(i) The expression for the profit is -p² + 14p - 33
(ii) The price per muffin that maximizes profit is $7.
What is the expression for profit in terms of p?(i) The expression for profit in terms of p can be calculated by subtracting the cost from the revenue. The revenue is obtained by multiplying the price per muffin (p) by the number of muffins sold (q):
Revenue = p * q
The cost per muffin is given as $3. Therefore, the profit (P) can be expressed as:
P = Revenue - Cost
P = (p * q) - (3 * q)
Since q = 11 - p, we can substitute this expression into the profit equation:
P = (p * (11 - p)) - (3 * (11 - p))
Simplifying further, we have:
P = 11p - p² - 33 + 3p
P = -p² + 14p - 33
(ii) To find the price that maximizes profit, we need to determine the value of p that corresponds to the maximum point of the profit function. In this case, the profit function is a quadratic equation.
To find the maximum point, we can calculate the vertex of the quadratic function using the formula:
p = -b / (2a)
In the quadratic equation P = -p² + 14p - 33, we can identify that a = -1, b = 14, and c = -33.
Using the vertex formula, we can find:
p = -14 / (2*(-1))
p = 7
Therefore, the price per muffin that maximizes profit is $7.
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2. Determine the points in C for which the following functions are holomorphic: (a) f(z) = z² (b) g(z) = x² - y² + 2xy (where z = x + iy)
There are no points in C for which the function g(z) is holomorphic.
The functions given are :
f(z) = z² and g(z) = x² - y² + 2xy (where z = x + iy)
We need to determine the points in C for which the functions are holomorphic.
(a) To check whether f(z) = z² is holomorphic or not, we will verify the Cauchy-Riemann equations (CRE) which are:
u x = v y and v x = - u y
Let us assume that f(z) = u(x, y) + iv(x, y)
Substituting in f(z) = z², we have f(z) = (x + iy)²= x² + 2ixy - y²
Now comparing with u(x, y) + iv(x, y), we get :
u(x, y) = x² - y² and v(x, y) = 2xy
Now applying the CRE, we get :
u x = 2xv
y = 2xu
y = - 2yv
x = 2y
We can see that both the CRE are satisfied.
Hence, f(z) = z² is holomorphic for all values of z in C.
(b) Similarly, for g(z) = x² - y² + 2xy (where z = x + iy), we have g(z) = u(x, y) + iv(x, y)
Substituting in g(z) = x² - y² + 2xy, we have g(z) = x² - y² + 2ixy
Now comparing with u(x, y) + iv(x, y), we get :
u(x, y) = x² - y² and v(x, y) = 2xy
Now applying the CRE, we get :
u x = 2xv
y = 2xu
y = - 2yv
x = 2x
Since the CRE are not satisfied, g(z) = x² - y² + 2xy (where z = x + iy) is not holomorphic at any point in C.
Therefore, we can say that there are no points in C for which the function g(z) is holomorphic.
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evalute sin 60. cos 30 sin +sin 30 .cos 60 what is the value of sin(30-60) what can you conclude
Answer:
See explanations below
Step-by-step explanation:
evaluate sin 60. cos 30 sin +sin 30 .cos 60 what is the value of sin(30-60) what can you conclude
According to the trigonometry identity
Sin 30 = 1/2
Sin 60 = √3/2
Cos 30 = √3/2
Cos 60 = 1/2
sin 60. cos 30 +sin 30 .cos 60
= √3/2(√3/2) + 1/2(1/2)
= √9/4 + 1/4
= 3/4 + 1/4
= 4/4
= 1
sin(30-60) = sin30cos60 - cos30sin60
sin(30-60) =1/2(1/2) - √3/2(√3/2)
sin(30-60) = 1/4 - √9/4
sin(30-60) = 1/4 - 3/4
sin(30-60) = (1-3)/4
sin(30-60) = -2/4
sin(30-60) = -1/2
hence the former fraction gives a positive values while the later gives a negative
simplify this number 300mm:9m
Answer:
1 : 30
Step-by-step explanation:
300mm:9m
We need to change the meters to mm
1 meter is 1000 mm
so 9 m is 9000 mm
300mm:9000mm
Divide both sides by 300
300mm/300 : 9000/300
1 : 30
Please help!! I'm super confused:(
Answer: Singing
Mode: The value that appears the most in a list
Also by the way, the frequency table is most likely just a table of every single value and how many times they appear in the list. Its super easy all you have to do is count them
Has anyone done the Alg1B Portfolio - Unit 6 for connections academy
Answer:
I have
Step-by-step explanation:
please help :c
Angela bought a total of 3 dozen cookies for Easter. Each cookie was 0.65 (INCLUDING TAX!) Of the cookies, 3/4 of them were shaped like Easter eggs. What was the cost of the Easter egg shaped cookies?
please add the work along with the question
Answer: $17.55
Step-by-step explanation:
36 x 0.65 = $23.40 total for all cookies
($23.40/1) x (3/4) = (70.2/4)
70.2 divided by 4 = $17.55
or
$23.40 x .75 = $17.55
Max is mixing oil and gas for his moped. He uses 3.75 liters of gas and 1.5 liters of oil. How many liters of gas are used per liter of oil?
Answer:
2.5 liters of gas is used with per liter of oil
Step-by-step explanation:
Max is missing oil and gas for his moped.
Amount of gas used = 3.75 liters
Amount of oil required = 1.5 liters
Ratio in which gas and oil are mixed = [tex]\frac{3.75}{1.5}[/tex]
= [tex]\frac{2.5}{1}[/tex]
That means if he uses 1 liter of oil then the gas required for the mixture = 2.5 liters
Therefore, 2.5 liters of gas is used per liter of oil.
PLZ HELP ME, ASAP I WILL CROWN BRAINLIEST!!!!
Which sequences are geometric? Check all that apply. –2, –4, –6, –8, –10, … 16, –8, 4, –2, 1 –15, –18, –21.6, –25.92, –31.104, … 4, 10.5, 17, 23.5, 30, … 625, 125, 25, 5, 1, …
Answer:
16,-8,4,-2,1
-15,-18,-21.6,25.92,-31.104...
625,125,25,5,1
Step-by-step explanation:
Answer: 16, –8, 4, –2, 1. has a common ratio,r = (-1/2)
-15, –18, –21.6, –25.92, –31.104 has a common ratio, r = (1.2)
625, 125, 25, 5, 1 has a common ratio, r= (1/5)
Step-by-step explanation: just took the test
It is assumed that the average Triglycerides level in a healthy person is 130 unit. In a sample of 20 patients, the sample mean of Triglycerides level is 122 and the sample standard deviation is 20. Calculate the test statistic value.
The test statistic value can be calculated using the formula (sample mean - population mean) / (sample standard deviation / sqrt(sample size)).
To calculate the test statistic value, we use the formula (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). In this case, the sample mean is 122, the population mean is 130, the sample standard deviation is 20, and the sample size is 20.
Test Statistic Value = (122 - 130) / (20 / sqrt(20))
= (-8) / (20 / 4.47)
= -8 / 4.47
≈ -1.79
Therefore, the test statistic value is approximately -1.79.
The test statistic value (t) measures the difference between the sample mean and the assumed population mean in terms of the standard error. It helps us determine the likelihood of obtaining such a sample mean if the population mean is indeed equal to the assumed value.
In this case, the test statistic value is -1.79, indicating that the sample mean is 1.79 standard errors below the assumed population mean of 130. The negative sign indicates that the sample mean is lower than the assumed value.
To determine the significance of this difference, we would compare the test statistic to critical values from the t-distribution or calculate the p-value associated with the observed test statistic. This would allow us to make conclusions about the statistical significance of the difference between the sample mean and the assumed population mean.
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Let C denotes any closed contour lying in the open disk |z| < 3. Consider the function f(z) : = (8²-16)5* Calculate the contour integral of the function f(z) over the contour C. 2622
The contour integral of the function f(z) over the contour C is zero because the function f(z) is analytic inside and on the contour C.
How to determine contour integral?In this case, the function f(z) = (8² - 16)5 = 64 × 5 = 320 is a constant function. Constant functions are always analytic within their domain. Therefore, f(z) is analytic within the region enclosed by the contour C.
According to Cauchy's Integral Formula, the contour integral of a function over a closed contour C is given by:
∮C f(z) dz = 2πi × sum of the residues of f(z) at its isolated singularities within C.
Since f(z) is a constant function, it does not have any singularities. Therefore, all the residues of f(z) are zero.
Hence, the contour integral of f(z) over the contour C is zero:
∮C f(z) dz = 0.
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Part A:
A group of students are going on an overnight camping trip. One tent holds 7 students, and 4 tents hold 28 students. Determine how many students fit in six, nine, and ten tents.
Part B:
Create a table of values to represent the relationship between the number of tents and the number of students.
Part C:
Write the equation to represent the relationship between the number of tents and the number of students.
Part A:
Six Tents Holds 42 People
Nine Tents Holds 63 People
10 Tents Holds 70 People
Part B:
Number of tents|||Number of people
1 |||7
6 |||42
9 |||63
10 |||70
Part C:
Tents=x
1x=7 people
What is AB?
Triangle ACB is right angle triangle. The length of AC is 12 and BC is 35.
Answer:
the answer is 35
Step-by-step explanation:
because if BC is 35 that means AB will have to be that same because it's a triangle
The required value of AB is 33 units for the given right triangle.
What is the right triangle?A right triangle is defined as a triangle in which one angle is a right angle or two sides are perpendicular.
Triangle ACB is a right-angle triangle. The length of AC is 12 and BC is 35.
Pythagoras's theorem states that in a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.
Assume BC is the hypotenuse,
Since this is a right triangle, use the formula AB² + AC² = BC² and substitute values of AC = 12 and BC = 35.
AB² + AC² = BC²
AB² + (12)² = (35)²
AB² + 144 = 1225
AB² = 1225 -144
AB² = 1081
AB = 32.8785
Rounded to two decimal places,
AB = 33 units
Therefore, the required value of AB is 33 units.
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Find the radian measure of an angle of -340
Answer:
-1.889 rad
Step-by-step explanation:
180 degrees = [tex]\pi[/tex]
-340 x [tex]\frac{\pi }{180}[/tex]
-340[tex]\pi[/tex]/180
-34[tex]\pi[/tex]/18
-1.889 rad
Who can help , desperately need
Answer:
1/4 is x and 1 is b
Step-by-step explanation:
Determine volume of a cylindre r2 + y2 = 4 inside a sphere r2 + y2 +22 = 16
The volume of the cylinder inside the given sphere is 8 cubic units.
How to determine the volume of the cylinder inside the given sphere?To determine the volume of the cylinder inside the given sphere, we need to find the limits of integration and set up the integral.
Let's analyze the equations:
Cylinder equation:[tex]r^2 + y^2 = 4[/tex]
Sphere equation: [tex]r^2 + y^2 + 2^2 = 16[/tex]
From the equations, we can see that the cylinder is centered at the origin (0, 0) with a radius of 2 and an infinite height along the y-axis. The sphere is centered at the origin as well, with a radius of 4.
To find the limits of integration, we need to determine where the cylinder intersects the sphere. By substituting the cylinder equation into the sphere equation, we can solve for the values of r and y:
[tex](2^2) + y^2 + 2^2 = 16\\4 + y^2 + 4 = 16\\y^2 = 8[/tex]
y = ±√8
We can see that the cylinder intersects the sphere at y = √8 and y = -√8. Since the cylinder has infinite height, the limits of integration for y will be from -√8 to √8.
Now we can set up the integral to calculate the volume of the cylinder:
V = ∫∫∫ dV
= [tex]\int_0^ 2 \int_{\sqrt -8} ^ {\sqrt 8}\int _{\sqrt-(16 - r^2 - y^2)} ^{\sqrt (16 - r^2 - y^2)} dz dy dr[/tex]
Since the integrand is equal to 1, we can simplify the integral to:
V = [tex]\int_0 ^ 2 \int _{-\sqrt8} ^ {\sqrt8} 2\sqrt{(16 - r^2 - y^2)}[/tex] dy dr
Evaluating this integral will give us the volume of the cylinder inside the sphere.
To evaluate the integral and calculate the volume, we can integrate the given expression with respect to y first and then with respect to r.
[tex]\int_0 ^ 2 \int _{-\sqrt8} ^ {\sqrt8} 2\sqrt{(16 - r^2 - y^2)}[/tex]
Let's begin by integrating with respect to y:
[tex]\int_{-\sqrt8} ^ {\sqrt8} 2\sqrt(16 - r^2 - y^2) dy[/tex]
We can simplify the integrand using the trigonometric substitution y = √8sinθ:
dy = √8cosθ dθ
y = √8sinθ
Replacing y and dy in the integral:
[tex]\int _{-\pi /2} ^{\pi/2} 2\sqrt(16 - r^2 - (\sqrt 8sin\theta)^2) \sqrt 8cos\theta d\theta[/tex]
= 16[tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(1 - (r/4)^2 - sin^2\theta)[/tex]cosθ dθ
To simplify the integral further, we can use the trigonometric identity [tex]sin^2\theta + cos^2\theta = 1:[/tex]
16[tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(1 - (r/4)^2 - sin^2\theta)[/tex]cosθ dθ
= 16 [tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(r^2/16)[1 - cos^2\theta][/tex]cosθ dθ
= 4r[tex]\int _{-\pi/2} ^ {\pi/2}[/tex] sinθ cosθ dθ
= 4r [tex][ -cos^2\theta/2[/tex] ]| [-π/2 to π/2 ]
= 4r [ [tex]-cos^2(\pi/2)/2 + cos^2(-\pi/2)/2[/tex] ]
= 4r [ -1/2 + 1/2 ]
= 4r
Now, we can integrate with respect to r:
[tex]\int_0 ^ 2[/tex] 4r dr
= 2[tex]r^2[/tex]| [0 to 2]
= 2[tex](2^2 - 0^2)[/tex]
= 2(4)
= 8
Therefore, the volume of the cylinder is 8 cubic units.
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An auditor is determining the appropriate sample size for testing inventory valuation using MUS. The population has 2.620 inventory items valued at $12.625.000. The tolerable misstatement is $500.000 at a 10% ARIA. No misstatements are expected in the population. Calculate the preliminary sample size. (Confidence factor: 2,31)
The preliminary sample size is undefined since the projected misstatement is zero.
In determining the appropriate sample size for testing inventory valuation using MUS, the following steps are taken;
Plan the audit- Identify the tolerable misstatement. Assess inherent and control risk. Estimate population deviations. Determine the preliminary sample size. Select the sample to perform the audit procedures. Evaluate the results.Given that the population has 2,620 inventory items valued at $12,625,000 and the tolerable misstatement is $500,000 at a 10% ARIA, we can calculate the preliminary sample size using the formula;
Preliminary sample size = (Confidence Factor2 × Tolerable Misstatement)/Projected misstatement.
Considering that no misstatements are expected in the population, the projected misstatement will be zero.
Thus; the Preliminary sample size = (2.31 × 500,000)/0. Preliminary sample size = (2.31 × ∞) / 0. The preliminary sample size is undefined.
In conclusion, the preliminary sample size is undefined since the projected misstatement is zero.
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what is the x 19 + 13x = 32
Answer:
1
Step-by-step explanation:
1. Start Time: 3:30 P.M.
End Time: 7:00 P.M.
Elapsed Time:
Answer:
7:00 = 6:60
Step-by-step explanation:
6:60 - 3:30 = 3 hours and 30 minutes
A farmer wants to seed and fence a section of land. Fencing costs $27 per yard. Grass seed costs $2 per square foot. How much does it cost to fence and seed the pasture? (No links)
Answer:
she have 29 seed for the pasture
HELP ME!!!!!! Correct answers only!!!!!
Answer: 22.94 cubic meters
I'm pretty sure it is this.
I think it's 25.23 cubic meters if not then I don't know
Answer: 891ft^3
Step-by-step explanation:
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!
Answer:
One solution
I actually do not think you're going to give me brainliest
Drag the correct steps into order to solve the equation 5x5 + 5 = 10 for x
Answer:
x = 1
Step-by-step explanation:
5x^5 + 5 = 10
Start the solution by subtracting 5 from both sides:
5x^5 = 5
Dividing both sides by 5 yields x^5 = 1.
Taking the 5th root of both sides, we get x = 1
Can someone plz help me I beg u
Answer:
28.26
Step-by-step explanation:
The formula for finding the circumference is C=2pi(radius) and the radius is half of the diameter, which in our case is 4.5. So 2*3.14*4.5=28.26
[Help asap, question is in image, will mark brainliest]
Answer:
V = 339.12
Step-by-step explanation:
Volume of a cone:
V = πr²(h/3)
Given:
r = 6
h = 9
Work:
V = πr²(h/3)
V = 3.14(6²)(9/3)
V = 3.14(36)(3)
V = 113.04(3)
V = 339.12
Find the volume and total area of the right circular cone.
To find the volume and total area of the right circular cone, we will use the formulas below. Volume of the right circular cone: $$V = \frac{1}{3}πr^2h$$
Total surface area of the right circular cone:$$A = πr^2 + πrl$$, Where r is the radius, l is the slant height and h is the height of the cone.π (pi) is a mathematical constant that is approximately equal to 3.14159 and is used to calculate the circumference and area of a circle. The radius of the right circular cone is 3.5 cm and its height is 7 cm. To calculate the slant height, we will use the Pythagorean theorem which states that the square of the hypotenuse (l) is equal to the sum of the squares of the other two sides:$$l^2 = r^2 + h^2$$$$l = \sqrt{r^2 + h^2} = \sqrt{3.5^2 + 7^2} \approx 7.98\ cm$$
Volume of the right circular cone:$$V = \frac{1}{3}πr^2h = \frac{1}{3}π(3.5)^2(7) \approx 89.75\ cm^3$$. Total surface area of the right circular cone:$$A = πr^2 + πrl = π(3.5)^2 + π(3.5)(7.98) \approx 91.86\ cm^2$$. Hence, the volume of the right circular cone is approximately 89.75 cm³ and the total surface area is approximately 91.86 cm².
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Find the height of a prism whose volume is 108 cm3
and the area of the base is 15 cm2.
Answer:17
Step-by-step Experlation: The volume is 108 cm. 3,2,15.a square prism with a base edge of 9.5 inches and a height of 17 ... amount for a landowner
Question 6: Integration (12 marks) a. Which of the following definitions best describes the result of integrating a positive function f(x)? A The value of f(x) when == 0 B. The area between the curve of f(x) and the x-axis. C. The difference between the minimum of f(x) and the maximum of f(x). D. The gradient of f () at the point where x = 0. (1 mark) b. Which of the following is the general antiderivative of the function f(x) = 23+8x?? A 10x4 + 24x2 B. 2x° (x2 + 4) C. 2x6 + 8x4 D. 32° +2x4+C (1 mark) Which of the following statements is true for an odd function 9(2) ? 1 C. A. B. В S 0-2500 S = g(x) = 0 5 (2) = 0 Soo-a C. D (1 mark) d. By using the substitution 4x + 2 = u, show that the expression below is true. 1 +1 dx +C (4x + 2) 1600 + 8 (5 marks) e. Find the volume of the solid obtained when the area bounded by the curves below is rotated about the x-axis. Use the result shown in part (e) to assist you. Sa+gads 1 O x 1 = 0 (4 marks)
(a) The area between the curve of f(x) and the x-axis best describes the result of integrating a positive function f(x).
(b) The general antiderivative of the function f(x) = 23 + 8x is 2x³ + 4x² + C.
Hence, option (C) is the correct answer.
(c) An odd function satisfies f(-x) = -f(x). Thus, for an odd function f(x), the integral from -a to a is equal to zero because f(x) and -f(x) will have opposite signs, and the areas will cancel each other out. Hence, option (A) is the correct answer.
(d) To use the substitution u = 4x + 2, we need to find dx in terms of du.du = d/dx (4x + 2) dx= 4dxIntegrating both sides gives ∫du/4 = ∫dx/ (4x + 2). Therefore, the given expression becomes, ∫ 1/(4x + 2) dx = (1/4)∫du/u= (1/4)ln|u|+C= (1/4) ln|4x + 2| + C. Hence, (1/4) ln|4x + 2| + C is true by using the substitution 4x + 2 = u.
(e) The given function can be graphed as below: [tex]\int_0^1 (x^2 + 1) dx = \frac{4}{3}[/tex] We need to use the disk method to find the volume of the solid generated by rotating the region bounded by the curves about the x-axis. We need to consider an elemental area, find its volume, and integrate it over the region of interest. We know that the volume of the disk is given by V = πr²h, where r is the radius and h is the height of the disk. Let us consider an elemental area, A of the region rotated about the x-axis. If we rotate this area through a small angle, θ, then the area of the sector generated is given by d A = πr²dθ/2π = r²dθ/2. The radius of the disk is x, and the height is given by g(x) - f(x). Thus, V = ∫[g(x) - f(x)]²πx²dx.In this case, we have g(x) = x + 1 and f(x) = x². Substituting these values, V = π∫(x + 1 - x²)² x² dx. The limits of integration are from 0 to 1.
Therefore, V = π∫[x⁴ - 2x³ + x² + 2x + 1]dx= π[x⁵/5 - x⁴/2 + x³/3 + x² + x]₀¹= π[(1/5) - (1/2) + (1/3) + 1 + 1]
The volume of the solid obtained is, V = π[(8/15) + 2] = (14π/15).
Hence, the volume of the solid obtained when the area bounded by the curves below is rotated about the x-axis is (14π/15).
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please help help help help !!!!! ASAP
Factorise
3у^2 - 54y + 243
Answer:
3(y-9)^2
Step-by-step explanation:
Answer:
See in the picture mark brainliest if correct
555555555555 plzzz help