - 4x = 8
Step 2: Simplify both sides of the equation by distributing and combining like terms. This gives us:
-6x - 12 - 4x = 8
-10x - 12 = 8
Step 3: Add 12 to both sides of the equation to isolate the variable on one side. This gives us:
-10x = 20
Step 4: Divide both sides of the equation by -10 to solve for x. This gives us:
x = -2
However, we need to check our solution to make sure it is not an extraneous solution. If we plug x = -2 back into the original equation, we get:
-6 - (4(-2))/(-2+2) = (8)/(-2+2)
-6 - (-8)/0 = (8)/0
Since we cannot divide by zero, our solution of x = -2 is an extraneous solution and there are no valid solutions to this equation. Therefore, the answer is no solution.
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If the velocity of an orbiting body were increased, its orbital path would change
into a parabola or hyperbola. If this were to happen, what would happen to the
object and its gravitational pull of the Sun?
The gravitational pull of the Sun on the object would decrease as the distance between them increased.
What is the gravitational pull?If the velocity of an orbiting body were increased, its orbital path would change from an elliptical orbit to a parabolic or hyperbolic orbit, depending on the extent of the increase in velocity.
The amount of gravitational pull on the object would decrease as the distance between the object and the Sun increased. This is because gravitational force decreases with distance according to the inverse-square law.
In summary, if the velocity of an orbiting body were increased to the point where its orbital path changed into a parabolic or hyperbolic orbit, the object would eventually escape the Sun's gravitational pull and move away from it in a straight line.
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The bases of a trapezoid are 8 centimeters and 12 centimeters, and the height is h centimeters. Which equation can be used to represent A, the area of the trapezoid in square centimeters?
Answer: A = 1/2(8+12)h
Step-by-step explanation: The formula for area of a trapezoid is 1/2(b1+b2)h so if you just sub in all the variables "a" would be correct
An equation for Area of trapezoid is,
A = (8 + 12) h / 2
We have,
The bases of a trapezoid are 8 centimeters and 12 centimeters, and the height is h centimeters.
We know that,
Area of trapezoid is,
A = (a + b) h / 2
Where, a and b are bases and h is height of trapezoid.
Here,, a = 8 cm,
b = 12 cm
Hence, Area of trapezoid is,
A = (8 + 12) h / 2
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solve 2t+3=12
and working please xx
2 Three different forces act on an object. They are: -- F1 = F2 = F3 = < -2, -3 > Find the net force Fnet on the object (the sum of the forces) Fnet = Find what fourth force, FA would need to be add
So the fourth force FA that would need to be added to make the net force zero is < 6, 9 >.
The net force on an object is the sum of all the forces acting on it. In this case, there are three different forces acting on the object: F1, F2, and F3. Each of these forces has a magnitude of < -2, -3 >. To find the net force Fnet, we simply add up all the forces:
Fnet = F1 + F2 + F3
Fnet = < -2, -3 > + < -2, -3 > + < -2, -3 >
Fnet = < -6, -9 >
To find the fourth force FA that would need to be added to make the net force zero, we simply need to find a force that is equal and opposite to Fnet. That is:
FA = -Fnet
FA = -< -6, -9 >
FA = < 6, 9 >
So the fourth force FA that would need to be added to make the net force zero is < 6, 9 >.
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Feb 23, 9:26:36 AM Express your answer as a polynomial in standard form. f(x)=x+5 g(x)=3x^(2)-7x+4 Find: g(f(x))
The value of g(f(x)) is 3x^(2)+23x+44.
To find g(f(x)), we need to substitute the expression for f(x) into the expression for g(x). This means that wherever we see an "x" in the expression for g(x), we will replace it with the expression for f(x).
g(f(x)) = 3(f(x))^(2)-7(f(x))+4
Now we can substitute the expression for f(x) into the equation:
g(f(x)) = 3(x+5)^(2)-7(x+5)+4
Next, we need to simplify the expression by expanding the squared term and distributing the -7:
g(f(x)) = 3(x^(2)+10x+25)-7x-35+4
g(f(x)) = 3x^(2)+30x+75-7x-35+4
Finally, we can combine like terms to get the polynomial in standard form:
g(f(x)) = 3x^(2)+23x+44
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Pythagorean Theorem rounding 8 and 17?
The length of the missing side is 15.
What is a theorem?
A statement that can be proven true using well-known mathematical procedures and arguments is known as a theorem. A theorem, in general, is an application of a general principle that joins it to a larger theory. Proof is a procedure used to demonstrate a theorem's correctness.
A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the area of the square whose side is the hypotenuse.
The hypotenuse is 17. The length of one leg is 8.
The length other leg is
√(17² - 8²)
= √(289 - 64)
= √(225)
= 15
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A grocer wants to mix two kinds of coffee. One kind sells for $0. 90
per pound, and the other sells for $2. 40
per pound. He wants to mix a total of 28
pounds and sell it for $1. 45
per pound. How many pounds of each kind should he use in the new mix? (Round off the answers to the nearest hundredth. )
He should use 11.11 pounds of the $0.90 per pound coffee, and 16.89 pounds of the $2.40 per pound coffee for a total cost of 28 pounds.
x + y = 28
0.90x + 2.40y = 38.20
x = 11.11, y = 16.89
The grocer wants to mix 28 pounds of two kinds of coffee, one selling for $0.90 per pound and the other for $2.40 per pound, and sell it for $1.45 per pound. To determine how many pounds of each kind to use in the new mix, we can set up a system of equations. Let x represent the quantity of pounds of coffee priced at $0.90 per pound and y the quantity of pounds of coffee priced at $2.40 per pound. Thus, x + y = 28 (the total number of pounds) and 0.90x + 2.40y = 38.20 (the total cost of the 28 pounds, at $1.45 per pound). Solving this system of equations yields x = 11.11 pounds of the $0.90 per pound coffee and y = 16.89 pounds of the $2.40 per pound coffee. Therefore, the grocer should use 11.11 pounds of the $0.90 per pound coffee, and 16.89 pounds of the $2.40 per pound coffee for a total of 28 pounds.
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Please answer fast !
Question 5(Multiple Choice Worth 2 points)
(Identifying Functions LC)
The mapping diagram represents a relation where x represents the independent variable and y represents the dependent variable.
A mapping diagram with one circle labeled x values containing values negative 3, negative 1, 1, 3, and 5 and another circle labeled y values containing values 0, 2, and 5 and arrows from negative 3 to 0, negative 1 to 2, 1 to 0, 3 to 2, and 5 to 5.
Is the relation a function? Explain.
No, because for each input there is not exactly one output
No, because for each output there is not exactly one input
Yes, because for each input there is exactly one output
Yes, because for each output there is exactly one input
The correct answer is "No, because for each input there is not exactly one output".
What are Functions?A function is a mathematical rule that assigns a unique output value for each input value. It is a set of ordered pairs where the first element is the input and the second element is the output.
The relation is not a function because for the input value of 1, there are two possible output values: 0 and 2. In a function, each input can have only one output value. However, in this relation, the input value of 1 is associated with two different output values, so it violates the definition of a function. Therefore, the correct answer is "No, because for each input there is not exactly one output".
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Val rented a bicycle while she was on vacation. She paid flat rental fee for 55. 0
The equation that can be used to determine the number of days is $123 = $55 + $8.50d
What is the equation?An equation is an expression that has an equal to sign. A linear equation is an equation that has a single variable raised to the power of one. The form of a linear equation is:
y = mx + b
Where:
m = slope b = interceptThe form of the linear equation that can be used to determine the number of days is:
Total cost = flat fee + (cost per day x number of days)
$123 = $55 + ($8.50 x d)
$123 = $55 + $8.50d
d = ($123 - $55) / 8.50
d = 8 days
Here is the complete question:
Val rented a bicycle while she was on vacation. She paid a flat rental fee of $55.00, plus $8.50 each day. The total cost was $123. Write an equation you can use to find the number of days she rented the bicycle.
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Someone please answer and explain a & b. Image attached, thanks.
Jada has earned a total of 400 points so far.
141 + 87 + 81 + 91 = 400
There are a total of 450 points possible so far.
150 + 100 + 100 + 100 = 450
Jada has 400/450 or about 88.89% of the possible points, so no, she does not have 90%.
Adding in a 100-point test, the total number of points would become 550 and 90% of 550 is 495 points.
0.90 x 550 = 495
To finish the class with an A, Jada needs to have 495 points. She currently has 400 points. This means she needs a 95 on the final to finish the class with an A.
1. When analyzing population density in Canada it was found the density was 14 people/km2. Given this, how many people do expect would be found within an area of 1400 m by 1.3 km? (Please show work).
2. A cubed object has a density of. 25 g/m3. Given this would what be the mass of the cube for the object that has the side lengths of 56 cm. (Please show work).
The number of people we would expect to be found within an area of 1400 m by 1.3 km is approximately 25.
The mass of the cube for the object that has the side lengths of 56 cm is 0.043904 g.
1. To find the number of people in an area of 1400 m by 1.3 km, we need to first convert the measurements to the same unit. Since the population density is given in people/km2, we will convert the measurements to kilometers.
1400 m = 1.4 km
Now we can multiply the two measurements to find the area:
1.4 km x 1.3 km = 1.82 km2
Next, we can use the population density to find the number of people in this area:
14 people/km2 x 1.82 km2 = 25.48 people
Therefore, we would expect to find approximately 25 people in an area of 1400 m by 1.3 km.
2. To find the mass of the cube, we first need to find the volume. Since the side lengths are given in centimeters, we will convert them to meters:
56 cm = 0.56 m
Now we can find the volume of the cube by multiplying the side lengths:
0.56 m x 0.56 m x 0.56 m = 0.175616 m3
Next, we can use the density to find the mass of the cube:
0.25 g/m3 x 0.175616 m3 = 0.043904 g
Therefore, the mass of the cube is 0.043904 g.
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21 How many solutions does the equation 2 + 6(x-4)= 3x - 18 + 3x have? A) O B 1 (c) 2 D) Infinite
The difference in length of a spring on a pogo stick from its non-compressed length when a teenager is jumping on it after θ seconds can be described by the function f of theta equals 2 times cosine theta plus radical 3 period
Part A: Determine all values where the pogo stick's spring will be equal to its non-compressed length. (5 points)
Part B: If the angle was doubled, that is θ became 2θ, what are the solutions in the interval [0, 2π)? How do these compare to the original function? (5 points)
Part C: A toddler is jumping on another pogo stick whose length of their spring can be represented by the function g of theta equals 1 minus sine squared theta plus radical 3 period At what times are the springs from the original pogo stick and the toddler's pogo stick lengths equal? (5 points)
The solutions for θ in the interval [0, 2π) where cos(θ) = -√3/2 are θ = 2π/3 and θ = 4π/3.
The graph of f(θ) is shifted and stretched when compared to the graph of f(2θ).
What is Trigonometry?
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and the trigonometric functions that describe those relationships.
Part A:
To find when the pogo stick's spring will be equal to its non-compressed length, we need to solve for when f(θ) = 0.
f(θ) = 2cos(θ) + √3 = 0
2cos(θ) = -√3
cos(θ) = -√3/2
The solutions for θ in the interval [0, 2π) where cos(θ) = -√3/2 are θ = 2π/3 and θ = 4π/3.
Part B:
If we double the angle, θ becomes 2θ, and the function becomes:
f(2θ) = 2cos(2θ) + √3
Using the double angle formula for cosine, we can rewrite this as:
f(2θ) = 2(2cos²(θ) - 1) + √3
f(2θ) = 4cos²(θ) - 2 + √3
Substituting cos(θ) = -√3/2, we get:
f(2θ) = 4(-3/4) - 2 + √3
f(2θ) = -3 + √3
So the solutions for 2θ in the interval [0, 2π) where f(2θ) = 0 are:
2θ = π/6 and 2θ = 11π/6
Dividing by 2, we get the solutions for θ:
θ = π/12 and θ = 11π/12
These solutions are different from the solutions in Part A, and the graph of f(θ) is shifted and stretched when compared to the graph of f(2θ).
Part C:
To find when the lengths of the springs are equal, we need to solve the equation f(θ) = g(θ).
2cos(θ) + √3 = 1 - sin²(θ) + √3
2cos(θ) = 1 - sin²(θ)
Using the identity sin²(θ) + cos²(θ) = 1, we can rewrite this as:
2cos(θ) = cos²(θ)
cos(θ)(cos(θ) - 2) = 0
The solutions for θ in the interval [0, 2π) where the lengths of the springs are equal are:
θ = 0, θ = π/3, θ = 2π/3, θ = π, θ = 4π/3, θ = 5π/3
We can check that f(θ) = g(θ) at each of these values.
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Two  parallel lines, M and N Are cut by the transversal as shown suppose M1 equals 70 
The measure of angle 2 is 70° and the measure of angle 3 is also 70°°.
What is a transversal?We know when a transversal intersects two parallel lines at two distinct points,
Two pairs of interior and alternate angles are formed such that the measure of interior angles are same and the measure of alternate angles is also the same.
From the given information ∠1 and ∠3 are pairs of alternate interior angles.
Therefore, m∠1 = m∠B = 70°.
We also know that the vertically opposite angles are equal,
Here ∠1 and ∠2 are vertically opposite angles,
Therefore, m∠1 = m∠2 = 70°.
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The polynomial p(x) had the following roots: 5,-6,3-2i. Write the polynomial in factored form.
The polynomial in factored form is: (x-5)(x+6)(x-(3-2i))(x-(3+2i))
Since complex roots always come in conjugate pairs, we know that if 3-2i is a root, then 3+2i must also be a root. Therefore, we can write the polynomial in factored form by multiplying together the factors (x-5), (x+6), (x-(3-2i)), and (x-(3+2i)).
Alternatively, we can write the polynomial in factored form using the formula for a polynomial with complex roots:
p(x) = (x-r1)(x-r2)(x-r3)(x-r4)
where r1, r2, r3, and r4 are the roots of the polynomial. In this case, r1 = 5, r2 = -6, r3 = 3-2i, and r4 = 3+2i. Substituting these values into the formula gives us:
p(x) = (x-5)(x+6)(x-(3-2i))(x-(3+2i))
Therefore, the polynomial in factored form is written in the following form: (x-5)(x+6)(x-(3-2i))(x-(3+2i)).
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A rabbit culture begins with 10 rabbits that double in amount at the end of every month. How many rabbits are grown during the 12th month.
Using geometric sequences, 20,480 rabbits are grown during the 12th month.
What is the geometric sequences?A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the preceding term by a fixed non-zero number called the common ratio.
At the end of the first month, there are 10 x 2 = 20 rabbits.
At the end of the second month, there are 20 x 2 = 40 rabbits.
Similarly, at the end of the third month, there are 40 x 2 = 80 rabbits.
This is a geometric sequence with a common ratio of 2 and a first term of 10. The number of rabbits at the end of the 12th month would be:
10 x 2^12 = 10 x 4096 = 40,960 rabbits
To find the number of rabbits grown during the 12th month, we need to subtract the number of rabbits at the end of the 11th month from the number of rabbits at the end of the 12th month:
[tex]40,960 - (10 x 2^{11}) = 40,960 - 20,480 = 20,480[/tex]
Hence, 20,480 rabbits are grown during the 12th month.
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on a map, Orlando is 178 mm due south of Niagara Falls, Denver is 273 mm from Orlando, and Denver is 235 mm from. niagara falls. find the angle at niagara falls.
Niagara Falls' inclination is roughly 69.8 degrees because Orlando is 178 millimetres due south of the falls, which puts it at a slight angle to Orlando.
What is angle ?Two rays or lines that intersect at the angle's vertex form an angle, a geometric figure in mathematics. Angle sizes range from 0 degrees (a zero angle) to 360 degrees, and they are typically measured in degrees or radians (a full rotation). Mathematical disciplines like geometry, trigonometry, and calculus all use angles as a fundamental concept. They're used to explain how things are positioned in space.
given
The Law of Cosines can be used to calculate Niagara Falls' slope. Let's call the distances from Niagara Falls, Denver, and Orlando, as well as the distance from Denver to Orlando, "a," "b," and "c," respectively. Then:
235mm for a and 178mm for b.
c = 273 mm
For any triangle with side a and angle A across from side a, the following is true in accordance with the Law of Cosines:
A² = 2bc cos - b² + c²
Angle A is the angle at Niagara Falls, and we're trying to find it. The formula above is rearranged to give us:
cos A = (b²+c²-a²)/2bc
If we substitute our current numbers, we get:
cos A is equal to (2 x 178 x 273) / (178 x + 273 - 2352). cos A = 0.3506
Inverse cosine of both sides is calculated, and the result is:
[tex]A = cos^{-1} (0.3506) (0.3506)[/tex]
69.8 degrees A
Niagara Falls' inclination is roughly 69.8 degrees because Orlando is 178 millimetres due south of the falls, which puts it at a slight angle to Orlando.
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Complete question:
On a map, Orlando is 178 mm due south of Niagara Falls, Denver is 273 mm from Orlando, and Denver is 235 mm from Niagara Falls. Find the angle at Niagara Falls.
leading coefficient is negative, zero of 3 has multiplicity of 3, zero of 2 has multiplicity of 2, zero of 4 has multiplicity of 1
The polynomial with the given information can be represented as: P(x) = a(x-3)3(x-2)2(x-4). Where a is the leading coefficient. Since the leading coefficient is negative, we can assume that a = -1. Therefore, the polynomial can be written as: P(x) = -(x-3)3(x-2)2(x-4)
This polynomial satisfies all the given conditions:
- The leading coefficient is -1, which is negative.
- The zero of 3 has a multiplicity of 3, as indicated by the exponent of (x-3)3.
- The zero of 2 has a multiplicity of 2, as indicated by the exponent of (x-2)2.
- The zero of 4 has a multiplicity of 1, as indicated by the exponent of (x-4).
Therefore, the polynomial P(x) = -(x-3)3(x-2)2(x-4) is the correct answer.
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