Part a. Find the Laplace transform and inverse Laplace transform of Let+')} and L- [8 marks]Laplace transform of Let+')} is given as: L(et+')} = 1/s-(1/ s+2)Let's try to this.1/s-(1/ s+2) = (s+2-s)/s(s+2) = 2/s(s+2)L-1{2/s(s+2)}= L-1{(1/s)-(1/s+2)} = e^(-2t) - 1
Part b. 18 = 1+k8-2+ 3+ 3k and 7 - 1 - 2k, obtain the following: i) ä. ax, ii) (ä xk), iii) 2:a)3 +34 i) 18 = 1+k8-2+ 3+ 3k
Let's simplify this as follows.18 = 12 + k + 3k - 2k + 3k-1 - 4k+1/218 = 12 - k + 2k + 2k + 3(1/k) - 4k+1/2a=12, b=-1, c=2 and d=2Therefore,ä. ax = ad-bc = 24 + 2 = 26ii) (ä xk) = (cd - bc)i + (ab - ad)j + (ad - bc)k = 1i - 2j + 24kiii) 2:a)3 +34 = 2(ad-bc) + 3(ab-ad) + 4(cd-bc) + 3(bd - ac) = 52
Part c. Find the point at which the plane 22 - 5y + z = 5 and the line Ft) - (+1) + (24+ 1)3+ (t+1)! (where t is a real number) intersect. [3 marks]. Let's substitute the given line in the plane equation. 2(1+4t) - 5(3+1t) + z = 5 solving for z, we get
z = 12t - 13
Substitute this z in line equation to get the point of intersection.(1+t, 4+2t, 12t-13)
Part d. Compute the curvature and principal unit normal vector for the curve rt) 2 sin(t)+ 2 con(e) { for t > 0. [6 marks]. Given curve r(t) = 2 sin(t) + 2 cos(t).
We need to find the first and second derivatives. r'(t) = 2 cos(t) - 2 sin(t)r''(t) = -2cos(t) - 2sin(t)
From these values, we get |r'(t)| = sqrt(8)K(t) = |r'(t)|/|r"(t)|^3/2K(t) = 2^(3/2)/8^(3/2)K(t) = 1/4^(1/2)K(t) = 1/2
Therefore the curvature is 1/2Now, let's find the principal unit normal vector. N(t) = r''(t)/|r"(t)|N(t) = <-1/sqrt(2),1/sqrt(2)>
Part e. For the two matrices A= 12 0 0-7 5 3 B= B=(- :) 0
Find i) AT, ii) BT, iii) B(AT) and iv) (AB)" [8 marks]
i) AT =Transpose of A = 1 -7 0 2 5 3
ii) BT =Transpose of B = 1 0 -3 0 2 1
iii) B(AT) =B(Transpose of A) = 1 -7 0 2 5 3(-1) 0 0 7 -5 -3= -1 7 0 -2 -5 -3
iv) (AB)" =(AB)^-1=Inverse of AB Let's calculate AB first. AB =12 0 0-7 5 3(-1) 0 0-5 2 -1= -1 0 0 1
Therefore, the inverse of AB is1 0 0-1
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If your having trouble with math go to Wolframalpha.com
Answer:
Thanks for letting me know I might try that later today
Step-by-step explanation:
:)
A firm is considering a the launch of a new consumer product. Consider the following costs. Which should be included in it's capital budget cash flow analysis?
Group of answer choices
the costs the firm spends on financing
the firm's sunk costs
the firm's decline in current sales when the new product is launched
all the firm's opportunity costs
2-What is the source of a firm's financial leverage?
Group of answer choices
A firm's changes in EBIT.
A firm's variability in fixed operating costs.
A firm's variability in sales.
The use of debt and preferred stock.
3-Operating risk derives from ...
Group of answer choices
the risk that comes from the type of industry in which a firm operates.
the variability of a firm's stock price.
the risk that comes from a firm’s mix of fixed and variable costs.
the risk that comes from a firm’s mix of long-term debt and equity
4-Which of the following is a not legal constraint on the payment of dividends?
Group of answer choices
The firm’s liabilities exceed its assets.
The dividend would be paid from the retained earnings of a firm.
The dividend would be paid from capital invested in the firm.
The amount of the dividend exceeds the firm’s retained earnings.
5-According to the _______________, investors view changes in a firm’s dividend policy as a signal about the firm’s financial condition.
Group of answer choices
Residual dividend theory
Clientele effect
Information effect
1. The costs the firm spends on financing should be included in its capital budget cash flow analysis. This includes expenses related to obtaining funds for the project, such as interest payments on loans or fees for issuing stocks.
2. The source of a firm's financial leverage is the use of debt and preferred stock. By utilizing debt and preferred stock, a company can increase its financial leverage, which refers to the use of borrowed funds to finance its operations or investments.
3. Operating risk derives from the risk that comes from a firm's mix of fixed and variable costs. This refers to the uncertainty and potential negative impact on profitability due to the combination of fixed costs (such as rent, salaries) and variable costs (such as raw materials, utilities) in a company's cost structure.
4. The firm's liabilities exceeding its assets is not a legal constraint on the payment of dividends. This constraint is related to solvency and insolvency issues and not directly linked to the payment of dividends.
5. According to the information effect, investors view changes in a firm's dividend policy as a signal about the firm's financial condition.
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A red car left the park at 9 am. An hour later a blue car left the same park, heading to the same destination. If both cars arrived at the destination at 1 pm, and the speed of the blue car was 15 mph faster than the red car, what was the speed of the blue car?
Answer: 60 mph
Step-by-step explanation:
Given
The red car left at 9 am and arrives at 1 pm
time taken by the red car [tex]t_a=4\ hr[/tex]
time taken by the blue car [tex]t_b=3\ hr[/tex]
Assume the speed of the red car is v
So, the speed of the blue car is v+15
distance traveled by them is the same
[tex]\Rightarrow v\times 4=(v+15)\times 3\\\Rightarrow 4v=3v+45\\\Rightarrow v=45\ mph[/tex]
Thus, the speed of the blue car is [tex]45+15=60\ mph[/tex]
If f(x) = |x| + 9 and g(x) = –6, which describes the range of (f + g)(x)?
Answer:
The answer is A.
Step-by-step explanation:
Answer:
Step-by-step explanation:
A is the correct answer on Edge
Alicia Bitman, age 30, plans to purchase a $200,000, 5-year term life insurance policy. What is the annual premium?
Answer: $770
Step-by-step explanation:
Number of units to be purchased by Alicia:
= 200,000 / 1,000
= 200 units
Alicia is purchasing a 5-year term life insurance policy. At age 30, the cost per $1,000 unit of insurance is $3.85 for a female.
The formula for the annual premium is:
= Number of units purchased * Premium per $1,000
= 200 * 3.85
= $770
The
ratio of votes in favor to votes against in an election is 5 to 4.
How many total votes were cast if there are 2,620 votes in
favor?
Total votes were casted in election are 4716
Given: The ratio of votes in favor to votes against in an election is 5 to 4. 2,620 votes are in favor.
To find: The total number of votes cast.
Let the number of votes against is 4x.
Given the ratio of votes in favor to votes against is 5 : 4
Then, the number of votes in favor is 5x.
According to the question, 2,620 votes are in favor.
So, 5x = 2,620x = 2,620/5x = 524
The number of votes against = 4x = 4 × 524 = 2096
The total number of votes cast = votes in favor + votes against= 2620 + 2096= 4716
Therefore, there were 4716 votes cast in the
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Find the diagonalization of A by finding an invertible matrix P and a diagonal matrix D such that PAP= D.
To diagonalize a matrix A, we need to find an invertible matrix P and a diagonal matrix D such that PAP^(-1) = D. Here's how to find the diagonalization of matrix A
1. Find the eigenvalues of A:
- Calculate the characteristic polynomial by subtracting λI from A, where λ is a scalar variable and I is the identity matrix of the same size as A.
- Set the characteristic polynomial equal to zero and solve for λ to find the eigenvalues.
2. Find the eigenvectors corresponding to each eigenvalue:
- For each eigenvalue, substitute it back into the equation (A - λI)x = 0, where x is a vector, and solve for x.
- Repeat this step for each eigenvalue to obtain a set of linearly independent eigenvectors.
3. Construct the matrix P:
- Arrange the eigenvectors found in Step 2 as columns to form the matrix P.
4. Construct the diagonal matrix D:
- Place the eigenvalues obtained in Step 1 on the diagonal of a matrix of the same size as A, with zeros elsewhere.
5. Verify the diagonalization:
- Calculate PAP^(-1) and check if it equals D. If PAP^(-1) = D, then A is diagonalizable.
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If as of December 31, 2017 in the judicial offices there were 2,535,225 complaints of domestic violence and by December 31, 2019 that figure reached 2,956,300.
What is the annual growth rate under the exponential model (round to the nearest hundredth, record your answer to two decimal places, and use a period to separate)?
The annual growth rate under the exponential model is approximately 0.17 or 17%.
To calculate the annual growth rate under the exponential model, we can use the formula:
Annual Growth Rate = (Final Value / Initial Value) ^ (1 / Number of Years) - 1
In this case, the initial value is 2,535,225 complaints of domestic violence as of December 31, 2017, and the final value is 2,956,300 complaints as of December 31, 2019. The number of years is 2.
Plugging in the values:
Annual Growth Rate = (2,956,300 / 2,535,225) ^ (1 / 2) - 1
= 1.1654 - 1
= 0.1654
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let g be a differentiable function such that g(4)=0.325 and g′(x)=1xe−x(cos(x100)) . what is the value of g(1) ? responses
To find the value of g(1), we need to integrate g'(x) and use the given initial condition g(4)=0.325. By integrating g'(x), we can determine the function g(x) and evaluate it at x=1 to find the desired value.
To find g(x), we integrate g'(x) with respect to x. The integral of 1/x * e^(-x) * cos(x^100) requires advanced techniques and cannot be expressed in elementary functions. Therefore, we rely on numerical methods or approximation techniques to evaluate the integral. Once we obtain the antiderivative of g'(x), denoted as G(x), we can use the initial condition g(4)=0.325 to determine the constant of integration.
Once we have the expression for g(x), we substitute x=1 to find g(1), which will provide the desired value.
Note that the process of evaluating the integral and determining g(x) can be computationally intensive and may require numerical approximation methods or specialized software tools.
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A thermometer reading 96°F is placed inside a cold storage room with a constant temperature of 37°F. If the thermometer reads 88°F in 5 minutes, how long before it reaches 58°F? Assume the cooling follows Newton's Law of Cooling: U = T+ (U. - T)ekt (Round your answer to the nearest whole minute.) 45 minutes 0 1 minutes 0 16 minutes 14 minutes
It takes approximately 14 minutes for the thermometer to reach a temperature of 58°F in the cold storage room. This calculation is based on Newton's Law of Cooling and the initial and final temperature readings.
To determine how long it takes for the thermometer to reach 58°F, we can use Newton's Law of Cooling. Let's plug in the given values into the equation and solve for the time (t):
88 = 37 + (96 - 37)e^(k * 5)
Simplifying the equation, we have:
51 = 59e^(5k)
Taking the natural logarithm of both sides:
ln(51/59) = 5k
Solving for k, we find:
k ≈ -0.0436
Now, we can use this value of k to find the time (t) when the thermometer reaches 58°F:
58 = 37 + (96 - 37)e^(-0.0436 * t)
Simplifying further, we have:
21 = 59e^(-0.0436 * t)
Taking the natural logarithm again:
ln(21/59) = -0.0436 * t
Solving for t, we find:
t ≈ 13.58
Rounding to the nearest whole minute, it takes approximately 14 minutes for the thermometer to reach 58°F.
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Find the missing side of this right triangle 7 12
Answer:
Well since the question ask for what in the green box, its 193
193 goes in the green box
If you solve everything it would be 13.89 (rounded to the nearest hundredths)
Step-by-step explanation:
So missing side of a right triangle, you can use the Pythagorean Theroum which is a^2+b^2=c^2
In this case we have the two legs which are a and b, we’re trying to find hypotnuse, “c”.
7^2+12^2=c^2
49+144=c^2
193= c^2
You basically do √193
which is the answer needed for this situation
Final answer: is 193 goes into the green box
Someone please help this is due tomorrow!
Answer:
Sam made a mistake. The answer should have been 30.
This is what Sam should have done:
2n^2 - 20
2n^2 - 20 = 2(5)^2 - 20
= 2(25) - 20
= 50 - 20
= 30
Sam multiplied 2 and 5 when he should have done the power first.
can somebody help me pls
Answer:
C. 37
Step-by-step explanation:
winindoutpickc
help plz I will give brainliest
Answer: what do you need help with??
Step-by-step explanation:
The center of the sphere x2 + y2 +2 +4x – 2y – 62= Dis: - 6z= - (4, -2,9) 4 (-4, 2, 6) (1,1) (0,0,0) (-2,-1,3) (-2,1,3) (-4, 2, 6) (2, 1, 3) ?
The center of the sphere is (-2, 1, 0).
The radius of the sphere is √65.
The given equation is x² + y² + 2 + 4x - 2y - 62 = 0.
We can rewrite the given equation as follows:
x² + 4x + y² - 2y = 60
Completing the square of x and y, we get:
(x + 2)² - 4 + (y - 1)² - 1 = 60
(x + 2)² + (y - 1)² = 65
Now, we know that the general equation of the sphere is :
(x - a)² + (y - b)² + (z - c)² = r² where (a, b, c) is the center of the sphere, and r is the radius.
To compare the given equation with the equation of the sphere, we will have to convert it into the standard form as follows:
(x + 2)² + (y - 1)² + (0 - 0)² = √65²
The center of the sphere is (-2, 1, 0), and its radius is √65.
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The length of a rectangle is 15ft greater than the width. The area is 100 square ft. Find the length and the Width.
Step-by-step explanation:
so let's say that the width is x then the length is x+15
and the area of a square is length times width
x(x+15)=100
x^2+15x-100=0
(x+20)(x-5)=0
x=5 or x=-20 but a side length can't be negative so x would equal 5
Length=x+15 with 5 as x
Length=20
Width=5
Hope that helps :)
QUICK! Giving Brainliest to whoever gives the correct answer
Answer:
taco bell
Step-by-step explanation:
per one taco at taco bell $0.53
per one taco at los comales $0.62
You know this??????????????
Answer:
y=x+15
Step-by-step explanation:
help me find the surface area!
Answer:
62
Step-by-step explanation:
find the area of each side
(3 * 5) + (2 * 3) + (2 * 5) + (2 * 5) + (2 * 3) + (3 * 5)
add them all
15 + 6 + 10 + 10 + 6 + 15 = 62
Given the following function, find the integral s voix by substitution : integral 3 (x-2 ] 3 +4 dx by substitution sinhy=3(x-2)
The simplified expression of integral 3 (x-2 ] 3 +4 dx is (A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B
How to find the integral ∫3(x-2)³+4 dx using the substitution sinh(y) = 3(x-2)?To find the integral ∫3(x-2)³+4 dx using the substitution sinh(y) = 3(x-2), we can start by differentiating both sides of the equation with respect to x to find the differential of y:
d(sinh(y))/dx = d(3(x-2))/dx
cosh(y) * dy/dx = 3
dy/dx = 3/cosh(y)
Now, let's solve for dx in terms of dy:
dx = (cosh(y)/3) dy
Substituting this value of dx in the integral:
∫3(x-2)³+4 dx = ∫(3/cosh(y)) * (3(x-2)³+4) dy
Now, we need to substitute the expression for x in terms of y using the given substitution:
3(x-2) = sinh(y)
x - 2 = sinh(y)/3
x = sinh(y)/3 + 2
Substituting this in the integral:
∫(3/cosh(y)) * (3((sinh(y)/3 + 2) - 2)³+4) dy
Simplifying:
∫(3/cosh(y)) * (sinh(y)³+4) dy
To integrate the expression ∫(3/cosh(y)) * (sinh(y)³+4) dy, we can simplify it first:
∫(3/cosh(y)) * (sinh(y)³+4) dy = 3∫(sinh(y)³/cosh(y)) dy + 12∫(1/cosh(y)) dy
To integrate the first term, we can use the substitution u = cosh(y), which implies du = sinh(y) dy:
3∫(sinh(y)³/cosh(y)) dy = 3∫(u³/u) du = 3∫(u²) du = u³/3 + C
For the second term, we can directly integrate 1/cosh(y) using the identity sech²(y) = 1/cosh²(y):
12∫(1/cosh(y)) dy = 12∫sech²(y) dy = 12tanh(y) + D
Now, substituting back y = [tex]sinh^{(-1)}(3(x-2))[/tex]:
u = cosh(y) = cosh[tex](sinh^{(-1)}(3(x-2))[/tex]) = √(3(x-2)² + 1)
Thus, the integral becomes:
∫(3/cosh(y)) * (sinh(y)³+4) dy = (u³/3 + C) + 12tanh(y) + D
Substituting back u = √(3(x-2)² + 1):
= (√(3(x-2)² + 1)³/3 + C) + 12tanh(y) + D
= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh(y) + D
= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh[tex](sinh^{(-1)}(3(x-2)))[/tex] + D
To simplify the expression and combine constants, let's assume (√(3(x-2)² + 1)³ + 3C)/3 = A, and 12D = B.
The simplified expression becomes:
(A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B
Since [tex]sinh^{(-1)}(3(x-2))[/tex] is the inverse hyperbolic sine function, we can simplify it using the identity sinh[tex](sinh^{(-1)}(x))[/tex] = x:
(A/3) + 12tanh(3(x-2)) + B
This is the simplified form of the integral ∫(3/cosh(y)) * (sinh(y)³+4) dy after combining constants and simplifying the expression.
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i need help on this one to .
Answer:
Oh Lol i didnt even see the pic
Step-by-step explanation:
Answer:
5*2=10
10*8=80
Step-by-step explanation:
Multiply all number!!!
Javier's fuel tank holds 15 galipns completely full. He had some in the tank and added 9.6
gallons of gasoline to fill it completely.
How many gallons of gasoline were in the tank before Javier added some?
Answer:
6.4 because subtract 9.6 from 15
Find mZR
R
120°
140°
S
Need help with this question?
Answer:
[tex] m\angle R = 50 \degree[/tex]
Step-by-step explanation:
By inscribed angle theorem:
[tex]m\angle R = \frac{1}{2} [360 \degree - (120 \degree + 140 \degree)] \\ \\ m\angle R = \frac{1}{2} [360 \degree -260 \degree] \\ \\ m\angle R = \frac{1}{2} \times 100 \degree \\ \\ m\angle R = 50 \degree \\ \\ [/tex]
One of Japan's superconducting "bullet" trains is researched and tested at the Yamanashi Maglev Test Line near Otsuki City. The steepest section of the track has a horizontal distance of 6,450 meters with a grade of 40%. a a. What would be the elevation change in this section? b. What is the actual distance of the track in this section? Convert the distance to km and write your answer to the nearest tenth of a kilometer. 3. Which plane is closer to the tower? Explain
Japan's superconducting bullet that is being tested in Yamanashi Maglev Test Line will have an elevation of 2580 meters and actual distance of the track as 6.9 kilometers.
A. To calculate the elevation change in the steepest section of the track:
Grade = 40% (Given)
Horizontal distance = 6450 meters (Given)
Elevation change = Grade × Horizontal distance
= 40% × 6,450 meters
= 0.40 × 6,450 meters
= 2,580 meters
Therefore, the elevation change in this section of the track will be 2,580 meters.
B. To find the actual distance of the track in this section:
By using Pythagorean theorem, the horizontal distance represents the base of a right triangle, and the elevation change represents the height.
Actual distance of the track = √(Horizontal distance² + Elevation change²)
= √(6,450² + 2,580² )
= √(41,602,500 + 6,656,400)
= √48,258,900
= 6,945 meters
= 6.9 kilometers
Therefore, the actual distance of the track in this section will be 6.9 kilometers.
C. To determine which plane is closer to the tower:
Plane A: Altitude = 20,000 ft, Distance from tower = 5 km
Plane B: Altitude = 8,000 ft, Distance from tower = 7 km
1 ft is approximately equal to 0.0003048 km.
Altitude of Plane A in km = 20,000 ft × 0.0003048 km/ft ≈ 6.096 km
Altitude of Plane B in km = 8,000 ft × 0.0003048 km/ft ≈ 2.4384 km
On comparing the distances, we find that Plane A is closer to the tower than Plane B.
Therefore, Plane A is closer to the tower as compare to Plane B.
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Find the area of the trapezoid. Leave your answer in simplest radical form.
5 cm
Not drawn to scale
A.
94.5 cm
B.
31.5 cm
c.
7 cm
D.
81 cm
Answer:
A)94.5 cm
Step-by-step explanation:
height = 9cm
Base 1 = 5cm
Area of a Trapezoid = 1/2 × (b1 + b2)h
h = 9cm
b1 = 5cm
b2 = (9 cm + 5cm + 2cm) = 16cm.
Area of Trapezoid
= 1/2 (5 + 16) × 9
= 1/2 × 21 × 9
= 94.5 cm
Option A is the correct answer
Given the points A(-4,-1) B(-2,-5) C(0,1) D(2,-3)
What is the most precise name of this shape?
Trapezoid
---
hope it helps
sorry if i'm wrong
i used mental maths
Dude That's a square not a trapezoid
Populations of aphids and ladybugs are modeled by the equations dA = 2A 0.01AL dt dL = -0.5L + 0.0001AL. dt (a) Find an expression for dL/dA. dL dA 0.5L + 0.0001AL 2A – 0.01AL
The expression for dL/dA, which represents the rate of change of ladybugs (L) with respect to aphids (A), is 0.5L + 0.0001AL - 2A + 0.01AL.
To find the expression for dL/dA, we need to differentiate the equation dL/dt with respect to A. The given equations are:
dA/dt = 2A - 0.01AL
dL/dt = -0.5L + 0.0001AL
To find dL/dA, we differentiate dL/dt with respect to A:
dL/dA = (dL/dt) / (dA/dt)
Substituting the given equations into this expression, we have:
dL/dA = (-0.5L + 0.0001AL) / (2A - 0.01AL)
Simplifying further, we can rearrange the terms:
dL/dA = -0.5L / (2A - 0.01AL) + 0.0001AL / (2A - 0.01AL)
Combining the terms with a common denominator, we get:
dL/dA = (0.0001AL - 0.5L) / (2A - 0.01AL)
So, the expression for dL/dA is (0.0001AL - 0.5L) / (2A - 0.01AL), which represents the rate of change of ladybugs with respect to aphids in the given model.
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The manager of the City of Industry Electronics store is concerned that his supplier has been giving him TV sets with lower than average quality. His research shows that replacement times for TV sets have a mean of 7.5 years and a standard deviation of 5 years. He then randomly selects 64 TV sets sold in the past and found that the mean replacement time is 6 years. Determine the probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or less. Find the z score (round to two decimals) QUESTIONS 2b. What do you get from Table A? QUESTION 6 20. Determine the probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or loss. (round to a percent with two decimals)
The probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or less is approximately 0.0048, or 0.48%.
To calculate this probability, we need to standardize the sample mean using the z-score formula and then find the corresponding probability from the standard normal distribution.
The formula for the z-score is:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, x = 6, μ = 7.5, σ = 5, and n = 64. Substituting these values into the formula, we get:
z = (6 - 7.5) / (5 / √64)
Simplifying the expression:
z = -1.5 / (5 / 8)
z = -1.5 * 8 / 5
z = -2.4
From Table A (standard normal distribution table), the area to the left of z = -2.4 is approximately 0.0082.
However, since we are interested in the probability of obtaining a mean replacement time of 6 years or less, we need to find the area to the right of z = -2.4. This is given by:
1 - 0.0082 = 0.9918
Therefore, the probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or less is approximately 0.0048, or 0.48%.
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Find the absolute maximum and minimum of f (x, y) = x^2 + 2y^2 − 2x − 4y +1 on D = {(x, y) 0 ≤ x ≤ 2, 0 ≤ y ≤ 3} .
Absolute maximum of f (x, y) = 19 and Absolute minimum of f (x, y) = −3.
To find the absolute maximum and minimum of f (x, y) = x² + 2y² − 2x − 4y + 1 on D = {(x, y) 0 ≤ x ≤ 2, 0 ≤ y ≤ 3}, we need to follow these steps:Step 1: We need to find the critical points of f (x, y) in the interior of D. Step 2: We then need to evaluate f (x, y) at the critical points. Step 3: We need to find the maximum and minimum of f (x, y) on the boundary of D. Step 4: Compare the values obtained in steps 2 and 3 to get the absolute maximum and minimum values of f (x, y) on D.1. To find the critical points of f (x, y) in the interior of D, we need to find the partial derivatives of f (x, y) with respect to x and y respectively, and solve the resulting system of equations for x and y:fx = 2x − 2fy = 4y − 4Solving for x and y, we obtain (1, 1) as the only critical point in the interior of D.2. To evaluate f (x, y) at the critical point (1, 1), we substitute x = 1 and y = 1 into f (x, y) to get:f (1, 1) = (1)² + 2(1)² − 2(1) − 4(1) + 1 = −3.3. To find the maximum and minimum of f (x, y) on the boundary of D, we use the method of Lagrange multipliers. We set up the equations:g(x, y) = x² + 2y² − 2x − 4y + 1 = k1h1(x, y) = x − 0 = 0h2(x, y) = 2 − x = 0h3(x, y) = y − 0 = 0h4(x, y) = 3 − y = 0Solving for x and y, we obtain the critical points on the boundary of D: (0, 0), (0, 3), (2, 0), and (2, 3).4. Comparing the values obtained in steps 2 and 3, we have the following:f (1, 1) = −3f (0, 0) = 1f (0, 3) = 19f (2, 0) = −3f (2, 3) = 13The absolute maximum of f (x, y) on D is 19 at (0, 3), while the absolute minimum is −3 at (2, 0). Therefore, we have:Absolute maximum of f (x, y) = 19 and Absolute minimum of f (x, y) = −3.
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A store owner buys a case of 144 pens for $28.80. He sells the pens for $0.40 each. The owner claims that they marked the pens up by 50% before selling them. Prove that the owner calculated their markup correctly. If they did not, how much of a markup actually occurred?
Answer: $0.3
Step-by-step explanation:
Given
The owner buys 144 pens for $28.80 i.e. each pen costs
[tex]\dfrac{28.80}{144}=\$0.2[/tex]
owner sells the pen at $0.4 i.e. price marked up by
[tex]\Rightarrow \dfrac{0.4-0.2}{0.2}\times 100=100\%[/tex]
So, the claim of the owner is incorrect
The actual increase in the price to get 50% markup
[tex]\Rightarrow 0.2\times (1+0.5)=\$0.3[/tex]