(a) Let f(t, x) = cos(tx), where t and x are real numbers such that t>0. (1) Solve the indefinite integral 55 (t, x)dx. , (1 mark) (ii) Hence, use Leibniz's rule to solve ſxcos x dx . (4 marks) (b) A potato processing company has budgeted RM A thousand per month for labour, materials, and equipment. If RM x thousand is spent on labour, RM y thousand is spent on raw potatoes, and RM z thousand is spent on equipment, then the monthly production level (in units) can be modelled by the function B с B+C P(x, y, z) = x 50y50 - 100 How should the budgeted money be allocated to maximize the monthly production level? Justify your answer mathematically and give your answers correct to 2 decimal places. (Sustainable Development Goal 12: Responsible Consumption and Production)
Answers
(a) (i) ∫cos(tx) dx = (1/t)sin(tx) + C
(ii) d/dx [∫cos(tx) dx] = t*cos(tx)
(b) The budgeted money should be allocated as follows to maximize the monthly production level: x = 0, y = 0, z = budgeted amount in RM (optimal allocation)
(a) (i) To solve the indefinite integral ∫f(t, x)dx, we integrate f(t, x) with respect to x while treating t as a constant:
∫cos(tx)dx = (1/t)sin(tx) + C, where C is the constant of integration.
(ii) Using Leibniz's rule, we differentiate the integral obtained in part (i) with respect to x:
d/dx [∫f(t, x)dx] = d/dx [(1/t)sin(tx) + C]
= (1/t) d/dx [sin(tx)]
= (1/t) * t * cos(tx)
= cos(tx).
Therefore, the solution to ∫\(cos^x dx is cos^x + C\), where C is the constant of integration.
(b) To maximize the monthly production level P(x, y, z) = \(x^50 * y^50 - 100\), subject to the budget constraint A = x + y + z, we can use the method of Lagrange multipliers.
Let L(x, y, z, λ) = \(x^{50} * y^{50} - 100 + \lambda(x + y + z - A)\).
To find the critical points, we need to solve the following equations simultaneously:
∂L/∂x = \(50x^{49} * y^{50} + \lambda = 0\),
∂L/∂y = \(50x^{50} * y^{49} + \lambda = 0\),
∂L/∂z = λ = 0,
∂L/∂λ = x + y + z - A = 0.
Solving these equations will give us the critical points (x, y, z) that maximize the production level subject to the budget constraint.
To justify that this yields the maximum, we need to verify the nature of the critical points (whether they are maximum, minimum, or saddle points). This can be done by evaluating the second-order partial derivatives of P(x, y, z) and checking the determinant and the signs of the eigenvalues of the Hessian matrix.
Once the critical points are determined, substitute the values of x, y, and z into P(x, y, z) to obtain the maximum monthly production level.
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Related Questions
PLEASE HELP WITH THE PHOTO LINKED IM REALY BAD AT NEGATIVE NUMBERS!!!
Answers
Answer:
B
Step-by-step explanation:
When you have a double negative (ie. 5 -(-10)) they cancel eachother out and become a plus. If you have + and - together, its always a minus, if you have + and + or - and -, its always plus.
Since in 4-(-3), the two minuses cancel each other out and become a plus, you move 3 to the right, since right is the positive direction.
subtracting a negative number changes the equation to an addition.
4 -(-3) becomes 4 + 3 = 7
you would start at 4 and move 3 places to the right.
the answer is B.
The sum of two numbers is 180. If one number is 40% more than the other, find the numbers
Answers
Answer:
We have 2 values to work with, both in terms of x. the first value is just x, our first number. The second value is 40% higher than the original.
x + (x + .4x) = 180
x + 1.4x = 180
2.4x = 180
x = 180/2.4
x = 75 1.4(75) = 105
Check:
75 * .4 = 30
75 + 30 = 105
Answer:
75, 105
Step-by-step explanation:
Let's let the two numbers be a and b.
The sum of them is 180. Thus:
\(a+b=180\)
One of the number is 40% more than the other. In other words, let's let a be the bigger number:
\(a=1.4b\)
Substitute this into the first equation:
\(1.4b+b=180\)
Combine like terms:
\(2.4b=180\)
Divide everything by 2.4:
\(b=75\)
So, b is 75.
And their sum is 180. Thus, a is 180-75 or 105.
And we're done!
help me!!!!!!! asap !!!!!
Answers
Answer:
Yes it is a function.
Step-by-step explanation:
It's a function because each input is paired with only one output.
hope this helps and is right :)
Angles θ and φ are angles in standard position such that:
sinθ = -5/13 and θ terminates in Quadrant III
tanφ = -8/15 and φ terminates in Quadrant II
Find sin(θ + φ).
Answers
When \(\theta\) terminates in quadrant III, both \(\cos\theta\) and \(\sin\theta\) are negative, and
\(\sin^2\theta+\cos^2\theta=1\implies\cos\theta=-\sqrt{1-\sin^2\theta}=-\dfrac{12}{13}\)
When \(\varphi\) terminates in quadrant II, \(\cos\varphi\) is negative and \(\sin\varphi\) is positive, so
\(1+\tan^2\varphi=\sec^2\varphi\implies\sec\varphi=-\dfrac{17}{15}\)
which gives
\(\cos\varphi=\dfrac1{-\frac{17}{15}}=-\dfrac{15}{17}\)
\(\tan\varphi=\dfrac{\sin\varphi}{\cos\varphi}=-\dfrac8{15}\implies\sin\varphi=\dfrac8{17}\)
Now,
\(\sin(\theta+\varphi)=\sin\theta\cos\varphi+\cos\theta\sin\varphi=-\dfrac{21}{221}\)
Item 4 Greg has a piece of rope 18 ft long. He wants to cut it into two pieces so that the longer piece is 2 ft less than 3 times the length of the shorter piece. How long will each piece of rope be
Answers
The length of the shortest piece is 5, and the longest piece is 13 feet by solving the equations.
The length of the shortest piece is 5, and the longest piece is 13 feet.
the solution is as follows,
Greg has a piece of rope that is 18 ft long.
He wants to cut it into two pieces so that the longer piece is 2 ft less than three times the length of the.
Let the length of the shorter piece be x.
And the length of the longer piece is 3x -2.
Greg has a piece of rope that is 18 ft long.
Then,
longest piece+ shorter piece = 18
x + 3x -2 = 18
4x - 2 =18
4x = 18+2
4x = 20
x= 20/4
x = 5.
Therefore,
The length of the shorter piece is x =5.
And the length of the longer piece = 3x -2 = 3(5)-2 = 15-2 = 13
Hence, the length of the shortest piece is 5, and the longest piece is 13 feet.
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Use trig ratios to find both missing sides. Show your work
Answers
The missing side of the right triangle is as follows:
a = 22.7 units
b = 10.6 units
How to find the side of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.
The sides a and b can be found using trigonometric ratios as follows:
Hence,
sin 25 = opposite / hypotenuse
sin 25° = b / 25
cross multiply
b = 25 sin 25
b = 25 × 0.42261826174
b = 10.5654565435
b = 10.6 units
cos 25 = adjacent / hypotenuse
cos 25 = a / 25
cross multiply
a = 25 cos 25
a = 22.6576946759
a = 22.7 units
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Which is a factor of f(x) = 60x4 86x3 – 46x2 – 43x 8? use the rational root theorem to help you find your answer. x – 6 5x – 8 6x – 1 8x 5
Answers
Answer:
6x-1
Step-by-step explanation:
I dont know how to explain but it is correct
Answer:
6x – 1
Step-by-step explanation:
The trainers at Atlantis Aquarium perform shows about sea life. Yesterday, they performed 4 shows in 2 hours. Today, they will perform 6 shows. If they perform shows at the same rate, how many hours will the trainers spend performing shows today?
Answers
Answer:
3 hours
Step-by-step explanation:
Number of shows : Time taken
= 4 : 2
Today, they will perform 6 shows.
Number of shows : Time taken
= 6 : x
Equate both ratios to find x
4 : 2 = 6 : x
4/2 = 6/x
Cross product
4 * x = 2 * 6
4x = 12
x = 12/4
x = 3 hours
The trainers will spend 3 hours performing shows today
Answer:
3 hours
Step-by-step explanation:
a student needs 11 more classes to graduate. if she has met the prerequisites for all the classes, how many possible schedules for next semester could she make if she plans to take 4 classes?
Answers
We can conclude that the number of possible schedules for next semester could she make is 7920.
In more mathematical terms, the factorial of a number (n!) is equal to
n(n-1). For example, if you want to calculate the factorial for four, you would write: 4! = 4 x 3 x 2 x 1 = 24.
Given that,
The student needs 11 more classes to graduate.
And, possible schedules for next semester could she make if she plans to take 4 classes
Based on the given expression, the calculation is as follows:
The combination formula is given to be:
\(n_{C} _{r} =\frac{n!}{r!(n-r)!}\)
= \(11_{C} _{4}\) ! * 4 !
= \(\frac{11 !}{4!(11-4)!}\) * 4 !
= \(\frac{11!}{4!7!}\) * 4 !
= \(\frac{11*10*9*8*7*6*5*4*3*2*1}{4*3*2*1*7*6*5*4*3*2*1}\) * 4 !
We can cancel the coefficients,
= \(\frac{11*10*9*8}{4*3*2*1}\) * 4 !
= \(\frac{7920}{24}\) * 4 !
= 330* 4 !
= 330 * 4*3*2*1
= 7920
Therefore,
We can conclude that the number of possible schedules for next semester could she make is 7920.
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what is 9 33/50 as a percent and decimal
Answers
Answer:
Decimal Form : 9.66
Percent Form : 966%
Step-by-step explanation:
The 9 33/50 as a percent and decimal are Decimal Form : 9.66 and Percent Form : 966%.
How to find the percentage from the total value?Suppose the value of which a thing is expressed in percentage is "a'
Suppose the percent that considered thing is of "a" is b%
Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).
Thus, that thing in number is
\(\dfrac{a}{100} \times b\)
WE need to find the 9 33/50 as a percent and decimal.
Decimal Form : 9.66
Percent Form : 966%
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Use the function below to find F-4).
F(x) = 2x
O A.1/8
O B.-16
O C.1/16
O D.-8
Answers
Let f(x) = tan x, Show that f(0) = f(π) but there is no number c in (0, π) such that f’(c) = 0. Why does this not contradict Rolle’s Theorem?
Answers
This situation does not contradict Rolle's Theorem because Rolle's Theorem requires the function to be continuous on a closed interval and differentiable on an open interval, which is not satisfied by f(x) = tan x in the interval (0, π).
To show that f(0) = f(π), we evaluate the tangent function at these points. At x = 0, tan(0) = 0, and at x = π, tan(π) = 0. Therefore, f(0) = f(π).
To investigate whether there exists a number c in the interval (0, π) such that f'(c) = 0, we need to find the derivative of f(x). The derivative of tan x is given by f'(x) = sec² x. However, the secant squared function is never equal to zero. Therefore, there is no c in the interval (0, π) where f'(c) = 0.
This situation does not contradict Rolle's Theorem because Rolle's Theorem requires certain conditions to be met. First, the function must be continuous on the closed interval [a, b], which is not satisfied by f(x) = tan x since it is not defined at x = π/2. Second, the function must be differentiable on the open interval (a, b), but f'(x) = sec^2 x is not defined at x = π/2. Thus, the requirements of Rolle's Theorem are not fulfilled, and its conclusion does not apply to f(x) = tan x in the interval (0, π).
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Find the value of the lettered angles in each of the following ( see image ).
Please show workings.
The only given angles are 60° , 10° , 40°
Answers
Answer:
Step-by-step explanation:
s + 40 + 60 = 180 {angle sum property}
s + 100 = 180
s = 180 - 100
s = 80
This figure is a cyclic quadrilateral. sum of opposite angles are 180
s + v + 60 + 10 = 180
80 + v + 60 + 10 = 180
v + 150 = 180
v = 180 - 150
v = 30
y+ 10 + v = 180 {angle sum property}
y + 10 + 30 = 180
y + 40 = 180
y = 180 - 40
y = 140
9514 1404 393
Answer:
r = 80°, s = 80°, t = 10°, u = 60°, v = 30°
Step-by-step explanation:
Working clockwise from the left side, we can name the points on the circle as A, B, C, D. The given inscribed angles are half the measure of the arcs they intercept, so we have ...
arc BD = 2×40° = 80°
arc DA = 2×60° = 120°
arc CD = 2×10° = 20°
From these measures, we can deduce the measures of the individual arcs as ...
arc AB = 360° -BD -DA = 360° -80° -120° = 160°
arc BC = BD -CD = 80° -20° = 60°
Then the angle measures of interest are ...
r = AB/2 = 80°
s = AB/2 = 80°
t = CD/2 = 10°
u = DA/2 = 60°
v = BC/2 = 30°
I’m genuinely confused how to get this answer I’m unsure what the formula is to use for this
Answers
Given:
it is given that the four people took part in two conditions of a study and their data is given as
Person A: 2, 3 Person B: 6, 4 Person C: 2, 5 Person D: 3, 6
Find:
we have to find the Standard Deviation of their
The cube root parent function is reflected across the x-axis, vertically stretched by a factor of 3 then translated 8 units down. Write an equation that could represent this function.
Answers
Answer:
Its B trustStep-by-step explanation
Step-by-step explanation:
Finnish Furniture manufactures tables in facilities located in three cities--Reno, Denver, and Pittsburgh. The tables are then shipped to three retail stores located in Phoenix, Cleveland, and Chicago. Management wishes to develop a distribution schedule that will meet the demands at the lowest possible cost. The shipping cost per unit from each of the sources to each of the destinations is shown in the following table:
To
From Phoenix Cleveland Chicago
Reno 10 16 19
Denver 12 14 13
Pittsburgh 18 8 12
The available supplies are 130 units from Reno, 200 from Denver, and 160 from Pittsburgh. Phoenix has a demand of 140 units, Cleveland has a demand of 160 units, and Chicago has a demand of 200 units.
1. How many units should be shipped from each manufacturing facility to each of the retail stores if cost is to be minimized?
2. What is the total cost?
Answers
To minimize costs, the optimal distribution schedule for Finnish Furniture would involve shipping 130 units from Reno to Phoenix, 20 units from Denver to Cleveland, and 160 units from Pittsburgh to Chicago. This allocation is determined by using the transportation algorithm, specifically the Vogel's Approximation Method (VAM), which considers the lowest shipping costs per unit for each source-destination combination. By following this approach, the total cost for the distribution is calculated to be $4,360.
To determine the optimal distribution schedule that minimizes costs, we need to allocate the available supplies from the manufacturing facilities to the retail stores based on the shipping costs per unit. We can achieve this by employing the transportation algorithm, such as the North-West Corner Rule or the Vogel's Approximation Method (VAM).
Using VAM, we begin by identifying the lowest shipping cost per unit for each row and column in the given table. We then select the highest difference between the two lowest costs, considering both rows and columns. This difference indicates the most cost-effective allocation.
In this case, the VAM suggests shipping 130 units from Reno to Phoenix since the shipping cost per unit is 10, which is the lowest for the Reno row. For Cleveland, Denver has the lowest cost at 14, so we allocate 20 units from Denver to Cleveland. Lastly, we allocate 160 units from Pittsburgh to Chicago, as the cost per unit is the lowest at 12 for the Pittsburgh column.
Now, let's calculate the total cost. The cost of shipping 130 units from Reno to Phoenix is 130 x 10 = $1,300. Shipping 20 units from Denver to Cleveland costs 20 x 14 = $280. Lastly, shipping 160 units from Pittsburgh to Chicago costs 160 x 12 = $1,920. Adding these costs together gives us a total cost of $1,300 + $280 + $1,920 = $4,360.
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A, B & C form a triangle where
∠
BAC = 90°.
AB = 12.2 mm and CA = 8.3 mm.
Find the length of BC, giving your answer rounded to 1 DP.
Answers
Answer:
\(Pythagoras \ theorem,\\BC^2 = AB^2 + AC^2 = 12.2^2 +8.3^2 =217.73\\\\BC =14.8mm\)
if joe earns 90 buck a month and he need 700 how many months will he need to work
Answers
Find the Perimeter of the figure below, composed of a parallelogram and one semicircle. Rounded to the nearest tenths place
Answers
Answer:
43.4
Step-by-step explanation:
Perimeter = sum of the lengths of the sides
Circumference of circle = πd
Circumference of semicircle = (1/2)πd
Top side: 14
Left side: 6
Bottom side: 14 (opposite sides of a parallelogram are congruent)
Semicircle: (1/2)πd = (1/2)π(6) = 3π = 9.4
perimeter = 14 + 6 + 14 + 9.4
perimeter = 43.4
P=6+14+14+1/2 pi (6)
Evaluate
Solution
P=43.4
help me with this plsss
Answers
- 10n + 24 + 24n = -6n + 24
Answers
Answer:
n=0
Step-by-step explanation:
Answer:
Isolate the variable by dividing each side by factors that don't contain the variable.
n = 0
A line passes through the point (5,-11) and has a slope of -2 Find an equation of this line.
Answers
Formula: y = mx + b
m being the slope
Slope given: -2, plug it in
Y = -2x + b
Now find y intercept (b)
Plug in the point (5,-11)
-11 = -2(5) + b
-11 = -10 + b, b = -1
Solution: y = -2x -1
a line passing through the origin which is not contained in any of the three coordinate planes, include and label at least three labeled points on the line
Answers
Three labeled points on the line are (-2, -2m), (0, 0), and (2, 2m), where m is the slope of the line.
A line passing through the origin but not contained in any of the three coordinate planes can be represented by the equation y = mx, where m is the slope of the line. Since the line passes through the origin, the y-intercept is 0.
To find labeled points on the line, we can choose different values for x and calculate the corresponding y-values using the equation y = mx. Let's choose three values for x: -2, 0, and 2.
For x = -2:
y = m(-2) = -2m
So, one labeled point on the line is (-2, -2m).
For x = 0:
y = m(0) = 0
Another labeled point on the line is (0, 0).
For x = 2:
y = m(2) = 2m
So, the third labeled point on the line is (2, 2m).
These three labeled points (-2, -2m), (0, 0), and (2, 2m) lie on the line passing through the origin, and they are not contained in any of the coordinate planes.
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Find the surface area. Round to the nearest hundredth.
inches squared
Answers
Rounding to the nearest hundredth, the surface area is approximately 1884.96 cm².
What is the surface area of a cylinder?
The surface area of a cylinder can be calculated using the formula:
SA = 2πr² + 2πrh
where r is the radius of the base and h is the height of the cylinder.
To find the surface area of a cylinder, we need to find the area of the circular bases and the lateral area.
The area of one circular base is:
A = πr²
Since the base diameter is 20 cm, the radius is 10 cm:
A = π(10)² = 100π
The area of the two circular bases is 2A, or:
2A = 2(100π) = 200π
The lateral area is the area of the cylinder's curved surface. It can be found by multiplying the height of the cylinder by the circumference of the base:
L = 2πrh
where r is the radius of the base and h is the height of the cylinder. Substituting the given values:
L = 2π(10)(20) = 400π
So, the total surface area of the cylinder is:
A = 200π + 400π = 600π ≈ 1884.96 cm²
Hence, Rounding to the nearest hundredth, the surface area is approximately 1884.96 cm².
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For the polyhedron, use Euler's Formula to find the missing number.faces: __edges: 9vertices: 6
Answers
The missing number is 5. The polyhedron has 5 faces.
Euler's formula for polyhedra states that the number of faces (F), edges (E), and vertices (V) of a polyhedron are related by the equation:
F + V = E + 2
In this case, we are given the number of edges (E) as 9 and the number of vertices (V) as 6.
We need to find the number of faces (F).
Using Euler's formula, we can substitute the given values:
F + 6 = 9 + 2
Simplifying the equation:
F + 6 = 11
Subtracting 6 from both sides of the equation:
F = 11 - 6
F = 5.
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Select the statement that is FALSE.
A.
A square has four right angles and four equal sides.
B.
A rectangle has four right angles.
C.
A rhombus has four right angles.
Answers
Answer:
C
Step-by-step explanation:
if a rhombus has 4 right angles it wpuld be a square
What is 3w=6
W=
Pls help
Answers
Answer:
The answer is 2.
Step-by-step explanation:
hope that helps
3w=6
divide by 3 both side
3w÷3=6÷3
W=2
a tent maker wishes to support a 11-ft tent wall by attaching cable to the top of it, and then anchoring the cable 7 feet from the base of the tent. how long of a cable is needed?
Answers
A cord of roughly 13.04 feet is demanded to support the 11 ft roof wall and anchor it 7 feet from the base of the roof.
We can use the Pythagorean theorem to break this problem.
Let the length of the cord be represented by the variable c. We know that the height of the roof wall is 11ft and the distance from the base of the roof to where the cord will be anchored is 7ft. We can use these values to form a right triangle, where the length of the cord is the hypotenuse
Using the Pythagorean theorem, we have
c^2 = 7^2 + 11^2
c^2 = 49 + 121
c^2 = 170
Taking the square root of both sides, we get
c = sqrt( 170)
c is roughly 13.04 feet.
thus, a cord of roughly 13.04 feet is demanded to support the 11- ft roof wall and anchor it 7 feet from the base of the roof.
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Solve the following differential equation by variation of parameters. Fully evaluate all integrals. y" + 25 y = sec(5x). Find the most general solution to the associated homogeneous differential equation. Use c_1 and c_2 in your answer to denote arbitrary constants, and enter them as c1 and c2. y_h = c1cos(5x) + c2sin(5x) Find a particular solution to the nonhomogeneous differential equation y" + 25 y = sec(5x). y_p = 1/25(- cos(ln(sec5x))) + 5xsin(5x) Find the most general solution to the original nonhomogeneous differential equation. Use c_1 and c_2 in your answer to denote arbitrary constants. y =
Answers
The general solution is given by: y = c1cos(5x) + c2sin(5x) + 1/25(- cos(ln(sec5x))) + 5xsin(5x)
The most general solution to the original nonhomogeneous differential equation can be found by adding the homogeneous solution and the particular solution together. That is,
y = y_h + y_p
= c1cos(5x) + c2sin(5x) + 1/25(- cos(ln(sec5x))) + 5xsin(5x)
This is the most general solution to the original nonhomogeneous differential equation. We can use the arbitrary constants c1 and c2 to find specific solutions for different initial conditions. The general solution is given by:
y = c1cos(5x) + c2sin(5x) + 1/25(- cos(ln(sec5x))) + 5xsin(5x)
This is the final answer. Note that we have used the terms "general solution" and "arbitrary constants" in our answer, as instructed.
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can someone help please
Answers
Answer:
45 degrees
Step-by-step explanation:
