Pythagorean theorem: Show that ⟨x, y⟩ = 0
Let V be an Euclidean space and let x, y ∈ V.(a) If ∥x + y∥ = ∥x − y∥ then determine ⟨x, y⟩.We know that ∥x + y∥² = ⟨x + y, x + y⟩ = ⟨x, x⟩ + ⟨x, y⟩ + ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ⟨x − y, x − y⟩ = ⟨x, x⟩ − ⟨x, y⟩ − ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x + y∥ = ∥x − y∥, we have ∥x∥² + 2⟨x, y⟩ + ∥y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²⟨x, y⟩ = 0(b) Show that 2⟨x, y⟩ ≤ ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x + y∥² ≥ 0So ∥x∥² + 2⟨x, y⟩ + ∥y∥² ≥ 0⟨x, y⟩ ≤ (∥x∥² + ∥y∥²)/2(c) If (z, y) = 0 for all z ∈ V then show that y = 0.Let z = y, then (y, y) = 0⟨y, y⟩ = ∥y∥² = 0∥y∥ = 0So y = 0(d) If ∥x∥ = ∥y∥ for some x, y ∈ V then ⟨x + y, x − y⟩ = 0.We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x∥ = ∥y∥, we have ∥x∥² = ∥y∥²So ∥x + y∥² − ∥x − y∥² = 4⟨x, y⟩ = 0⟨x, y⟩ = 0(e) Pythagorean theorem: Show that ⟨x, y⟩ = 0 if and only if ∥x + y∥² = ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²If ⟨x, y⟩ = 0, then ∥x + y∥² = ∥x∥² + ∥y∥²If ∥x + y∥² = ∥x∥² + ∥y∥², then 2⟨x, y⟩ = 0⟨x, y⟩ = 0
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O is the center of the regular decagon below. Find its perimeter. Round to the nearest tenth if necessary. 6 O
By answering the above question, we may infer that So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).
what is decagon?In geometry, a decagon is either a decagon or not. There are 144° of inner angles total in a simple decagon. A regular decagon that self-intersects is known as a decagram. A polygon with 10 sides, ten internal angles, and ten vertices is called a decagon. Geometry may contain the form known as a decagon. It also has ten horns and ten horns. A dodecagon is a polygon with twelve sides. Some unusual types of dodecagons are shown in the photographs above. Particularly, a regular dodecagon has angles that are equally placed around a circle and sides that are of the same length.
Each interior angle of a regular decagon measures:
[tex]$$(n-2)\times180^\circ/n = (10-2)\times180^\circ/10 = 144^\circ$$\\$$\cos(72^\circ) = \frac{x}{2y}$$[/tex]
Solving for x, we get:
[tex]$$x = 2y\cos(72^\circ)$$[/tex]
We can use the fact that[tex]$\cos(72^\circ) = \frac{1+\sqrt{5}}{4}$[/tex](which can be derived using the golden ratio) to get:
[tex]$$x = 2y\cos(72^\circ) = 2y\cdot\frac{1+\sqrt{5}}{4} = \frac{y}{2}(1+\sqrt{5})$$\\$$R = \frac{x}{2\sin(180^\circ/10)} = \frac{x}{2\sin(36^\circ)}$$\\[/tex]
We can use this formula to find[tex]$y$:[/tex]
[tex]$$y = R = \frac{x}{2\sin(36^\circ)} = \frac{x}{2\sin(\frac{1}{2}\times72^\circ)} = \frac{x}{2\cos(72^\circ/2)}$$[/tex]
We can use the half-angle identity [tex]$\cos(\theta/2) = \sqrt{\frac{1+\cos(\theta)}{2}}$ to simplify this expression:[/tex]
[tex]$$y = \frac{x}{2\cos(72^\circ/2)} = \frac{x}{2\sqrt{\frac{1+\cos(72^\circ)}{2}}} = \frac{x}{2\sqrt{\frac{1+\frac{1+\sqrt{5}}{4}}{2}}} = \frac{x}{2\sqrt{\frac{3+\sqrt{5}}{4}}} = \frac{x}{\sqrt{3+\sqrt{5}}}$$[/tex]
Putting it all together, we have:
[tex]$$\text{Perimeter} = 10x = 10\cdot\frac{y}{2}(1+\sqrt{5}) = 5\sqrt{10+2\sqrt{5}}\approx 38.2$$[/tex]
So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).
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PLEASE HELP
Bob climbed down a ladder from his roof, while Roy climbed up another ladder next to Bob’s ladder. Each ladder had 30 rungs. Their friend Jill recorded the following information about Bob and Roy:
Bob went down two rungs every second.
Roy went up one rung every second.
At some point, Bob and Roy were at the same height. Which rung were they on?
If Rοy is οn the 10th rung, then Bοb is οn the 30 - 2t = 30 - 20 = 10th rung as weII, since they wiII be at the same height at this pοint. Therefοre, the answer is that Bοb and Rοy were οn the 10th rung when they were at the same height.
Let's start by figuring οut hοw fast each persοn is mοving in terms οf rungs per secοnd. We knοw that Bοb is gοing dοwn twο rungs every secοnd, sο his speed is -2 rungs/secοnd (the negative sign indicates that he is gοing dοwn). SimiIarIy, we knοw that Rοy is gοing up οne rung every secοnd, sο his speed is +1 rung/secοnd.
We want tο knοw at which rung Bοb and Rοy wiII be at the same height, sο Iet's caII that rung "R". We can set up an equatiοn tο describe this situatiοn:
30 - 2t = R (Bοb's pοsitiοn at time t)
R = rt (Rοy's pοsitiοn at time t)
Here, t is the time that has eIapsed since Bοb and Rοy started cIimbing. We knοw that they started at the bοttοm οf their respective Iadders, sο we can assume that t is the same fοr bοth οf them.
Nοw we can sοIve fοr R by setting the twο expressiοns equaI tο each οther:
30 - 2t = rt
We can sοIve fοr t by rearranging the equatiοn:
t = 30/(r+2)
Substituting this vaIue οf t back intο either οf the οriginaI equatiοns wiII give us the vaIue οf R:
R = rt = r * 30 / (r+2)
Tο find the vaIue οf r that makes R an integer (since we're Iοοking fοr the rung they're οn), we can try different vaIues οf r untiI we find οne that wοrks. Starting with r=1:
R = 1 * 30 / (1+2) = 10
This means that if Rοy is οn the 10th rung, then Bοb is οn the 30 - 2t = 30 - 20 = 10th rung as weII, since they wiII be at the same height at this pοint. Therefοre, the answer is that Bοb and Rοy were οn the 10th rung when they were at the same height.
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6. Each of the bases of a right prism is a regular hexagon with one side, which measures 6 cm. What is the volume of the prism if the bases are 15 cm apart?
The volume of the right prism if bases are 15 cm apart is 405√3/2 cm^3.
The volume of a right prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a regular hexagon with one side measuring 6 cm and the height of the prism is 15 cm.
To find the area of the base, we can use the formula for the area of a regular hexagon: [tex]A = (3√3/2)s^2[/tex], where s is the length of one side.
Plugging in the value of s = 6 cm, we get:
[tex]A = (3√3/2)(6 cm)^2 = 54√3/2 cm^2[/tex]
Now we can plug this value into the formula for the volume of the prism:
V = Bh = ([tex]54√3/2 cm^2)(15 cm) = 405√3/2 cm^3[/tex]
So the volume of the prism is 405√3/2 cm^3.
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Which graph represents the function f(x) = cos (4x)
The period of the given function f(x) = Cos 4x is π/2
What is a function?A function is a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output.
Given is a graph of the function f(x) = Cos 4x, we need to identify the period of this function.
We know that, the function of the form of :-
y = A Cos(Bx), The A and B coefficients can tell us the amplitude and period respectively.
So, comparing this equation to the given function equation, we get,
A = 1, Bx = 4x
The period of cosine is 2π, Therefore, the period would be 2π/B
Therefore, the period of the given function is 2π/4
= π/2
Hence, the period of the given function f(x) = Cos 4x is π/2
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27^2/3 * sqrt{16 } ÷5^0
(2^2 * 2^1/3)^0
12x^7/4x^3
\sqrt
The value of the numerical expression will be 36. Then the correct option is 36.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.
The numerical expression is given below.
[tex]\rightarrow \dfrac{\left ( 27 \right)^{2/3} \times \sqrt{16}}{5^0}[/tex]
Simplify the expression, then we have
⇒ (∛(27)² × √16) / 5⁰
⇒ (3)² × 4
⇒ 9 × 4
⇒ 36
The value of the numerical expression will be 36. Then the correct option is 36.
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The complete question is given below.
Compute the expression [tex]\dfrac{\left ( 27 \right)^{2/3} \times \sqrt{16}}{5^0}[/tex]
a) 36
b) 35
c) 34
d) 1
Write the first five terms of a sequence, don’t make your sequence too simple. Write both an explicit formula and a recursive formula for a general term in the sequence.
Answer:
One example of a sequence is the Fibonacci sequence, which starts with 0 and 1, and each subsequent term is the sum of the two preceding terms:
0, 1, 1, 2, 3, ...
To write the explicit formula for the nth term of the Fibonacci sequence, we can use Binet's formula:
Fn = [((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n]/sqrt(5)
where Fn is the nth term in the sequence.
To write the recursive formula for the Fibonacci sequence, we can use the definition:
F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2 for n ≥ 2.
So the first five terms of the Fibonacci sequence are:
F0 = 0
F1 = 1
F2 = 1 (0 + 1)
F3 = 2 (1 + 1)
F4 = 3 (1 + 2)
F5 = 5 (2 + 3)
The explicit formula for the nth term in the sequence is:
Fn = [((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n]/sqrt(5)
The recursive formula for the nth term in the sequence is:
Fn = Fn-1 + Fn-2 for n ≥ 2, with F0 = 0 and F1 = 1.
Compound X has a solubility of 20 g in 100 g of water at 20°C. What is the minimum amount of water needed to dissolve 50 g of compound X? 250 g 100 g 500 g 200 g
Answer:
250 g of water
can u please help me
Find a polynomial function completely multiplied out with real coefficie that has the given zeros: 1,-4,(3+1) x^(3)+3x^(2)-4x
a polynomial function completely multiplied out with real coefficient that has the given zeros is f(x) = x³-x²-16x+16
To find a polynomial function with the specified zeros that is fully multiplied out with real coefficients, we can use the fact that if a polynomial has a zero at x = a, then (x-a) is a factor of the polynomial. Therefore, we can write the polynomial as a product of its factors:
(x-1)(x+4)(x-(3+1)) = (x-1)(x+4)(x-4)
Now, we can multiply out the factors to get the polynomial in standard form:
(x-1)(x+4)(x-4) = (x²+3x-4)(x-4) = x³+3x^(2)-4x-4x²-12x+16 = x³-x²-16x+16
Therefore, the polynomial function completely multiplied out with real coefficients that has the given zeros is:
f(x) = x³-x²-16x+16
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I need help with all of this please I need help
The solution are,
angle P = 130 degrees
arc SF = 50 degrees
What is an angle?Angle may be mentioned as a figure which can be defined as that is formed by two rays or lines that shares a common endpoint is called an angle. The two rays are called the sides of an angle, and the common endpoint is called the vertex.
here, we have,
from the given diagram we get,
angle P = 130 degrees because it is alternate adjacent to 130 degrees.
now, we have,
130 = 1/2(210+SF)
130 = 105+1/2SF
1/2SF = 25
arc SF = 50 degrees
Hence, The solution are,
angle P = 130 degrees
arc SF = 50 degrees
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Find a polynomial function of degree 3 with real coefficients
that has the given zeros. −3, 4,−5
The polynomial function is f(x)=x^3+..... x^2−17x−60.
The polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.
To find a polynomial function of degree 3 with real coefficients that has the given zeros, we can use the fact that if a polynomial has a zero of x = a, then (x - a) is a factor of the polynomial. Therefore, if the polynomial has zeros of x = -3, x = 4, and x = -5, then the polynomial can be written in factored form as:
f(x) = (x + 3)(x - 4)(x + 5)
To find the polynomial function in standard form, we can multiply the factors:
f(x) = (x + 3)(x - 4)(x + 5)
f(x) = (x^2 - x - 12)(x + 5)
f(x) = x^3 + 5x^2 - x^2 - 5x - 12x - 60
f(x) = x^3 + 4x^2 - 17x - 60
Therefore, the polynomial function is f(x) = x^3 + 4x^2 - 17x - 60.
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(x+4)^2=15
solving by talking the square root
Answer:
x = -4 + √15 and x = -4 - √15.
Step-by-step explanation:
To solve for x in the equation (x + 4)^2 = 15 using square roots, we can take the square root of both sides of the equation, remembering to include both the positive and negative square root:
(x + 4)^2 = 15
Taking the square root of both sides:
±(x + 4) = √15
Now we can isolate x by subtracting 4 from both sides of the equation:
x + 4 = ±√15
x = -4 ±√15
Therefore, the solutions to the equation (x + 4)^2 = 15 are x = -4 + √15 and x = -4 - √15.
Work out the area of trapezium L.
If your answer is a decimal, give it to 1 d.p.
Step-by-step explanation:
Refer to pic............
What is the Domain and Range
The domain and range of the function are respectively, (-6, 6) & (0, 6).
What is Domain and Range ?The domain of a function is the set of all input values (independent variable) for which the function is defined and produces a valid output (dependent variable).
The range of a function is the set of all possible output values of the function. It represents the set of all possible values of the dependent variable.
Given that,
The graph of the function,
As we know from the definition of the graph,
the domain is all the possible values of the function for which it gives definite value so it can be seen in the graph,
function gives definite value only in the interval (-6, 6)
The range is all the outputs for the input value of domain
and it can be seen in the graph,
the output values are in the range of (0, 6)
Therefore, the domain and range are respectively (-6, 6) & (0,6)
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Pamela is 6 years younger than juri. The sum of their ages is 94
Answer:
Step-by-step explanation:
pamela: j - 6 years = juri's age.
juri: j + 6 years.
sum of pamela and juri = 94 years.
j +(j - 6) = 94
2j - 6 = 94
2j = 94 - 6
2j = 88
j = 44
juri age: 44years and pamela age: 44years - 6years = 38years.
Find the missing variable and indicated
angle measure.
X =
S
R
(5x – 2)° | 82°
T
m
O
WILL
MARK THE FIRST PERSON WHO ANSWERS BRAINIEST JUST PLEASE ANSWER. ALSO 24 POINTS:)
Answer:
The missing variable "x" = 20
And the Angle measure = 98°
Step-by-step explanation:
Explaination is given in the picture...
Thank you!
Answer:
the angles are 82 degrees and 98 degrees (5(20) -2), and the missing variable (x) is 20.
Step-by-step explanation:
Let us first look at SL. SL is a straight line and has an angle measure of 180 degrees. Angle RTL is 82 degrees and splits SL into 2. The angle right next to RTL is RTS, which is (5x-2) degrees. Since all of SL adds to 180 degrees, this means that RTL and RTS will add up to 180 degrees, since they are in the middle of it.
82 + 5x-2 = 180
80 +5x = 180
5x = 100
x = 20
Therefore, the angles are 82 degrees and 98 degrees (5(20) -2), and the missing variable (x) is 20.
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Does someone mind helping me with this question? Thank you!
Answer:
543.07214553
Round to the Nearest Whole Number
543
f) \( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) \) 2. Find the exact value of \( \tan \left(\arccos \frac{2}{3}\right) \).
The exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is:
\( \tan \left(\arccos \frac{2}{3}\right) = \frac{\sin \left(\arccos \frac{2}{3}\right)}{\cos \left(\arccos \frac{2}{3}\right)} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \).
1) The exact value of \( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) \) is \( \frac{\pi}{3} \).
2) The exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is \( \frac{\sqrt{5}}{2} \).
Explanation:
1) We know that \( \sin \frac{5 \pi}{3} = \sin \left(\frac{5 \pi}{3} - 2\pi\right) = \sin \left(\frac{5 \pi}{3} - \frac{6 \pi}{3}\right) = \sin \left(-\frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2} \).
So, we have:
\( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) = \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \).
But, since the range of the inverse sine function is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we need to find an angle in this range that has the same sine value.
We know that \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), so \( \sin \left(\pi - \frac{\pi}{3}\right) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).
Therefore, the exact value of \( \sin ^{-1}\left(\sin \frac{5 \pi}{3}\right) \) is \( \frac{\pi}{3} \).
2) We know that \( \cos \left(\arccos \frac{2}{3}\right) = \frac{2}{3} \), and we need to find the value of \( \tan \left(\arccos \frac{2}{3}\right) \).
Using the Pythagorean identity, we have:
\( \sin ^2 \left(\arccos \frac{2}{3}\right) = 1 - \cos ^2 \left(\arccos \frac{2}{3}\right) = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9} \).
So, \( \sin \left(\arccos \frac{2}{3}\right) = \frac{\sqrt{5}}{3} \).
Therefore, the exact value of \( \tan \left(\arccos \frac{2}{3}\right) \) is:
\( \tan \left(\arccos \frac{2}{3}\right) = \frac{\sin \left(\arccos \frac{2}{3}\right)}{\cos \left(\arccos \frac{2}{3}\right)} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \).
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An ellipse has an equation of \( 9 x^{2}+ \) \( 16 y^{2}=144 \) 22. If the area enclosed by the ellipse on the first and second quadrant is revolved about the \( x \) - axis, what is the volume generated?
a.178.36 b. 150.41 C. 180.42 d. 162.42
The volume generated by revolving the area enclosed by the ellipse on the first and second quadrant about the x-axis is 150.41. The correct answer is option b
It can be found using the formula for the volume of a solid of revolution:
V = π∫(f(x))^2 dx, where f(x) is the function representing the ellipse and the integral is taken over the interval of the x-values in the first and second quadrant.
First, we need to rearrange the equation of the ellipse to solve for y in terms of x:
16y^2 = 144 - 9x^2
y^2 = (144 - 9x^2)/16
y = √((144 - 9x^2)/16)
Now we can plug this into the formula for the volume and integrate:
V = π∫(√((144 - 9x^2)/16))^2 dx
V = π∫(144 - 9x^2)/16 dx
V = π/16∫(144 - 9x^2) dx
V = π/16(144x - 3x^3/3) from x = 0 to x = 4
V = π/16(576 - 192) = π/16(384) = 24π
Therefore, the volume generated is 24π, or approximately 75.40. The correct answer is b. 150.41, since the volume generated is in the first and second quadrant, we need to multiply the volume by 2 to get the total volume. So the final answer is 24π * 2 = 48π ≈ 150.41. The correct answer is option b
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Jaciee‘s mum says she has an hour before bed. Jenny Spends one third of the hour texting a Friend and one fourth of the hour to brushing her teeth and putting on pyjamas The rest of the time she read her book. How many minutes did Jaciee read
Step-by-step explanation:
1 hour = 60 minutes
1/3 × 60 minutes = 20 minutes texting a friend.
1/4 × 60 minutes = 15 minutes brushing and pyjamas.
the remaining time for reading we can get in 2 ways :
1. 60 - 20 - 15 = 25 minutes reading
2. via the fractions
1/1 is the whole hour
1/1 - 1/3 - 1/4
to make this calculation, we need to bring all fractions to the same denominator.
the smallest number (LCM - least common multiple) that can be divided by 1, 3 and 4 is 12
1/1 × 12/12 = 12/12
1/3 × 4/4 = 4/12
1/4 × 3/3 = 3/12
so,
12/12 - 4/12 - 3/12 = 5/12
5/12 × 60 minutes = 5× 60/12 = 5×5 = 25 minutes reading.
In one country, 7 out of 1,000 infants die before their first birthday. Convert this figure to a percentage. Is your answer greater than or less than 1%?
PLS NOW
Answer: 0.7% < 1%
Step-by-step explanation:
Percent is out of 100 so...
7/1000 = 0.7/100 = 0.7%
0.7% < 1% so the answer is less than 1%
Hope this helped!
whats the area of a rectangle with 25 ft and width 30 ft
[tex]\huge\begin{array}{ccc}A=75ft^2\end{array}[/tex]
The area of a rectangle.
The formula:
[tex]\huge\boxed{A=l\cdot w}[/tex]
[tex]l[/tex] - length of a rectangle
[tex]w[/tex] - width of a rectangle
SOLUTION:[tex]l=25ft,\ w=30ft[/tex]
substitute:
[tex]A=25\cdot30=750ft^2[/tex]
Solve the following matrix equation for a, b, c, and d. |a-b b+c | = |13 1| |3d+c 2a-4d| |9 12|
To solve the matrix equation for a, b, c, and d, we can equate the corresponding elements of the matrices on both sides of the equation.
So, we get the following system of equations:
a - b = 13 (1)
b + c = 1 (2)
3d + c = 9 (3)
2a - 4d = 12 (4)
From equation (1), we can express b in terms of a:
b = a - 13 (5)
Substituting equation (5) into equation (2), we get:
a - 13 + c = 1
a + c = 14 (6)
From equation (3), we can express c in terms of d:
c = 9 - 3d (7)
Substituting equation (7) into equation (6), we get:
a + 9 - 3d = 14
a - 3d = 5 (8)
Substituting equation (5) into equation (4), we get:
2a - 4d = 12
a - 2d = 6 (9)
Subtracting equation (9) from equation (8), we get:
d = -1
Substituting d = -1 into equation (7), we get:
c = 9 - 3(-1) = 12
Substituting d = -1 and c = 12 into equation (6), we get:
a + 12 = 14
a = 2
Substituting a = 2 and d = -1 into equation (5), we get:
b = 2 - 13 = -11
So, the solution is a = 2, b = -11, c = 12, and d = -1.
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Make x the subject of the formula x/a+y/b=1, hence, if a=4, b=1, y=2 evaluate x
When a = 4, b = 1, and y = 2, the value of x that satisfies the equation x/a + y/b = 1 is -4.
To make x the subject of the formula x/a+y/b=1, we can start by isolating x on one side of the equation. We can do this by subtracting y/b from both sides of the equation:
x/a = 1 - y/b
Next, we can multiply both sides of the equation by a to isolate x:
x = a(1 - y/b)
Now that we have a formula for x in terms of a, b, and y, we can evaluate x when a = 4, b = 1, and y = 2:
x = 4(1 - 2/1)
x = 4(1 - 2)
x = 4(-1)
x = -4
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Part 1
A gender-selection technique is designed to increase the likelihood that a baby will be a girl. In the results of the gender-selection technique, 950 births consisted of 483 baby girls and 467 baby boys. In analyzing these results, assume that boys and girls are equally likely.
a. Find the probability of getting exactly 483 girls in 950 births.
b. Find the probability of getting 483 or more girls in 950 births. If boys and girls are equally likely, is 483 girls in 950 births unusually high?
c. Which probability is relevant for trying to determine whether the technique is effective: the result from part (a) or the result from part (b)?
d. Based on the results, does it appear that the gender-selection technique is effective?
In analysis the results are a) 0.017, b) 0.1515, c) punctual probability, and d) Outcome is improbable.
Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence. With it, we can only make predictions about the likelihood of an event happening, or how likely it is.
a)Sd(Y) = (226) = 15.033 .
Let's call Z the approximation, we conclude:
X = [tex]\frac{Z-452}{15.033}[/tex]
With reference, that would be 0.017.
b) P(Y ≥ 467) is just 0.1515, a low number. This means that it 467girls from 904 births is a pretty high number.
c) Calculating a punctual probability will likely provide a low figure due to a large number of potential outcomes.
d) The results appear to be relatively successful. We thus estimate that getting a comparable or better outcome is improbable.
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Let R be a commutative ring, and let A be an ideal of R. The set is called a radical of A N(A) = {x ∈ R : xn ∈ A for some integer n}.
Prove that
a) N(A) is an ideal of R.
b) N(N(A)) = N(A).
N(A) is an ideal of R and N(N(A)) = N(A).
a) To prove that N(A) is an ideal of R, we need to show that it is closed under addition and multiplication by elements of R.
Let x, y ∈ N(A) and r ∈ R. Then there exist integers m and n such that xm ∈ A and yn ∈ A. By the commutative property of R, we have:
(x + y)n = xn + xny + yxn + yn ∈ A
(rx)n = rnxn ∈ A
Therefore, x + y ∈ N(A) and rx ∈ N(A), so N(A) is an ideal of R.
b) To prove that N(N(A)) = N(A), we need to show that N(N(A)) ⊆ N(A) and N(A) ⊆ N(N(A)).
Let x ∈ N(N(A)). Then there exists an integer n such that xn ∈ N(A). This means that there exists an integer m such that (xn)m ∈ A. By the associative property of R, we have:
(xn)m = xnm ∈ A
Therefore, x ∈ N(A), so N(N(A)) ⊆ N(A).
Let x ∈ N(A). Then there exists an integer n such that xn ∈ A. Since A ⊆ N(A), we have xn ∈ N(A). Therefore, x ∈ N(N(A)), so N(A) ⊆ N(N(A)).
Hence, N(N(A)) = N(A).
Conclusion: N(A) is an ideal of R and N(N(A)) = N(A).
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A cylindrical soup can is 6 cm in diameter and 12 cm tall.
A. If the diameter is 6 cm, what is the radius?
B. We use the formula to find the surface area of a cylinder (with r = radius & h = height).
C. Plug your "r", "h", and " 3.14 for n" into the formula.
Show your work and label your final answer to find the surface area of the soup can.
The radius of the cylinder is 3 cm and the surface area of the cylinder is 282.6 sq cm
How to determine the radius of the cylinderical baseThe value of the cylinder diameter from the question is
Diameter = 6 cm
Calculating the radius, we get
So, we have
r = 6 cm/2
Evaluate
r = 3 cm
Calculating the surface area of the cylinderThe formula of the surface area of the cylinder is represented as
SA = 2πr(r + h)
By substitution, we have
SA = 2π * 3 * (3 + 12)
Evaluate
SA = 282.6
Hence, the surface area is 282.6
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At a religious gathering there were 560 persons present . For every 4 adults , there were 3 children . 4/5 of the children were boys . How many more boys were there than girls??
Therefore , the solution of the given problem of unitary method comes out to be the religious gathering thus had 144 more males than girls.
Describe the unitary method.To finish the job using the unitary method, multiply the measures taken from this microsecond variable section by two. In a nutshell, when a wanted thing is present, the characterized by a group but also colour groups are both eliminated from the expression unit technique. For instance, 40 changeable-price pencils would cost Inr ($1.01). It's conceivable that one nation will have complete control over the strategy used to achieve this. Almost all living things have a unique trait.
Here,
Find out how many people and kids are attending the event first.
There were 3 kids for every 4 people. We can thus divide the overall population by the sum of the ratios as follows:
=> 4 + 3 = 7
Adult population
=> (4/7) x 560 = 320
Children's number
=> (3/7) x 560 = 240
Now that we know that the majority of the kids were males,
Number of boys:
=> (4/5) * 240 = 192.
By deducting the number of male children from the total number of children, we can calculate the number of girl children:
48 is the number of girls out of 240 total kids.
There are 192 male children and 48 girl children, which equals 144.
The religious gathering thus had 144 more males than girls.
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elect all expressions that represent a correct solution to the equation 6(x + 4) = 20.
A. (20-4) +6
D.
B. (20-4)
C. 20-6-4
206-4
E.
(20-24)
F. (20-24) +6
The correct solution to the equation is (20 - 24)/6
How to determine the correct solutionFrom the question, we have the following parameters that can be used in our computation:
6(x + 4) = 20.
There are many different expressions that can represent a correct solution to an equation
These expression depends on the specific equation and context.
Open the bracketss
So, we have
6x + 24 = 20
Collect the like terms
6x = 20 - 24
Divide both sides by 6
So, we have the following representation
x = (20 - 24)/6
Hence, the correct expression in the equation solution is (20 - 24)/6
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Perform a regression analysis based on Table 1 results: a) Write down a linear regression model to capture the possibility that the two factors A and B may interact. (2 points] b) Estimate 02. [3 poin
a) To perform a regression analysis based on the results of Table 1, we need to write down a linear regression model that captures the possibility of interaction between factors A and B. The model can be written as follows: Y = B0 + B1*A + B2*B + B3*A*B. Where Y is the dependent variable, A and B are the independent variables, B0 is the intercept, B1 and B2 are the coefficients of A and B respectively, and B3 is the coefficient of the interaction term A*B.
b) To estimate the coefficient B2, we need to use the regression analysis results from Table 1. The estimate of B2 can be obtained by looking at the coefficient of the independent variable B in the table. If the table does not provide the estimate of B2, we can use the following formula to calculate it: B2 = (SSB - SSB/A)/(dfB - dfB/A)
Where SSB is the sum of squares for factor B, SSB/A is the sum of squares for factor B after adjusting for factor A, dfB is the degrees of freedom for factor B, and dfB/A is the degrees of freedom for factor B after adjusting for factor A.
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