1) cos (θ / 2) = √[(1 + cos θ) / 2], sin (θ / 2) = √[(1 - cos θ) / 2], tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) (x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°). The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) The linear function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
How to apply trigonometry on deriving formulas and transforming points
1) The following trigonometric formulae are used to derive the half-angle formulas:
sin² θ / 2 + cos² θ / 2 = 1 (1)
cos θ = cos² (θ / 2) - sin² (θ / 2) (2)
First, we derive the formula for the sine of a half angle:
cos θ = 2 · cos² (θ / 2) - 1
cos² (θ / 2) = (1 + cos θ) / 2
cos (θ / 2) = √[(1 + cos θ) / 2]
Second, we derive the formula for the cosine of a half angle:
cos θ = 1 - 2 · sin² (θ / 2)
2 · sin² (θ / 2) = 1 - cos θ
sin² (θ / 2) = (1 - cos θ) / 2
sin (θ / 2) = √[(1 - cos θ) / 2]
Third, we derive the formula for the tangent of a half angle:
tan (θ / 2) = sin (θ / 2) / cos (θ / 2)
tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) The formulae for the conversion of coordinates in rectangular form to polar form are obtained by trigonometric functions:
(x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) Let be the point (x, y) = (2, 3), the coordinates in polar form are:
r = √(2² + 3²)
r = √13
θ = atan(3 / 2)
θ ≈ 56.309°
The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°).
Let be the point (r, θ) = (4, 30°), the coordinates in rectangular form are:
(x, y) = (4 · cos 30°, 4 · sin 30°)
(x, y) = (2√3, 2)
The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) Let be the linear function y = 5 · x - 8, we proceed to use the following substitution formulas: x = r · cos θ, y = r · sin θ
r · sin θ = 5 · r · cos θ - 8
r · sin θ - 5 · r · cos θ = - 8
r · (sin θ - 5 · cos θ) = - 8
r = - 8 / (sin θ - 5 · cos θ)
The linear function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
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PLEASE HELP ME !” ASAP?
Answer:
39°
Step-by-step explanation:
Opposite angles of a parallelogram are congruent.
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
" The opposite angles of a Parallelogram are always equal to one another "
[tex] \qquad \large \sf {Conclusion} : [/tex][tex] \qquad \large \sf {Conclusion} : [/tex]
Angle M = Angle N = 39°what is the area of the figure bellow?
Answer:
The area of the figure is C. 48.5 cm²
There are 48 people coming to a family reunion. One fourth of them live out of state. How many live in-state?
Step-by-step explanation:
1/4×48
=12
So if 12people are out of state
48-12=36,is the number of people living in the state
Solve for y.......
[tex]6 = 2(y + 2)[/tex]
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
y = 1[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex] \qquad❖ \: \sf \:6 = 2(y + 2)[/tex]
[tex] \qquad❖ \: \sf \:(y + 2) = \cfrac{6}{2} [/tex]
[tex] \qquad❖ \: \sf \:y + 2 = 3[/tex]
[tex] \qquad❖ \: \sf \:y = 3 - 2[/tex]
[tex] \qquad❖ \: \sf \:y = 1[/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
Value of y = 1Question 8 of 10
Factor the polynomial: -5x³ - 10x² - 15x
OA. 5x(x²+2x-3)
OB. -5x(-x2 - 2x - 3)
C. -5x(x2 + 2x - 15)
OD. -5x(x2+2x+3)
In the diagram, is parallel to. Also, is drawn such that the length of is half the length of. If sin A = 0.5, then what is sin E?
sinE is 0.5
What are similar triangles?
Two triangles will be similar if the angles are equal (corresponding angles), and sides are in the same ratio or proportion (corresponding sides). Similar triangles may have different individual lengths of the sides of triangles, but their angles must be equal and their corresponding ratio of the length of the sides must be the same.
Clearly, given triangle AFB and triangle DFE are similar.
We know that Similar Triangles have the same corresponding angle
We can find sinE as show below:
From diagram clearly
∠A=∠E
and ∠B=∠D
Since, ∠A=∠E
Taking sin on both sides
sinA=sinE
Give, sinA=0.5
sinA=sinE=0.5
⇒ sinE=0.5
Hence, sinE is 0.5
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What is 6 wholes divided by 2/3 ?
Answer:
9
Step-by-step explanation:
6 ÷ 2/3 = 6 / (2/3 ) = (6/1) / (2/3) = 6/1 x 3/2 = 18/2 = 9
Tell if each pairs of segments are congruent
Yes/No
A. AB and BC
B. AC and BD
A. No, AB and BC are not congruent
B. No, AC and BD are not congruent
Calculating length of a segmentFrom the question, we are to determine if the given pairs of segments are congruent
To do this, we will determine the length of the line segments
Consider the given number line,
AB = -2 - (-5) = -2 + 5 = 3
BC = 0 - (-2) = 0 + 2 = 2
AC = 0 - (-5) = 0 + 5 = 5
BD = 5 - (-2) = 5 + 2 = 7
Now,
A. AB and BC
AB = 3 and BC =2
∴ AB and BC are not congruent.
Thus, No
A. AC and BD
AC = 5 and BD = 7
∴ AC and BD are not congruent.
Thus, No
Hence,
A. No, AB and BC are not congruent
B. No, AC and BD are not congruent
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According to indical equation solve 3×=1\81
Answer:
x = - 4
Step-by-step explanation:
using the rule of indices
[tex]\frac{1}{a^{n} }[/tex] = [tex]a^{-n}[/tex] , then
[tex]3^{x}[/tex] = [tex]\frac{1}{81}[/tex] = [tex]\frac{1}{3^{4} }[/tex] = [tex]3^{-4}[/tex] , that is
[tex]3^{x}[/tex] = [tex]3^{-4}[/tex] , thus
x = - 4
6) Evaluate 4x if x = -1/4
Answer: -1
Step-by-step explanation:
Evaluate 4x if x = -1/4
4(-1/4) = -1
Answer: -1
Step-by-step explanation:
4x and x = -1/4
plug x into the expression
[tex]4(-\frac{1}{4} )[/tex]
simplifies to:
[tex]-\frac{4}{4}[/tex]
4 divided by 4 is just 1, but it is negative so your final answer is
-1
13. A data set has a mean of x=3905 and a standard deviation of 110. Find the z-score for each of the following.
14. A random sample of 80 tires showed that the mean mileage per tire was 41,400mi, with a standard deviation of 4700mi.
Question 13
Part (a)
[tex]\frac{3840-3905}{110}=\boxed{0.5\overline{90}}[/tex]
By similar logic, the other answers are
(b) 2.6818181....
(c) 3.5909090...
(d) 1.1363636...
Question 14
Part (a)
[tex]\frac{46800-41400}{4700} \approx \boxed{1.15}[/tex]
Part (b)
[tex]-2.57=\frac{x-41400}{4700} \\ \\ -2.57(4700)=x-41400 \\ \\ x=41400-2.57(4700) \\ \\ x \approx \boxed{29300}[/tex]
HELPPPP PLSSSSSSSSS!!!!!!!!??!
Answer:
(3+3) x (3+1)
Step-by-step explanation:
nice
We have give 4 numbers that are 1,3,3,3. we have to apply operations on it to make it 24.
Solution :» (1 + 3) × (3 + 3)
» (4) × (6)
» 24
Here's our answer..!!
An insurance provider states that their customers save at least, on average, 300 dollars per year by switching to them, with a standard deviation of 150 dollars. Before we decide to switch to the new company and go through all of the hassle, we want to test the claim. So, we go out and sample 64 individuals who switched to the new insurance company and found them to have saved an average of 255 dollars per year. Do we have enough evidence at the α = 0.05 level to state that the insurance provider is false in their claim? Discussion Prompts Answer the following questions in your initial post: What are the hypotheses based on the words given in the problem? Should we use a Z or T distribution in this case? What is our Z or T statistic? What is the P-value? Based on your p-value and alpha, what conclusion will we make? Based on your results, would you switch to this company? Explain why or why not (Note: this can go beyond the use of statistics, but statistical analysis can help our decisions)
The solution to the question is mathematically given as
1)
H0: M [tex]\geqslant[/tex] 300
H0: M [tex]\geqslant[/tex] 300
2)
Z distribution.
3)
z=-2.4
4)
P=0.0082
5)
"H0" is rejected as a hypothesis with a level of significance of 0.05.
What is the hypothesis ?Generally, the equation for is mathematically given as
Solutions
we have, [tex]$u=300$$$\begin{aligned}& \sigma=150 \text { dollars } \\N &=64 \text { individuals } \\\bar{x} &=255 \text { dollare. } \\a=& 0.05 .\end{aligned}[/tex]
(1):
H0: M [tex]\geqslant[/tex] 300 dollars; customers save at least 300 dollars per year by switching to them.
H0: M [tex]\geqslant[/tex] 300 dollars:
(This is a left-tailed test)
(2):
when [tex]\sigma[/tex] is known, we use the Z test. we use Z distribution.
(3):
[tex]\begin{gathered}z=\frac{\bar{x}-\mu}{6 / \sqrt{n}}=\frac{255-300}{150 / \sqrt{64}}=\frac{-45}{18.75}=-2.4 \\z=-2.4\end{gathered}[/tex]
(4):
[tex]Pvalue $=P(z < -2 \cdot 4)$ $=0.0082 \quad\{$ wing $z$ tables $\}$ \\\\\text { Pvalue }=0.0082[/tex]
(5):
In conclusion, Since p-value = 0·0082 <0.05
This is statistically significant at the 0.05 level.
We thus "Refect H0" using a threshold of significance of 0.05.
"H0" is rejected as a hypothesis with a level of significance of 0.05.
There is not enough data to support the assertion that consumers may save at least $300 annually by switching insurance providers.
Therefore, we would not consider making the transition to this firm.
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The base of the following pyramid is a square. If the volume of the pyramid is 400 feet³, what is the missing length? Round the answer to the nearest hundredth. The S=11ft
The length of the missing side is 9.92 ft given that the base of the following pyramid is a square, base side length(s) is 11 ft and the volume of the square pyramid is 400 ft³. This can be obtained using the formula of finding the volume of the pyramid with base as square.
Calculate the length of the missing side:The formula of finding the volume of the pyramid with base as square is,
⇒ V = s²h/3
where V is the volume of the square pyramid, s is the side length of the base and h is the height of the square pyramid.
Here in the question it is given that,
it is a pyramid with base as squarethe side length of the base is 11 ftvolume of the square pyramid is 400 ft³that is, s = 11 ft, V = 400 ft³
By using the formula of finding the volume of the pyramid with base as square we get,
V = s²h/3
400 = (11)²h/3
By rearranging the equation we get,
h = 400 × 3/11²
h = 9.92 ft
Hence the length of the missing side is 9.92 ft given that the base of the following pyramid is a square, base side length(s) is 11 ft and the volume of the square pyramid is 400 ft³.
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Can someone help me with a step by step process of how to solve this? Calculus 2
If you fix a point on the curve in the given interval, and revolve that point about the [tex]x[/tex]-axis, it will trace out a circle with radius given by the function value [tex]y[/tex] for that point [tex]x[/tex]. The perimeter of this circle is then [tex]2\pi(8\sqrt x) = 16\pi \sqrt x[/tex].
The surface in question is essentially what you get by joining infinitely many of these circles at every point in the interval [0, 9].
So, the surface area is given by the definite integral
[tex]\displaystyle \int_0^9 16\pi \sqrt x \, dx = 16\pi\times\frac23 x^{3/2}\bigg|_{x=0}^{x=9} = \frac{32\pi}3 \left(9^{3/2} - 0^{3/2}\right) = \boxed{288\pi}[/tex]
a dust mite is 400 pm long under a microscope, it looks 100,000 pm long. what magnification scale was used
Answer: 250:1
Step-by-step explanation:
Divide the measurement of appearance by the actual measurement.
[tex]100,000/400=250[/tex]
The magnification scale is 250:1. The scale is used by a 250X magnifying lense.
P=2(L+W) Solve for W
Answer:
W = (P -2L)/2
Step-by-step explanation:
P = 2(L + W) Distribute the 2
P = 2L + 2W Subtract 2L from both sides
P - 2L = 2W Divide both sides by 2
(P-2L)/2 = W
Find the product. Write your answer in exponential form. 8^-2•8^-9
Answer:
8 ^-11
Step-by-step explanation:
It should be noted that in this case, since it's the product that we want to get, it's important to add the power together. This will be:
8^-2 × 8^-9
= 8^(-2 + -9)
= 8^-11
What is the volume of a cylinder with a radius of 3 inches and a height of 5 inches? Round to the nearest tenth if necessary.
The volume of the cylinder in discuss in which case, the radius of the cylinder is 3 inches and the height is inches is; V = 141.3in³.
What is the volume of the the described cylinder?The cylinder in discuss according to the task content has radius, 3 inches and height, 5 inches.
Since the volume of a cylinder by convention is given as; V = πr²h
Therefore, the volume of the cylinder in discuss is;
V = π (3²)×5
V = 3.14 × 45
V = 141.3in³
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2. The product of two consecutive even numbers is 168. What are the numbers?
Answer:
12 and 14
Step-by-step explanation:
12 * 14 = 168
Answer:
12 & 14
Step-by-step explanation:
the work: x = first even number, (x+2) = second even number
The equation
[tex]x(x+2)=168[/tex]
[tex]x^{2} +2x=168[/tex]
[tex]x^{2} +2x-168=0[/tex]
Factoring
[tex](x-12)(x+14)=0[/tex]
first factor
[tex]x-12=0\\x=12[/tex]
Second factor
[tex]x+14=0\\x=-14[/tex]
The solution is the positive number
x = 12 is the first even number
x + 2 = 12 + 2 = 14 is the second even number
Hope this helps
(5.01 LC) A square is shown below. which expression can be used to find the area, in square units, of the Shaded triangle in the Square?
(answer options are in picture)
The expression can be used to find the area, in square units, of the Shaded triangle in the Square is 1/2 ( 4 · 4 ) .
What is the area of a square?The area of a square can be calculated as the square of the sides of a given figure.
The area of a square = side x side
where s = side length.
If a diagonal is drawn to create 2 halves, then area of each half is 8.
So, The area of a square = 1/2 ( 4 · 4 ) = 16.
If a square has a side length of 4, then the area of the square is 16.
The expression can be used to find the area, in square units, of the Shaded triangle in the Square is 1/2 ( 4 · 4 ) .
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Cos3A ×cos2A =cos A ×cos 2A -sin4A×sin A=prove it
Answer:
Step-by-step explanation:
cosA×cos 2A-sin4A×sinA
=cosAcos2A-2sin2Acos2A sin A
=cos 2A(cosA-2sin2AsinA)
=cos 2A(cosA-2×2sinAcosAsin A)
=cos2A×cosA(1-4sin²A)
=cos 2AcosA(1-4(1-cos²A))
=cos2A×cosA(1-4+4cos²A)
=cos 2A(-3cosA+4cos³A)
=cos 2A(4cos³A-3cosA)
=cos 2A×cos3A
Rewrite the quadratic function in vertex form (which is also called a standard form) and give the vertex.
f(x)=x^2-3x
f(x)=
Enter your answer as a point (a,b)
Vertex:
Please explain how you arrived at that answer.
The vertex form of the equation f(x) = x^2 - 3x, is f(x) = (x - 3/2)^2 - 9/5
How to rewrite the quadratic function?The quadratic function is given as:
f(x) = x^2 - 3x
Differentiate the function
f'(x) = 2x - 3
Set the function to 0
2x - 3 = 0
Add 3 to both sides
2x = 3
Divide by 2
x = 3/2
Set x = 3/2 in f(x) = x^2 - 3x
f(x) = 3/2^2 - 3 * 3/2
Evaluate
f(x) = -9/5
So, we have:
(x, f(x)) = (3/2, -9/5)
The above represents the vertex of the quadratic function.
This is properly written as:
(h, k) = (3/2, -9/5)
The vertex form of a quadratic function is
f(x) = a(x - h)^2 + k
So, we have:
f(x) = a(x - 3/2)^2 - 9/5
In f(x) = x^2 - 3x,
a = 1
So, we have:
f(x) = (x - 3/2)^2 - 9/5
Hence, the vertex form of the equation f(x) = x^2 - 3x, is f(x) = (x - 3/2)^2 - 9/5
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After 3 minutes, a submarine had descended to −550 feet. After 8 minutes, the submarine had descended to −600 feet. Assuming a linear function, write an equation in the form d(t)=mt+b that shows the depth, d(t), after t minutes.
Answer:
d(t)=-10t-520
Step-by-step explanation:
First you have to find the gradient (m) using the formula [tex]m=\frac{d(t)2-d(t)1}{t2-t1}[/tex]
using the two points you have been given: 1(3,-550) and 2(8,-600)
so you substitute these points into the equation [tex]m=\frac{-600+500}{8-3}[/tex]
m=-10
Now that you have found the m-value you need to find the b-value by using one of the points you already have, I will use Point 2 (8,-600), and substitute it into the equation and solve
d(t)=-10t+b
-600=-10*8+b
-600=-80+b
-520=b
Therefore, the equation is d(t)=-10t-520
8v(v+3)(v+3) rewrite the expression factoring out (v+3)
If we rewrite the expression 8v(v+3)(v+3) then we will get [tex]v^{3}+6v^{2} +9v[/tex]
Given an expression 8v(v+3)(v+3).
We are required to rewrite the given expression.
Expression is a combination of numbers, symbols, fraction , determinants,indeterminants, coefficients. They are not mostly found in equal to form. It expresses a relationship of a line or something else.
The expression is 8v(v+3)(v+3). There are three terms in which variable is v.
To rewrite the expression we have to multiply all the three terms with each other which can be done as under:
=8v(v+3)(v+3)
=[tex](8v^{2}+24v)(v+3)[/tex]
=[tex]8v^{3}+24v^{2}+24v^{2}+72v[/tex]
=[tex]8v^{3}+48v^{2} +72v[/tex]
Now divide it by 8.
=[tex]v^{3}+6v^{2} +9v[/tex]
Hence if we rewrite the expression 8v(v+3)(v+3) then we will get [tex]v^{3}+6v^{2} +9v[/tex] .
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The volume of a sphere is 144π. What is its Surface Area to the nearest whole number?
HOW MANY DIFFERENT ARRANGEMENTS CAN BE MADE WITH THE NUMBERS
28535852
Answer:
1,680
Step-by-step explanation:
8 positions with basically 8 choices.
that is 8! arrangements.
but 2 is there 2 times.
8 is there 2 times.
5 is there 3 times.
only 3 is a single digit.
so, we need to eliminate every arrangement, where the 2s trade places, where the 5s trade places, and where the 8s trade places, because they are the same numbers.
that means we have to divide the 8! by 2!, then again by 2!, and again by 3!.
8! / (2! × 2! × 3!) = 8! / 24 = 1,680
Becca polled the members of her volleyball team to see what their opinions were regarding the awards banquet. Her results were:
A total of 16 people wanted the steakhouse option.
• Seafood buffet but not bsteakhouse: 8
Steakhouse but not seafood buffet: 9
.neither 4
Based on the total number of people who want the steakhouse option, the probability that a student will not want the seafoodbuffet if they want the steakhouse option is 24%
What is the probability of the students' choice?The probability that a student does not want the seafood buffet given that they want the steakhouse option is:
= Number of people who want steakhouse but not seafood / Total surveyed
Solving gives:
= 9 / (16 + 8 + 9 + 4)
= 9 / 37
= 24%
Full question is:
If a student wants the steakhouse option, what is the probability that they will not want the seafoodbuffet?
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Question: 2 Based on the mass of lactose of 0.28 grams in 1 mL of skim milk, calculate the percent composition of lactose in skim milk.
Considering the definition of mass volume percent, the percent composition of lactose in skim milk is 28%.
Mass volume percentThe concentration of solutions is the amount of solute contained in a given amount of solvent or solution. In other words, the relationship between the amount of solute and the amount of solvent is called concentration.
Mass volume percent (% m/V) is a measure of concentration that indicates the number of grams of solute in each 100 mL of solution.
In other words, mass-volume percent is an intensive property that is defined as the mass of solute (in grams) that there are in 100 mL of solution and it is calculated by applying the following equation:
[tex]Mass volume percent=\frac{mass solute (grams)}{volume solution (mL)}x100[/tex]
This is, mass volume percent is defined as the ratio of the mass of solute that is present in a solution, relative to the volume of the solution, as a whole.
Percent composition of lactose in skim milkIn this case, you know:
mass solute= 0.28 gramsvolume solution= 1 mLReplacing in the definition of mass volume percent:
[tex]Mass volume percent=\frac{0.28 grams}{1 mL}x100[/tex]
Solving:
mass volume percent= 28 %
Finally, the percent composition of lactose in skim milk is 28%.
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14. A photograph measures eight inches wide and ten inches long. The picture is enlarged to fit on a wall. If the new larger picture is 208 inches wide, how long is it?
Answer:
The file contains the solution to the question