Solve the equation. (Find all solutions of the equation in the interval [0, 2pi). Enter your answers as a comma-separated list.)
Answers
The solutions to the given equation in the interval $[0,2\pi)$ are:
[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]
How to deal with trigonometric equation?The given equation is:
[tex]$$\cos(2x) + \sin(x) = 0$$[/tex]
We can write [tex]$\cos(2x)$[/tex] as [tex]$1-2\sin^2(x)$[/tex]using double angle formula for cosine. Substituting this in the given equation, we get:
[tex]$$1-2\sin^2(x) + \sin(x) = 0$$[/tex]
Rearranging and factoring, we get:
[tex]$$(2\sin(x) - 1)(\sin(x) + 1) = 0$$[/tex]
So, either [tex]$2\sin(x) - 1 = 0$[/tex] or [tex]$\sin(x) + 1 = 0$[/tex].
Solving [tex]$2\sin(x) - 1 = 0$[/tex], we get:
[tex]$$\sin(x) = \frac{1}{2}$$[/tex]
which has solutions [tex]x=\frac{\pi}{6}$[/tex] and $[tex]x=\frac{\pi}{6}$[/tex] in the interval [tex]$[0,2\pi)$[/tex].
Solving [tex]$\sin(x) + 1 = 0$[/tex], we get:
[tex]$$\sin(x) = -1$$[/tex]
which has solution [tex]x=\frac{3\pi}{2}$[/tex]in the interval [tex]$[0,2\pi)$[/tex]
Therefore, the solutions to the given equation in the interval $[0,2\pi)$ are:
[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]
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Related Questions
f(n) = 45 . J |K 4 5 n-1 Complete the recursive formula of f(n). ƒ(1) = f(n) = f(n-1).
Answers
Answer:
It looks like there might be a typo in the expression given. Assuming that "J" and "K" are just placeholders, we can write the expression as:
f(n) = 45 * |4 - 5(n-1)|
To find the recursive formula for this sequence, we need to determine how each term relates to the previous term. We can start by looking at the first few terms of the sequence:
f(1) = 45 * |4 - 5(1-1)| = 45 * |4 - 5(0)| = 45 * |4| = 180
f(2) = 45 * |4 - 5(2-1)| = 45 * |4 - 5(1)| = 45 * |-1| = 45
f(3) = 45 * |4 - 5(3-1)| = 45 * |4 - 5(2)| = 45 * |-6| = 270
From this, we can see that the sign of the expression inside the absolute value changes with each term, alternating between positive and negative. Furthermore, the magnitude of this expression increases by 5 with each term. We can use these observations to write the recursive formula:
f(1) = 180
f(n) = f(n-1) + (-1)^(n-1) * 5 * 45 for n >= 2
This formula says that the first term in the sequence is 180, and each subsequent term is found by adding or subtracting 225 (5 * 45) from the previous term, depending on whether n is odd or even.
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calculate zeff for a valence electron in carbon using slater's rules. submit an answer to two decimal places.
Answers
The value of Zeff for a valence electron in carbon using slater's rule is equals to 3.60.
Effective nuclear charge (Zeff) for a valence electron in carbon using Slater's rules,
Consider the shielding effect of the inner electrons.
Slater's rules state that the effective nuclear charge felt by an electron in an atom = the actual nuclear charge minus the shielding constant for that electron.
The shielding constant for an electron in a particular shell is determined by summing the contributions of all the electrons in inner shells.
Slater assigned the following shielding constants for each shell,
Electrons in the same shell contribute 0.35
Electrons in the next inner shell contribute 0.85
Electrons in all inner shells beyond that contribute 1.00
Carbon has an atomic number of 6, so it has six electrons.
The first two electrons are in the inner shell, leaving four valence electrons in the outer shell.
Zeff for a valence electron in carbon,
Consider the contributions of the inner electrons.
Two electrons in the 1s orbital contribute
2× 0.35 = 0.70 to the shielding constant.
The two electrons in the 2s orbital contribute
2 × 0.85 = 1.70 to the shielding constant.
There are no electrons in the 2p orbitals,
This implies,
They do not contribute to the shielding constant.
Thus, the effective nuclear charge felt by a valence electron in carbon is,
Zeff
= 6 - 0.70 - 1.70
= 3.60
Rounding to two decimal places, we get,
Zeff = 3.60
Therefore, the Zeff for a valence electron in carbon is equals to 3.60.
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Find the dimensions of a rectangle with area 1,000 m^2 whose perimeter is as small as possible. (If both values are the same number, enter it into both blanks.)What is m (smaller value)What is m (Larger value)
Answers
10√10 is the dimensions of a rectangle with area 1,000 m² whose perimeter is as small as possible.
a. The smaller value is 10√10 m.
b. The larger value is 10√10 m.
We have to determine the dimensions of a rectangle with area 1,000 m² whose perimeter is as small as possible.
P = 2w + 2L
1000 = Lw
P = 2w + 2(1000/w)
P = 2w + 2000/w
P-prime = 2 -2000/w²
0 = 2 - 2000/w²
Add 2000/w² on both side, we get
2000/w² = 2
Multiply by w² on both side, we get
2000 = 2w²
Divide by 2 on both side
w² = 2000/2
w² = 1000
Taking square root on both side, we get
w = 10√10
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Some friends went out for a meal. The restaurant added a 10% service charge to the cost of the meal. The total bill was £126.50 including the service charge. What was the cost of the meal? Give your answer in pounds (£). Receipt Cost of the meal: £ Service charge: +10% Total: £126.50
Answers
Answer:
Let's start by setting up an equation to represent the problem. Let x be the cost of the meal:
x + 0.1x = 126.50
Simplifying the left side of the equation:
1.1x = 126.50
Dividing both sides by 1.1:
x = 115
Therefore, the cost of the meal was £115.
For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x) = 4x+9; g(x)=9x - 5
Answers
Answer:
(a) Find (f + g)(x)
To find (f + g)(x), we add the two functions f(x) and g(x):
(f + g)(x) = f(x) + g(x) = (4x + 9) + (9x - 5) = 13x + 4
The domain of (f + g)(x) is all real numbers, since there are no restrictions on x that would make (f + g)(x) undefined.
(b) Find (f - g)(x)
To find (f - g)(x), we subtract the function g(x) from f(x):
(f - g)(x) = f(x) - g(x) = (4x + 9) - (9x - 5) = -5x + 14
The domain of (f - g)(x) is all real numbers, since there are no restrictions on x that would make (f - g)(x) undefined.
(c) Find (f * g)(x)
To find (f * g)(x), we multiply the two functions f(x) and g(x):
(f * g)(x) = f(x) * g(x) = (4x + 9)(9x - 5) = 36x^2 + 11x - 45
The domain of (f * g)(x) is all real numbers, since there are no restrictions on x that would make (f * g)(x) undefined.
(d) Find (f / g)(x)
To find (f / g)(x), we divide the function f(x) by g(x):
(f / g)(x) = f(x) / g(x) = (4x + 9) / (9x - 5)
The domain of (f / g)(x) is all real numbers except x = 5/9, since this value would make the denominator of (f / g)(x) equal to zero, resulting in division by zero, which is undefined.
(e) Find f(g(x))
To find f(g(x)), we substitute g(x) into the expression for f(x):
f(g(x)) = 4g(x) + 9
Substituting the expression for g(x), we get:
f(g(x)) = 4(9x - 5) + 9 = 36x - 11
The domain of f(g(x)) is all real numbers, since there are no restrictions on x that would make f(g(x)) undefined.
(f) Find g(f(x))
To find g(f(x)), we substitute f(x) into the expression for g(x):
g(f(x)) = 9f(x) - 5
Substituting the expression for f(x), we get:
g(f(x)) = 9(4x + 9) - 5 = 36x + 76
The domain of g(f(x)) is all real numbers, since there are no restrictions on x that would make g(f(x)) undefined.
(g) Find f(f(x))
To find f(f(x)), we substitute f(x) into the expression for f(x):
f(f(x)) = 4f(x) + 9
Substituting the expression for f(x), we get:
f(f(x)) = 4(4x + 9) + 9 = 16x + 45
The domain of f(f(x)) is all real numbers, since there are no restrictions on x that would make f(f(x)) undefined.
(h) Find g(g(x))
To find g(g(x)), we substitute g(x) into the expression for g(x):
g(g(x)) = 9
Step-by-step explanation:
Answer:
Step-by-step explanation:
(a) Find f(g(x)).
To find f(g(x)), we first need to find g(x) and then substitute it into f(x).
g(x) = 9x - 5
f(g(x)) = f(9x - 5) = 4(9x - 5) + 9 = 36x - 11
Therefore, f(g(x)) = 36x - 11.
(b) Find g(f(x)).
To find g(f(x)), we first need to find f(x) and then substitute it into g(x).
f(x) = 4x + 9
g(f(x)) = g(4x + 9) = 9(4x + 9) - 5 = 36x + 76
Therefore, g(f(x)) = 36x + 76.
(c) Find f(f(x)).
To find f(f(x)), we need to substitute f(x) into f(x).
f(f(x)) = 4(4x + 9) + 9 = 16x + 45
Therefore, f(f(x)) = 16x + 45.
(d) Find g(g(x)).
To find g(g(x)), we need to substitute g(x) into g(x).
g(g(x)) = 9(9x - 5) - 5 = 81x - 50
Therefore, g(g(x)) = 81x - 50.
Domain of f(x) and g(x): Since both f(x) and g(x) are linear functions, their domains are all real numbers.
(e) Find the inverse of f(x).
To find the inverse of f(x), we need to switch the roles of x and f(x) and solve for f(x).
y = 4x + 9
x = 4y + 9
x - 9 = 4y
y = (x - 9) / 4
Therefore, the inverse of f(x) is f^(-1)(x) = (x - 9) / 4.
(f) Find the inverse of g(x).
To find the inverse of g(x), we need to switch the roles of x and g(x) and solve for g(x).
y = 9x - 5
x = 9y - 5
x + 5 = 9y
y = (x + 5) / 9
Therefore, the inverse of g(x) is g^(-1)(x) = (x + 5) / 9.
(g) Find the domain of f^(-1)(x).
The domain of f^(-1)(x) is the range of f(x). Since f(x) is a linear function, its range is all real numbers. Therefore, the domain of f^(-1)(x) is also all real numbers.
(h) Find the domain of g^(-1)(x).
The domain of g^(-1)(x) is the range of g(x). Since g(x) is a linear function, its range is all real numbers. Therefore, the domain of g^(-1)(x) is also all real numbers.
THERE ARE 2 PARTS PLEASE ANSWER BOTH RIGHT TY HELPP!! There are 12 red cards, 17 blue cards, 14 purple cards, and 7 yellow cards in a hat.
Part A. What is the theoretical probability of drawing a purple card from the hat?
Part B.
In a trial, a card is drawn from the hat and then replaced 1,080 times. A purple card is drawn 324 times. How much greater is the experimental probability than the theoretical probability?
Enter the correct answers in the boxes.
A. The theoretical probability of drawing a purple card from the hat is ______.
B. The experimental probability of drawing a purple card is ____%
greater than the theoretical probability.
Answers
Part A. The probability pf drawing a purple card out of the hat is 28%.
Part B. The experimental probability is 2% greater than the theoretical probability.
Define probability?The probability that a specific event will occur is known as probability. The ratio of favourable outcomes to all other possible outcomes serves as a stand-in for the likelihood that an event will occur.
In numerous disciplines, including mathematics, statistics, physics, economics, and computer science, uncertain events are described and understood using probability theory. It is used to analyse risks, make decisions, and forecast events.
Now in the given question,
Total cards in the hat = 12 + 17 + 14 + 7 = 50 cards
Total purple cards in the hat = 14
Probability of getting a purple card from the hat = 14/50
= 0.28
= 28%
Now similarly for the experiment,
Probability = 324/1080
= 0.3
= 30%
Therefore, the experimental probability is 30% - 28% = 2% greater than the theoretical probability.
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A class Eleventh Maths teacher Khushali wrote some sets in set builder form on a black board of class;A={x: xis a prime natural number and x is less than equal to 7 }
B ={y: y is an odd natural number and y E 7}
Where Universal set U = {l,2,3,4,5,6,7,8}
i)Write sets A and B in roster form
ii)Find A u B and A n B
iii)Find the number of all subsets of universal set U and number of relations from A to B
Answers
Step-by-step explanation:
i)
A = {2, 3, 5, 7}
as "1" can only be divided by one number (instead of the usual 2 numbers for prime numbers), if it's not part of that set.
out of U = {1, 2, 3, 4, 5, 6, 7, 8}
B = {1, 3, 5}
I am not sure what you mean by "y E 7".
I don't think you mean the E7 algebraic group.
I decided you mean y <> 7 (not equal to 7).
ii)
A u B (united) = {1, 2, 3, 5, 7}
A n B (elements in common) = {3, 5}
iii)
a set with n elements has 2^n subsets and (2^n) - 1 proper subsets (all subsets minus the equal one).
our U here has 8 elements, so the number of subsets is
2⁸ = 256.
the number of relations from A to B is 2^|A×B| = 2^(|A|·|B|).
|A| = 4
|B| = 3
so the number of relations from A to B are
2^(4×3) = 2¹² = 4096
remember, for the number of possible relations we have 4×3 = 12 possible combinations of elements of A and engender of B.
each of these combinations can be in the set of relations or not, which gives us 2 options per combination.
that gives us 2¹² relations.
the dcpromo wizard will guide you through which of the following installation scenarios? [check all that apply]
Answers
The Dcpromo wizard will guide you through e. All of the above installation scenarios
A utility in Active Directory called DCPromo (Domain Controller Promoter) installs and uninstalls Active Directory Domain Services and promotes domain controllers. Since Windows 2000, every version of Windows Server contains DCPromo, which creates forests and domains in Active Directory. It works with Windows Server and houses all network resources as a centralised security management solution.
The functionality aids in building a completely new forest structure. It allows for both the addition of a new domain tree to an existing forest and the addition of a child domain to an existing domain. Additionally, it degrades the domain controllers and ultimately deletes a domain or forest.
Complete Question:
The dcpromo wizard will guide you through which of the following installation scenarios? [check all that apply]
Creating an entirely new forest structure.
Adding a child domain to an existing domain.
Adding a new domain tree to an existing forest.
Demoting domain controllers and eventually removing a domain or forest
All of the above
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work out 40÷160000.
write your answer in standard form.
Answers
The value of expression 40 divided by 160000 would be equal to 0.00025
How can we interpret the division?When 'a' is divided by 'b', then the result we get from the division is part of 'a' that each one of 'b' items will get. Division can be interpreted as equally dividing the number that is being divided into total x parts, where x is the number of parts the given number is divided.
We need to find the expression of 40 divided by 160000
A negative divided by a negative is positive, then
40÷160000 = 0.00025
Therefore, The value of 40 divided by 160000 is; 0.00025
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Find g. Write your answer as a whole number or a decimal. Do not round.
Answers
The value of length of side g using the similar triangles is found as 20 ft.
Explain about the similar triangles?Triangles that are similar to one another in terms of shape, angle measurements, and proportion are said to be similar.If the single difference between two triangles is their size and perhaps the requirement to rotate or flip one of them, then they are similar.In the given figures:
DC || EA
So,
∠D = ∠A
∠C = ∠E
By Angle -Angle similarity both triangles are similar.
Thus,
Taking the ratios of their side, it will be also equal.
EA / DC = EB / BC
5 / 10 = g / 10
g = 10*10 / 5
g = 100 / 5
g = 20
Thus, the value of length of side g using the similar triangles is found as 20 ft.
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I need help with this
Answers
Angle PLE is measured at 90°. The options supplied do not contain the solution.
What measurement of an angle does not equal 90?The acute angle is defined as being less than 90 degrees. Right angles are 90 degrees in length. Angles that are obtuse have a greater angle than 90 degrees. Discover the various sorts of angles and examples of each.
As complementary angles, ZPLA and ZELA, we can infer that:
ZPLA + ZELA = 90°
Using the following expressions in place of ZPLA and ZELA, we obtain:
5x-2 + x+8 = 90
When we simplify the equation, we obtain:
6x + 6 = 90
6 is subtracted from both sides to yield:
6x = 84
Dividing by 6, we get:
x = 14
With the knowledge of x, we can determine the dimensions of ZPLA and ZELA:
ZPLA = 5x-2 = 5(14)-2 = 68°
ZELA = x+8 = 14+8 = 22°
Therefore, we can find the measure of angle PLE by subtracting the measures of ZPLA and ZELA from 180°:
PLE = 180 - ZPLA - ZELA
PLE = 180 - 68 - 22
PLE = 90
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Question 4
Let us assume an original starting value of $4 trillion in 2033, but that the actual rate of
decline after 2033 was 10% greater than the 7.6% rate. (Notice that this refers to a 10%
relative increase over the 7.6% rate of decline that was originally estimated in the
lesson, and not an absolute increase of 10 percentage points.) In the questions below,
consider how this would affect the estimated value of the funds in 2038?
What is the new estimated value of the trust funds in 2038? Round to the nearest
Answers
The new estimated value of the trust funds in 2038 would be $2.5 trillion.
What is funds?Funds are resources of monetary value that can be used to acquire goods, services, or to meet financial obligations. They are usually obtained through the sale of goods and services or by borrowing. Funds can come from a variety of sources, including individuals, businesses, governments, and other organizations. Funds can be used to purchase assets such as stocks, bonds, real estate, or other forms of investments.
This is because a 10% relative increase over the original estimated 7.6% rate of decline means that the actual rate of decline is 8.36% (7.6% x 1.10 = 8.36%). Applying this rate to the original starting value of $4 trillion, the value of the trust funds in 2038 would be $4 trillion x 0.9164 (which is the equivalent of 1 - 0.0836) = $2.5 trillion.
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The new estimated value of the trust funds in 2038 would be $2.4 trillion.
What is trust fund?A trust fund is a legal arrangement in which a person or organization (the trustor) transfers assets to another person or organization (the trustee) to be managed and distributed for the benefit of a third party (the beneficiary). The trustor sets out a set of instructions (the trust deed) on how the trust funds should be handled, managed, and distributed. The trustee is legally bound to abide by these instructions, and is responsible for the management and distribution of the trust funds according to the trust deed. Trust funds can be used for a variety of purposes, such as providing for a beneficiary's education, medical care, or retirement.
This is because a 10% increase in the rate of decline would result in a rate of decline of 8.36% (7.6% plus 10% of 7.6%).
Using this new rate of decline, the value of the trust funds in 2038 would be $4 trillion multiplied by (1 - 0.0836)⁵,
which is equal to $2.4 trillion.
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Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each. 40/111 39/111 30/111 Relative frequency 20/111 23/111 17/111 10/111 5/111 2/111 0 1 1 2 3 4 5 6 7 Number of T-shirts costing more than $19 each Find the percent of people that own at most five t-shirts costing more than $19 each. (Round your answer to a whole number.) % Additional Materials eBook
Answers
The percent of people that own at most 4 T-Shirts costing more than $19 each is 76%.
The number of people who own at most four T-Shirts costing more than $19 is ⇒ n = N(1) + N(2) + N(3) + N(4),
where , N(i) represents the number of people who own "i" T-Shirts costing more than $19 each.
From the frequency histogram , we substitute the values and get;
⇒ n = 5 + 17 + 23 + 39
⇒ n = 84.
So, percentage of people who own at most four T-Shirts costing more than $19 each is calculated as :
⇒ (n/111) × 100% ;
⇒ (84/111) × 100% = 75.68% ≈ 76%.
Therefore, 76% of people that own at most 4 t-Shirts costing more than $19 each.
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The given question is incomplete, the complete question is
Suppose 111 people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each. The following relative frequency histogram shows the results of the survey.
The percent of people who own at most four T-Shirts costing more than $19 each is ?
can you find the value of constant k?
k=?
Answers
The constant k that makes f(x) continuous everywhere is k = 49/41.
What is a continuous function?If a function's limit at a given position exists and is the same as the function's value there, the function is said to be continuous at that location. If a function is continuous over its whole domain, then it is continuous everywhere. As the name implies, a continuous function is one whose graph is continuous throughout without any pauses or leaps. To put it another way, we say that a function is continuous if we can draw the curve (graph) of the function without ever picking up the pencil.
For f(x) to be continuous everywhere, it must be continuous at x = 7.
Using the left limit we have:
k(7)² = 49k
Using the right limit we have:
(7x + k) = 49 + k
Setting the limits equal we have:
49k = 49 + k = 8k
49k = 8k + 49
41k = 49
k = 49/41
Hence, the constant k that makes f(x) continuous everywhere is k = 49/41.
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The position of a particle moving in the xy-plane is given by the parametric functions x (t) and y(t), where = t sin (nt") and (3t+1) . The position of the particle is (2,7) at time t = 3. What is the particle's position vector («(t), y(t)) ? dy dt 30 sin (mtº) + 2t* cos(xt) 180 (3t+1)' :) B 2n cos (*t) +2 - 27, 10 31+1 + 8 C (cos (t) +2 -1 - + 8) D (cos (182) +2 -1 - +10)
Answers
The position of the particle moving in the x-y plane is given by the parametric functions x(t) and y(t), where = t sin(n t'') and (3t+1). The position of the particle is (2) , 7) at time t = 3.
In mathematics, a parametric equation defines a set of quantities as a function of one or more independent variables called parameters. [1] Parametric equations are often used to represent the coordinates of points that constitute geometric objects such as curves or surfaces, called respectively parametric curves and parametric surfaces. In this case, the equations are collectively referred to as the object's parametric representation or parameter system or parametrization (or orthographic parametrization)
The velocity of the particle is zero at t = 1 second
v(x) = dx/dt
= d/dt (3t²- 6t)
= 6t−6.
At t = 1, v(x) = 0
v(y) =dy/dt
= d/dt (t²−2t)
=2t−2.
At t=1,
v(y) = 0
Hence v = [tex]\sqrt{v_x^2 + v_y^2}[/tex] = 0
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Solve for the unknown whole number in the following expressions
z ÷ 19= 4 R 16
Answers
Answer:
We can solve for the unknown whole number by using the formula:
Dividend = Divisor × Quotient + Remainder
In this case, the dividend is z, the divisor is 19, the quotient is the unknown whole number, and the remainder is 16. We can substitute these values into the formula and solve for the unknown whole number:
z = 19 × Quotient + 16
To isolate the variable (Quotient), we can subtract 16 from both sides:
z - 16 = 19 × Quotient
Then, we can divide both sides by 19 to solve for Quotient:
Quotient = (z - 16) ÷ 19
Therefore, the unknown whole number is (z - 16) ÷ 19.
Wich one of the following expressions is equivalent to 7/tan b+ 7 tan b
Answers
Therefore, the expression [tex]\frac{7}{Tanb} +7Tanb[/tex] is equivalent to function. [tex]7*secb*cscb[/tex].
What is trigonometry?Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It includes the study of trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant, as well as their properties and applications.
Trigonometry is useful in a wide range of fields, including engineering, physics, navigation, astronomy, and surveying. It is often used to solve problems involving triangles, such as determining the height of a tall object, finding the distance between two points, or calculating the trajectory of a moving object.
The origins of trigonometry can be traced back to ancient civilizations such as the Babylonians, Greeks, and Indians, who developed various methods for calculating angles and distances. Today, trigonometry is an important part of mathematics education and continues to be used extensively in many fields of study.
Given by the question.
To simplify the expression 7/tan b + 7 tan b, we need to first recall the following trigonometric identity:
tan(x) * cot(x) = 1
Using this identity, we can rewrite the expression as:
[tex]\frac{7}{Tanb} +7Tanb[/tex]
= [tex]\frac{7}{Tanb} +7Tan^{2} b*cotb\\[/tex]
=[tex]\frac{7}{Tanb} +7*(sin^{2}/cos^{2}b)*(cosb/sinb)[/tex]
= [tex]\frac{7}{Tanb} +7*(cosb/sinb)[/tex]
= [tex]7*(1/tanb+sinb/cosb[/tex]
[tex]=7*(cosb/sinb+sinb/cosb)\\=7*((cos^{2}b+sin^{2}b)/(sinb/cosb))\\ =7*(1/(sinb/cosb))\\=7*secb*cscb[/tex]
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(10
points
)
Let
S(t)= 1+e −t
1
. (a) Find
S ′
(t)
. (b) Which of the following equations hold true? Show why your choice is true. [Note: only one equation is true.] i.
S ′
(t)=S(t)
ii.
S ′
(t)=(S(t)) 2
iii.
S ′
(t)=S(t)(1−S(t))
iv.
S ′
(t)=−S(−t)
Answers
The derivative of S(t)= 1+e −t is S'(t) = S(t)(1 - S(t)). So, the correct answer is (iii).
To find S'(t), we can use the chain rule:
S'(t) = (d/dt) [1 + e^(-t/2)]^-2 * d/dt [1 + e^(-t/2)]
Using the chain rule again for the second derivative:
d/dt [1 + e^(-t/2)] = (-1/2)e^(-t/2)
d/dt [1 + e^(-t/2)]^-2 = -2(1 + e^(-t/2))^-3 * (-1/2)e^(-t/2) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3
Substituting into the expression for S'(t), we have:
S'(t) = [(1/2) e^(-t/2) / (1 + e^(-t/2))^3] * [1 - (1/2)e^(-t/2)]
S'(t) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
Taking the derivative of S(t), we have:
S'(t) = e^(-t/2) / (1 + e^(-t/2))^2
Comparing this to the given choices, we can see that:
S'(t) = S(t) is not true, since S(t) = 1 + e^(-t/2) and S'(t) is a different function.
S'(t) = (S(t))^2 is not true, since (S(t))^2 = (1 + e^(-t/2))^2 is a different function from S'(t).
S'(t) = S(t)(1 - S(t)) is true, since we can substitute S(t) and S'(t) from above and simplify:
S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
S(t)(1 - S(t)) = [1 + e^(-t/2)] * [1 - (1 + e^(-t/2))] = e^(-t/2) / (1 + e^(-t/2))
Therefore, S'(t) = S(t)(1 - S(t)) is true.
S'(t) = -S(-t) is not true, since S(-t) = 1 + e^(t/2) and -S(-t) is a different function from S'(t).
So the correct choice is (iii): S'(t) = S(t)(1 - S(t)).
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You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 3
Answers
Answer:
the answer is 12/13 simplified
Step-by-step explanation:
simplified
list all symmetry groups that are the symmetry groups of quadrilaterals and for each group sketch a quadrilateral
Answers
The quadrilaterals which have both line and rotational symmetry of order more than 1 are square, and rhombus
Symmetry is a fundamental concept in mathematics and geometry. It refers to the property of a shape that remains unchanged when it is transformed in a certain way.
Now, let's talk about quadrilaterals that have both line and rotational symmetry of order more than 1. One example of such a quadrilateral is a square.
Another example of a quadrilateral with both line and rotational symmetry of order more than 1 is a rhombus. A rhombus is a type of quadrilateral where all four sides are equal in length, and opposite angles are equal.
In summary, a square and a rhombus are examples of quadrilaterals that have both line and rotational symmetry of order more than 1.
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Complete Question:
Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
2 bags of dog food. How many days will 3/4 last?
Answers
Answer:
3/4 of a bag of dog food will last 3 days.
what are the coordinates of point p on the directed line segment from a to b such that p is the length of the line segment from a to b?
Answers
the coordinates of point P are ((Ax + Bx) / 2, (Ay + By) / 2).
How to find?
If we want to find the coordinates of point P on the directed line segment from A to B such that P is the length of the line segment from A to B, we can use the following formula:
P = (1 - t)A + tB
where A and B are the coordinates of the two endpoints of the line segment, t is a scalar between 0 and 1, and P is the coordinates of the point we are trying to find.
When t = 1, we get the coordinates of point B, and when t = 0, we get the coordinates of point A. When t is between 0 and 1, we get a point on the line segment from A to B.
To find the point P that is the length of the line segment from A to B, we set t = 1/2, which gives us:
P = (1 - 1/2)A + (1/2)B
= (1/2)A + (1/2)B
So the coordinates of point P are the average of the coordinates of A and B:
Px = (Ax + Bx) / 2
Py = (Ay + By) / 2
where (Ax, Ay) and (Bx, By) are the coordinates of points A and B, respectively.
Therefore, the coordinates of point P are ((Ax + Bx) / 2, (Ay + By) / 2).
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Complete Question:
What are the Cartesian coordinates of point P on the directed line segment from point A to point B such that point P is located at a distance equal to the length of the line segment from point A to point B?
Use the Chain Rule to find dz/dt. z = cos(x + 8y), x = 7t^5, y = 5/t
Answers
Answer:
We need to find dz/dt given:
z = cos(x + 8y), x = 7t^5, y = 5/t
Using the chain rule, we can find dz/dt by taking the derivative of z with respect to x and y, and then multiplying by the derivatives of x and y with respect to t:
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt
First, let's find dz/dx and dz/dy:
dz/dx = -sin(x + 8y)
dz/dy = -8sin(x + 8y)
Now, let's find dx/dt and dy/dt:
dx/dt = 35t^4
dy/dt = -5/t^2
Substituting these values, we get:
dz/dt = (-sin(x + 8y)) * (35t^4) + (-8sin(x + 8y)) * (-5/t^2)
Simplifying this expression, we get:
dz/dt = -35t^4sin(x + 8y) + 40sin(x + 8y)/t^2
Substituting x and y, we get:
dz/dt = -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2
Therefore, dz/dt is given by -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2.
A bird is diving for fish in the ocean. His height above the water varies sinusoidally with time at 4 seconds, he spots a fish from a maximum height of 112 ft above water. He dives and at 7 seconds, he is at a minimum height of 14ft under water. Write an equation of the bird's height above the water as a function of time.
Answers
The equation of the bird's height above the water as a function of time can be expressed as:
H(t) = A * sin (B * t + C) + D
Where:
A is the amplitude, which is the difference between the maximum and minimum
B is angular frequency (2πf)
C is the phase shift
D is the midline
The maximum height is 112 ft and the minimum height is 14 ft, so A = 98ft.
The frequency of the cycle is 4 seconds (1 cycle every 4 seconds).
Therefore, the angular frequency is 2π/4 = π/2
The bird was at maximum height of 112 ft at t=4s, so the phase shift C = 0.
The midline is the average of the maximum and minimum, so D = (112+14)/2 = 63 ft.
Therefore, the equation of the bird's height above the water as a function of time is:
H(t) = 98 * sin (π/2 * t + 0) + 63
Dylan has a pitcher with 1.65 L of orange juice. He pours out 0.2 L of the juice. Then he adds some sparkling water to the pitcher to make orangeade. He ends up with 1.9 L of orangeade. Solve the equation 1.65 - 0.2 + x= 1.9 to find the amount of sparkling water, x, Dylan adds to the pitcher.
please soon
edit nevermind I actually read the question and it's not that hard and I solved it so hehe
Answers
Answer:
Step-by-step explanation:
x= 0.45
Answer: 0.45 L of sparkling water
Step-by-step explanation:
1.65 - 0.2 = 1.45
1.9 - 1.45 = 0.45
0.45 L of sparkling water
If 5 is increased to 9, the increase is what percentage of the original number
Answers
Answer: It's a 80% increase
Step-by-step explanation:
Two cellphone companies are offering different rate plans. Rogers is offering $19.99 per month, which includes a
maximum of 200 weekday minutes plus $0.35 for every minute above the maximum. TELUS is offering $39.99 for a
maximum 300 weekday minutes, but it charges $0.10 for every minute above the maximum. Above how many minutes
would TELUS be the better choice?
Answers
A printing company bought a machine for $86.000. it's estimated life is 10 years with residual value of $6,000, Using the straight-line method, what's the book value of the machine at the end of year2?
Answers
Hence, at the end οf year 2, the machine's bοοk value is $70,000.
Examples οf residual value what?Example οf Calculating Residual Value fοr a Business-Owned Vehicle. Hence, if the cοrpοratiοn sells the car after 6 years, 60% οf the οriginal cοst οf the car is depreciated οver 6 years, and the residual value is 40%. Remaining Value Befοre Tax is anοther name fοr this value.
The cοst οf an asset is evenly dispersed acrοss the asset's useful life with the straight-line technique οf depreciatiοn. Cοmpute the annual depreciatiοn cοsts and deduct it frοm the machine's οriginal cοst tο arrive at the machine's bοοk value at the end οf year twο.
Yοu can cοmpute the yearly depreciatiοn expense as fοllοws:
Cοst οf Asset - Residual Value / Useful Life = Depreciatiοn Cοsts
Inputting the values prοvided yields:
Depreciatiοn Expense = ($86,000 - $6,000) / 10 = $8,000 per year
The machine's οriginal cοst must be subtracted frοm the bοοk value at the end οf year twο in οrder tο determine the machine's value:
Bοοk Value at End οf Year 2 = Cοst οf Asset - (Depreciatiοn Expense x Number οf Years)
Bοοk Value at End οf Year 2 = $86,000 - ($8,000 x 2) = $70,000
Thus, the machine's bοοk value at the end οf year twο is $70,000.
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The amount of money Sally earns in dollars, P, varies directly with the number, n, of necklaces she sells. When Sally sells 15 necklaces she earns $150.
Answers
Answer:
Equation: [tex]P=10n[/tex]
[tex]P=40[/tex] when n = 4
Step-by-step explanation:
Given
Direct Variation: P ∝ n
When n = 15, P = 150
Required
Determine the equation
Her earnings if she sold 4 necklaces
P ∝ n
Convert variation to equation
[tex]P=kn[/tex]
When P = 150, n = 15 (Substitute these values in the given equation)
[tex]150=k\times15[/tex]
Divide both sides by 15
[tex]\dfrac{150}{15} =\dfrac{k\times15}{15}[/tex]
[tex]10=k[/tex]
[tex]k=10[/tex]
Substitute k = 10 in [tex]P=kn[/tex]
[tex]P=10\times n[/tex]
[tex]P=10n[/tex] ------- Equation that relates P and n
When she sold 4 necklaces, n = 4
Substitute n = 4 in the formula above
[tex]P=10\times4[/tex]
[tex]P=40[/tex]
my notes use implicit differentiation to find an equation of the tangent line to the curve at the given point.
Answers
The equation of the tangent line to the curve at the point (2,4) is y = (-1/2)x + 5.
To use implicit differentiation to find an equation of the tangent line to the curve at the given point (2,4), we need an implicit equation of the curve. Let's assume the curve is given by the equation:
x² + y² = 16
We can use implicit differentiation to find the slope of the tangent line at any point on this curve. Taking the derivative of both sides with respect to x, we get:
2x + 2y (dy/dx) = 0
Simplifying for (dy/dx), we get:
dy/dx = -x/y
Now we can substitute the given point (2,4) into this equation to find the slope of the tangent line at that point:
dy/dx = -2/4 = -1/2
So the slope of the tangent line at (2,4) is -1/2. We can use this slope and the point-slope form of the equation of a line to find an equation of the tangent line. The point-slope form is:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point. Substituting the values, we get:
y - 4 = (-1/2)(x - 2)
Simplifying, we get:
y = (-1/2)x + 5
Therefore, the equation of the tangent line to the curve at the point (2,4) is y = (-1/2)x + 5.
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Complete question:
Use implicit differentiation to find an equation of the tangent line to the curve at the given point (2,4)