(a) Calculate sinh (log(5) - log(4)) exactly, i.e. without using a calculator. (3 marks) (b) Calculate sin(arccos(4/√65)) exactly, i.e. without using a calculator. (3 marks) (c) Using the hyperbolic identity cosh^2p - sinh^2x =1, and without using a calculator, find all values of cosh x, if tanh x = 1/4(4 marks)
Answers
The required answer is -
a) sinh(log(5) - log(4)) = 9/40
b) sin(arccos(4/√65)) = √(1 - (16/65))= √(49/65) = 7/√65
c) The value of cosh x is ± √(16/15) for tanh x = 1/4.
Explanation:-
a) Calculate sinh (log(5) - log(4)) exactly, i.e., without using a calculator.
Solution: sinh(x) = (e^x - e^-x)/2Therefore, sinh (log(5) - log(4)))= [e^(log(5) - log(4))] - [e^-(log(5) - log(4))]/2= (5/4 - 4/5)/2= (25-16)/40= 9/40
Therefore, sinh(log(5) - log(4)) = 9/40.
b) Calculate sin(arccos(4/√65)) exactly, i.e., without using a calculator.
Solution: Let θ = arccos(4/√65) ⇒ cos θ = 4/√65⇒ sin²θ + cos²θ = 1 [using the identity sin²θ + cos²θ = 1]⇒ sin²θ + (16/65) = 1⇒ sin²θ = 49/65⇒ sin θ = √(49/65) = 7/√65. sin(arccos x) = √(1 - x²).
Therefore, sin(arccos(4/√65)) = √(1 - (16/65))= √(49/65) = 7/√65.
c) Using the hyperbolic identity cosh²x - sinh²x = 1, and without using a calculator, find all values of cosh x, if tanh x = 1/4.
Solution: We know that tanh x = sinh x/cosh x⇒ 1/4 = sinh x/cosh x⇒ sinh x = cosh x/4Using the identity cosh²x - sinh²x = 1⇒ cosh²x - (cosh²x/16) = 1⇒ (15/16) cosh²x = 1⇒ cosh²x = 16/15⇒ cosh x = ± √(16/15)
Therefore, the value of cosh x is ± √(16/15) for tanh x = 1/4.
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Related Questions
a rectangle is inscribed in a right isosceles triangle with a hypotenuse of length 6 units. what is the largest area the rectangle can have?
Answers
Answer: The largest area of a rectangle is 32 square units. A rectangle has its base on x-axis and its upper two vertices on the parabola y= 12 - x^2.
Step-by-step explanation:
This is the answer
Solve the triangle using the law of sines. Show all work and round answers to the nearest tenth.
Answers
The unknown values in the triangle are ;
a = 45.3
angle B = 7°
angle A = 28°
What is sine rule?In a triangle, side “a” divided by the sine of angle A is equal to the side “b” divided by the sine of angle B is equal to the side “c” divided by the sine of angle C.
Therefore,
a/sinA = b/sinB = c/sinC
So, we use the Sine rule to find the values of unknown lengths or angles of the triangle.
b/sinB = c/sinC
7/sinB = 33/sin145
33sinB = 7 sin145
33 sinB = 4.02
sinB = 0.122
B = 7°
angle A = 180-(145+7)
A = 180- 152
A = 28°
a/sinA = c/sinC
a/sin28 = 33/sin145
asin145 = 33sin128
0.574a = 26
a = 26/0.574
a = 45.3
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if you take 3/4 of 1 3/5 how much do you have
Answers
Answer:
1 1/5
Step-by-step explanation:
1.6*3/4
4.8/4
1.2
1 1/5
Therefore, the answer is 17/20.
Hoped this helped.
\(BrainiacUser1357\)
a payroll clerk addresses five paychecks and envelopes to five different people and randomly inserts the paychecks into the envelopes. find the probability of each event. (a) exactly three paychecks are inserted in the correct envelopes. (no response) seenkey 1/12 (b) at least three paychecks are inserted in the correct envelopes.
Answers
The required probability for the given different events are as follow,
Probability that exactly three paychecks are inserted in the correct envelopes = 1/12
probability of at least three paychecks is 1/4.
Probability that exactly three paychecks are inserted in the correct envelopes, use the formula for permutations,
Probability 'P'
=[ (number of ways to choose 3 correct envelopes) x (number of ways to choose 2 incorrect envelopes) x (number of ways to arrange the paychecks in the correct envelopes) ] / (total number of ways to arrange the paychecks in the envelopes)
Number of ways to choose 3 correct envelopes out of 5 is
5C3 = 10
Number of ways to choose 2 incorrect envelopes out of 5 is
5C2 = 10
Number of ways to arrange the 3 correct paychecks in their correct envelopes is 3! = 6
Number of ways to arrange the 2 incorrect paychecks in their incorrect envelopes is 2! = 2.
Total number of ways to arrange the paychecks in the envelopes is
5! = 120.
Probability that exactly three paychecks are inserted in the correct envelopes is,
Probability
= (10 x 10 x 6 x 2) / 120
= 1/12
Probability that at least three paychecks are inserted in the correct envelopes,
Exactly 3 paychecks are inserted in the correct envelopes
Exactly 4 paychecks are inserted in the correct envelopes
All 5 paychecks are inserted in the correct envelopes
Probability of exactly 3 paychecks being correct is 1/12.
Number of ways to arrange the paychecks in the envelopes is 1.
P(exactly 4 correct)
= (number of ways to choose 4 correct envelopes) x (number of ways to choose 1 incorrect envelope) x (number of ways to arrange the paychecks) / (total number of ways to arrange the paychecks)
= 5C4 x 1 x 4! / 5!
= 1/5
Probability of all 5 paychecks being inserted in the correct envelopes is 1/5! = 1/120.
Required probability
= 1/12 + 1/5 + 1/120
= 1/4
Therefore, the probability of exactly and at least 3 paychecks being inserted in the correct envelopes is equal to 1/12 and 1/4 respectively.
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is it =,<, or > for each one?
Answers
Jordan and Dajia are both completing homework on Deltamath. Jordan started yesterday and completed three problems and is solving the problems at a rate of 2 questions per minute. Dajia waited until today to get started and is solving the problems at a rate of 3 questions per minute. After how many minutes will they have solved the same number of questions?
Answers
show that a is diagonalizable if (a − d)2 4bc > 0. a is not diagonalizable if (a − d)2 4bc < 0. [hint: see exercise 29 of section 5.1.]
Answers
To show that a matrix a is diagonalizable, we need to prove that a can be written as a product of two matrices P and D, where P is invertible and D is a diagonal matrix. In other words, we need to show that there exists a basis of eigenvectors for a.
Let λ be an eigenvalue of a with corresponding eigenvector x. Then, we have ax = λx, which can be rewritten as (a - λI)x = 0, where I is the identity matrix. Since x is nonzero, we must have det(a - λI) = 0, which gives us the characteristic equation of a.
Solving for λ in the characteristic equation, we get λ = d ± √(d^2 - 4bc)/(2b), where d is a diagonal entry of a. If (a - d)^2 - 4bc > 0, then both eigenvalues are real and distinct, which means a has a basis of eigenvectors and is diagonalizable.
On the other hand, if (a - d)^2 - 4bc < 0, then the eigenvalues are complex conjugates, which means a cannot be diagonalized over the real numbers. Therefore, a is not diagonalizable.
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A principal wishes to implement a decision that has to be a number between 0 and 1; that is, a decision d needs to be implemented where 0 sdS1. The difficulty for the principal is that she does not know what decision is appropriate given the current state of the economy, but she would like to implement a decision that exactly equals what is required given the state of the economy. In other words, if the economy is in state s (where 0 sS 1) the principal would like to implement a decision d s as the principal's utility Up (or loss from the maximum possible profit) is given by Up--s-d With such a utility function, maximising utility really means making the loss as small as possible. For simplicity, the two possible levels of s are 0.4 and 0.7, and each occurs with probability 0.5 There are two division managers A and B who each have their own biases. Manager A always wants a decision of 0.4 to be implemented, and incurs a disutility Ua that is increasing the further from 0.4 the decision d that is actually implement, specifically U-0.4-d.Similarly, Manager B always wants a decision of 0.7 to be implement, and incurs a disutility UB that is (linearly) increasing in the distance between 0.7 and the actually decision that is implemented - that is Ug--10.7 Each manager is completely informed, so that each of them knows exactly what the state of the economy s is (a) The principal can opt to centralise the decision but before making her decision given she does not know what the state of the economy is - she asks for recomm endation s from her two division mana gers. Centralisation means that the principal commits to implement a decision that is the average of the two recommendations she received from her managers. The recommendations are sent simultaneously and cannot be less than 0 or greater than 1 Assume that the state of the economy s = 0.7. What is the report (or recommendation) that Manager A will send if Manager B always truthfully reports s? (b) Again the principal is going to centralise the decision and will ask for a recommendation from both managers, as in the previous question. Now, however assume that both managers strategically make their recommendations. What are the recommendations rA and rB made by the Managers A and B, respectively, in a Nash equilibriunm
Answers
A. Manager A wants the decision to be 0.4, so they would recommend a decision of 0.4 to the principal.
B. The recommendations in the Nash equilibrium would be rA = 0.4 and rB = 0.7.
(a) If Manager B always truthfully reports the state of the economy (s = 0.7), Manager A would send a recommendation that minimizes their disutility Ua. In this case, Manager A wants the decision to be 0.4, so they would recommend a decision of 0.4 to the principal.
(b) In a Nash equilibrium, both managers strategically make their recommendations based on their own utility. Manager A wants to minimize their disutility Ua, which increases as the decision deviates from 0.4. Manager B wants to minimize their disutility UB, which increases as the decision deviates from 0.7.
To find the Nash equilibrium, we need to consider the recommendations made by both managers simultaneously. Let's denote the recommendations as rA (from Manager A) and rB (from Manager B). The principal's decision, d, would be the average of the recommendations, so d = (rA + rB) / 2.
Given that both managers strategically choose their recommendations, they will aim to minimize their disutility. In this case, Manager A would recommend a decision of 0.4 (as it minimizes Ua), and Manager B would recommend a decision of 0.7 (as it minimizes UB). Therefore, the recommendations in the Nash equilibrium would be rA = 0.4 and rB = 0.7.
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Mr. Selan loves the 99-cent menu at Wendy's (sadly prices went up a few years ago). He used to buy two chicken sandwich at 99-cents each. With a tax of 8.5%, how much would he pay for the two sandwiches? Round your answer to the nearest cent
Answers
Answer:
Mr. Selan would pay $2.15 for the sandwiches with tax.
Step-by-step explanation:
Given that:
Cost of each sandwich = 99 cents
Cost of two sandwiches = 99*2 = 198 cents
Sales tax = 8.5%
Sales tax = \(\frac{8.5}{100}*198=16.8\ cents\)
Sales tax = 17 cents
Total cost = 198 + 17 = 215 cents
Total cost = $2.15
Hence,
Mr. Selan would pay $2.15 for the sandwiches with tax.
x/4+7=27 what is x? how do i find x if it is in a fraction?
Answers
Answer:
x=80
Step-by-step explanation:
So normally when x is in the numerator, you multiply both sides of the equation by whatever the denominator is to cancel out the fraction. But in this case since you don't have the fraction alone, you normally want to isolate it first So let's do that
Original equation:
\(\frac{x}{4}+7=27\)
Subtract 7 from both sides
\(\frac{x}{4}=20\)
Multiply both sides by 4 to cancel out the denominator:
\(x=80\)
2. If f(x) = 2x2 – 7x -10, find the following: a) f(x-1) b) x, if f(x) = 5 Please helpppppp
Answers
Answer:
a) f(x-1) = 2x^2-11x-1
b) x = -1.5 or 5
Step-by-step explanation:
a) To get f(x-1)
We proceed to substitute x-1 for x as follows
f(x-1) = 2(x-1)^2 -7(x-1) -10
f(x-1) = 2(x^2 - 2x + 1) -7(x-1)-10
f(x-1) = 2x^2 -4x + 2 -7x + 7-10
f(x-1) = 2x^2-11x-1
b) What we have to do here is substitute 5 for f(x) and solve for x
5 = 2x^2 -7x-10
2x^2 -7x -10-5 = 0
2x^2 - 7x - 15 = 0
2x^2 -10x + 3x -15 = 0
2x(x-5) + 3(x-5) = 0
(2x + 3)(x-5) = 0
2x + 3 = 0 or x -5 = 0
2x = -3 or x = 5
x = -3/2 or x = 5
x = -1.5 or 5
Quadrilateral ABCD is inscribed in circle P as shown. Which statement is necessarily true?
Answers
Answer:
B. m∠A + m∠C = m∠B + m∠D
Step-by-step explanation:
To inscribe a quadrilateral in a circle, the sums of its opposite angles must be equal (each sum of its opposite angles equals 180 °)
Does combine mean add?
Answers
the areas of two squraes differe by 100 square units, an the perimeters of the two square differ by 1 units. what is the perimeter, in units, of teh smaller square
Answers
The perimeter of the smaller square is 12 units.
The area of a square is equal to the side length of the square squared. Therefore, the difference in area between two squares is equal to the difference in the side lengths of the squares squared. Since the difference in area between the two squares is 100 square units and the side lengths of the squares differ by 1 unit, we can set up the equation (x + 1)^2 - x^2 = 100, where x is the side length of the smaller square.
Expanding the right side of the equation and solving for x, we find that x = 3. The perimeter of a square is equal to the side length multiplied by 4, so the perimeter of the smaller square is 3 * 4 = 12 units.
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What are the 3 types of rigid transformations?
Answers
Three types of rigid transformations are Reflection, Rotation, and Translation. Rigid transformation, also called isometry.
A rigid transformation, also called isometry, is a transformation that doesn't change the size or shape of a geometric figure. The following are 3 types of rigid transformation:
1. Reflection
→ is the act of shifting an object's coordinates that flip it across a line without changing its shape or size. Horizontal (draw a figure to the left or right) or vertical (draw a figure to the up or down) reflections are possible. The result of reflection is a mirror image of the figure itself.
The figure is reflected across \(x-\) or \(y-\) axis, and then change \(x-\) or \(y-\) coordinate.
2. Rotation
→ is the non-modification of an object's size or shape by rotating it around an fixed point. A center of rotation is required to rotate an object. And the rotation did by using a degree.
The figure is rotated by a degree (ex: 90°), and then change \(x-\) or \(y-\) coordinate. Meanwhile, a point's center rotation stays at the same.
3. Translation
→ is sliding a figure in any direction without changing its size, shape, or orientation. Translation could be horizontal (make a figure left or right), or vertical reflections (make a figure up or down).
Vertical translation is shifting the graph along \(y-\) axis
Horizontal translation is shifting the graph along \(x-\) axis.
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What are the zeros of the function h (x) = x² + 3x - 8?
A
x = -8 and x = -2
OB
x= -8 and x = 2
cx = -2 and x = 8
OD x = 2 and x = 8
Answers
The following are the zeros for the function h (x) = x2 + 3x - 8: - x= -4 and x=2.
Describe functions.Given a collection of inputs X (domain) and a set of potential outputs Y (codomain), a function is more technically defined as a set of ordered pairings (x,y) where xX and yY with the caveat that there can only be one ordered pair with the same value of x. The function notation f:XY can be used to express that f is a function from X to Y.
The function's zero is a value of x that makes it equal to zero. In other words, the equation f(x) = 0 leads to a zero.
By putting h(x) equal to zero and figuring out x, we may determine the zeroes for the function h(x) = x2 + 3x - 8.
h(x) = x² + 3x - 8 = 0
We may factor the left side of the equation to find x:
x² + 3x - 8 = (x-2)(x+4) = 0
We set each factor to zero and solve for x to discover the zeroes:
x-2 = 0 or x+4 = 0
x = 2 or x = -4
Consequently, the function's zeros are x = 2 and x = -4.
So, A is the right response. x = -4 and x = 2
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The complete question is
What are the zeros of the function h (x) = x² + 3x - 8?
A. x = -4 and x = -2
B. x= -8 and x = 2
C. x = -2 and x = 8
D. x = 2 and x = 8
I got my ASVAB score back today... And I got a 21%... someone loves kill me now. But I didn't really have time to study for it, because my teachers offered for me to take it early and I did and now I have failed D:
Answers
Answer:
dang thats tough..
Step-by-step explanation:
idrk what an asvab is but it sounds horrible im
sorry that happened :(
-X + 5+ 4x
simplify the expression
Answers
Answer:
3x+5
Step-by-step explanation:
combine like terms, and get 3x+5
What is the area of the figure?
Answers
Answer:
21.5
Step-by-step explanation:
break the shape down to make two shapes: a triangle and rectangle.
triangle = A=bxh/2
3x5=15/2=7.5
Rectangle = A=bxh
2x7=14
7.5+14= 21.5
Hope this helps :)
Answer:
21.5
Step-by-step explanation:
(all steps provided in image attached)
Is (2,3) a solution of the system
Answers
Answer:
Yes
Step-by-step explanation:
I need the answer to this one .
Answers
Answer:
B. 6.5x10^5
Step-by-step explanation:
Hope this Helps :)
can someone help? i have a quiz tomorrow and i'm lowkey lost.
Answers
10 and 13 are the measure of the values of x and y respectively
Solving angles in a parallelogramThe given diagram is a parallelogram with unknown values x an y.
Since the measure of sum of angles in a triangle is 180 degrees, hence we will take the measure of the angles in the triangle EFY to have:
70 + 7x - 5 + 45 = 180
110 + 7x = 180
7x = 180 - 110
7x = 70
x = 10
Similarly for the triangle EDY;
45 + 70 + 5y = 180
115 + 5y = 180
5y = 180 - 115
5y = 65
y = 13
Hence the measure of the values of x and y are 10 and 13 respectively
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Can anyone please help me with this!!! :(
Answers
Answer:
ano ba gagawin
parang Ang Dali pero ano gagawin dyan
Hi there!
For this problem, pretty much add all of them and then subtract with 22 to find how much Jen ran on Friday.
Monday+Tuesday+Wednesday+Thursday:
Simplify all terms and find all common denominators:
Common denominator: 8
\(4 \frac{3}{4} = 4 \frac{6}{8}\)
\(5 \frac{1}{8} = 5 \frac{1}{8}\)
\(6 \frac{1}{4} = 6 \frac{2}{8}\)
Combine all terms:
\(4 \frac{6}{8} + 5 \frac{1}{8} + 6 \frac{2}{8} = 15 \frac{9}{8}\)
Simplify 15 9/8 or else can’t subtract with 22.
\(9/8 = 1 \frac{1}{8}\)
\(15 + 1 \frac{1}{8} = 16 \frac{1}{8}\)
Turn 22 into 21 8/8...
Subtract: 21 8/8 with 16 1/8
\(21 \frac{1}{8} - 16 \frac{1}{8} = 5 \frac{7}{8}\)
This means that 5 7/8 is how much mile Jen must run on Friday to reach her goal.
What percentage of videos on the streaming site are between 264 and 489 seconds? 0.15% 49.85% 95% 99.7%
Answers
Answer:
49.85%
Step-by-step explanation:
Streaming is defined as sending and receiving an video or an audio content in continuous flow over any network. It allows one to begin the playback and while it sends the rest of the data.
In the context, the percentage of the videos on a streaming site that is between 264 and 489 seconds is 49.85% which is found by a graph where the distribution of the lengths of the videos in seconds on a popular video streaming sites.
Find the perimeter of the triangle.
Answers
Answer:
13.83
Step-by-step explanation:
\(hf = \sqrt{({6 - 6 })^{2} +( {2 - 5)}^{2} }\)
\( = \sqrt{0 + {( - 3)}^{2} } \)
\( = \sqrt{0 + 9} = \sqrt{9} = 3\)
\(gf = \sqrt{ {(6 - 1)}^{2} + {(2 - 2)}^{2} } \)
\( = \sqrt{ {5}^{2} + {0}^{2} } \)
\( = \sqrt{ {5}^{2} } = 5\)
\(gh = \sqrt{ {(6 - 1)}^{2} + {(5 - 2)}^{2} } \)
\( = \sqrt{ {5}^{2} + {3}^{2} } \)
\( \sqrt{25 + 9} \)
\( = \sqrt{34} = 5.83\)
perimeter = 3+5+5.83 = 13.83
Find the area of the region bounded by the function y=5xln(2)−1 and the lines y=0 x=1 and x=e Online answer: Enter the area rounded to the nearest integer, if necessary.
Answers
The area of the region bounded by the function y = 5xln(2) - 1 and the lines y = 0, x = 1, and x = e is approximately 5ln(2) [(1/2) \(e^2\) - (1/2)] - (e - 1) square units.
To find the area of the region bounded by the given function and lines, we need to determine the limits of integration and set up the integral. First, we observe that the region is bounded by the x-axis (y = 0) and the curve y = 5xln(2) - 1. We can find the x-values where these two curves intersect by setting them equal to each other:
0 = 5xln(2) - 1
Solving this equation, we get x = (1 / (5ln(2))). The other bounds are given as x = 1 and x = e.
Next, we set up the integral to find the area bounded by the curves. The integral is given by:
\(\int\limits^e_1\) (5xln(2) - 1) dx
Evaluating this integral, we find the antiderivative of (5xln(2) - 1), which is [(5/2)\(x^2\)ln(2) - x]. Then, we substitute the upper and lower limits of integration into the antiderivative and subtract the lower value from the upper value:
[(5/2)\(e^2\)ln(2) - e] - [(5/2)\((1)^2\)ln(2) - 1]
5ln(2) [(1/2) \(e^2\) - (1/2)] - (e - 1)
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Complete the series 1,1,4,12,__
Answers
Hi how are you call 112 you will get nice
Factor the following expression completely.
16x^5-x^3
A. 1343 1) (4x + 1)
B. 13(41 – 1)
C. (413 – 22)
D. 13(16x2 – 0)
Answers
X^(16x^2-1) factor the expression
The answer to the graph
X^3(4x-1)(4x+1)
Janelle expanded the Expression as shown below what errors did she make select 3 options
Answers
Given:
\(-4(-3x+\frac{2}{7})=-12x-3\frac{5}{7}\)Correct answer is
\(-4(-3x+\frac{2}{7})=12x-1\frac{1}{7}\)Final options are:
The first term should be positive.
She did not distribute the -4 properly.
\(\text{She added }\frac{2}{7}\text{ to -4 instead of multiplying }\frac{2}{7}\text{ by -4.}\)hi, could you please answer this?
Answers
Answer:
He added -8 and -2 instead of multiplying them, 2nd term should be positive, the last term should be -1 1/2
Step-by-step explanation: