A company produces two types of solar panels per year: x thousand of type A and y thousand of type B. The revenue and cost equations,in millions of dollars, for the year are given as follows.
R (x,y) = 6x + 8y
C(x,y) = x^2 - 4xy + 6y^2 + 22x - 48y – 8,
Determine how many of each type of solar panel should be produced per year to maximize profit.
Answers
To maximize profit, x = 4 and y = 3 thousand of type A and type B solar panels, respectively, should be produced per year.
To determine the optimal production quantity of each type of solar panel, we need to maximize the profit function. Profit is calculated by subtracting the cost function from the revenue function.
Revenue function: R(x, y) = 6x + 8y
Cost function: C(x, y) = x^2 - 4xy + 6y^2 + 22x - 48y - 8
The profit function, P(x, y), can be obtained by subtracting the cost function from the revenue function:
P(x, y) = R(x, y) - C(x, y)
= (6x + 8y) - (x^2 - 4xy + 6y^2 + 22x - 48y - 8)
= -x^2 + 28x + 54y + 8
To find the maximum profit, we need to find the critical points of the profit function. Taking the partial derivatives of P(x, y) with respect to x and y, we get:
∂P/∂x = -2x + 28
∂P/∂y = 54
Setting these partial derivatives equal to zero and solving the resulting equations, we find:
-2x + 28 = 0 => x = 14
54 = 0 (no solution)
Since the partial derivative ∂P/∂y = 54 is a constant, it does not affect the critical point. Therefore, the critical point occurs at x = 14.
To determine if this critical point is a maximum or minimum, we can use the second partial derivative test. Taking the second partial derivatives of P(x, y), we get:
∂²P/∂x² = -2
∂²P/∂y² = 0
The second partial derivative ∂²P/∂x² = -2 is negative, indicating that the critical point is a maximum.
Hence, to maximize profit, x = 4 and y = 3 thousand of type A and type B solar panels, respectively, should be produced per year.
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Related Questions
(15) The ratio of boys to girls at the school is 3:5. If there are 160 students at 1 point the school, how many of them are girls? (Do not include any words in your answer.) Your answer
Answers
Answer:
100
Step-by-step explanation:
160 * (5/8) = 100
Find the value of x
6x-10
8x -20°
Answers
6x-10+8x-20=180
move the constant too the right adding 30 too 180 giving you
14x = 210
divide 14 on both sides giving you
x = 15
Sofia works at Cowboy Corner. She receives 10% commission on all of the cowboy hats she sells in a month. If Sofia sells $545 in total sales, what is her commission? (include steps or explanation in ur answer)
Answers
Answer:
$54.50
Step-by-step explanation:
10% = 10/100
=545 × 10/100
=5450/100
=54.5
James has 42.00 he bought his sister a gift that cost x dollars. how much money does james have now, in terms of x
Answers
Given data:
The amount of money James have A=42.00 .
The amount of money left after gift purchased.
\(\begin{gathered} L=A-x \\ =42-x \end{gathered}\)Thus, the amount of money left is 42-x.
find a function that models the area a of a circle in terms of its circumference c.
Answers
Answer:
A = C²/(4π)
Step-by-step explanation:
You want a formula for the area of a circle, given its circumference.
Circle relationsRelevant relations for a circle are ...
A = πr² . . . . . . . . A = area, r = radius
C = 2πr . . . . . . . . C = circumference
Area from circumferenceSolving the circumference equation for r, we find ...
r = C/(2π)
Substituting that into the area equation, we have ...
A = π(C/(2π))²
A = C²/(4π)
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The function that models the area A of a circle in terms of its circumference C is A = C²/(4π).
Explanation:The formula for the circumference of a circle is C = 2πr or C = πd, where r is the radius and d is the diameter. To find a function that models the area A of a circle in terms of its circumference C, we can rearrange the formula for the circumference to solve for the radius: r = C/(2π). Substituting this value of r into the formula for the area, we get A = π(C/(2π))² = C²/(4π).
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this visual representation shows the sets of real numbers. which statement is true regarding the number sets? integers and rational numbers have no numbers in common. integers and rational numbers have no numbers in common. all rational numbers are also integers. all rational numbers are also integers. some integers are also rational numbers, but not all integers are rational numbers. some integers are also rational numbers, but not all integers are rational numbers. all integers are also rational numbers.
Answers
The statement that is true regarding the number sets is that some integers are also rational numbers, but not all integers are rational numbers.
The visual representation of the sets of real numbers includes different subsets, such as integers and rational numbers. The statement clarifies the relationship between these sets. Integers are whole numbers, including positive, negative, and zero values. On the other hand, rational numbers are those that can be expressed as a fraction of two integers, where the denominator is not zero.
In the first part of the statement, it is correctly stated that integers and rational numbers have some numbers in common. This is because every integer can be expressed as a fraction with a denominator of 1. For example, 3 can be expressed as 3/1. Therefore, integers are a subset of rational numbers.
However, the statement also correctly notes that not all integers are rational numbers. Irrational numbers, such as the square root of 2 or pi, cannot be expressed as fractions and are not considered rational. Therefore, while some integers are rational numbers, there exist integers, like irrational numbers, that cannot be expressed as fractions and are not part of the rational number set.
In conclusion, the statement accurately describes that some integers are also rational numbers, but not all integers are rational numbers.
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Find the slope of a line perpendicular to the line whose equation is
x−y=3. Fully simplify your answer
Answers
The slope of a line perpendicular to the line whose equation is x−y=3 is -1.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.Since the equation of this line is perpendicular to the line x - y = 3, the slope is given by;
x - y = 3
y = x - 3
Slope, m₁ = 1
m₁ × m₂ = -1
1 × m₂ = -1
m₂ = -1/1
Slope, m₂ = -1
At data point (2, 4) and a slope of -1, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 4 = -1(x - 2)
y = -x + 2 + 4
y = -x + 6
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Type the correct answer in the box. Use numerals instead of words. If necessary, use/ for the fraction bar.
Given the figure, find the total area of the shaded region.
D
8-
6-
4-
2-
O
-2-
o
S
The area of the shaded region is
B
R
8
с
square units
Answers
The value of the total area of the shaded region are,
⇒ 42 units²
We have to given that;
Sides of rectangle are,
AB = 9
BC = 6
Hence, The area of rectangle is,
⇒ 9 x 6
⇒ 54 units²
And, Area of triangle is,
A = 1/2 × 4 × 6
A = 12 units²
Thus, The value of the total area of the shaded region are,
⇒ 54 - 12
⇒ 42 units²
So, The value of the total area of the shaded region are,
⇒ 42 units²
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help me due today it is math
Answers
Answer:
Exact form = 1/49
Step-by-step explanation:
Decimal form = 0.02
Hope this helps! :)
Answer:
\(\frac{1}{7}^{2} = \frac{1}{49}\)
Step-by-step explanation:
The power of the 2 means to multiply it two times itself.
Example: 7^2 = 7 * 7 = 49
Apply that to a faction.
\(\frac{1}{7}^{2} =\frac{1}{7} *\frac{1}{7} = \frac{1}{49}\)
find the circumference and the area of a circle with diameter 8 cm.
Answers
Answer:
Step-by-step explanation:
If the diameter is 8 cm, and the formula is A=pie*radius^2, the radius is 4.
The area would be 16*pie
The formula for finding the circumference 2*pie*radius, so the circumference is 8*pie.
Grade 12 Financial Mathematic
n-up Q:
Austin buys a house for R 1 785 000 on the 1st September 2021. The bank gives him an interest rate of
6% p.a. compounded monthly. Mr Austin makes payments of R11 000 at the end of each month.
a) What is the balance on the loan on the 31st December 2036?
b) How long will it take Mr Austin to payoff the loan?
Answers
Answer:
a)0.00
b)13year,8month,2weeks and 6day
(2x+4)(x-1) express as a trinomial
Answers
Answer:
\(\\\\\hookrightarrow 2x^{2} +2x-4\)
Step-by-step explanation:
\(\hookrightarrow (2x+4)(x-1) \\\\\hookrightarrow 2x(x-1)+4(x-1)\\\\\hookrightarrow 2x^{2} -2x+4x-4\\\\\hookrightarrow 2x^{2} +2x-4\)
Answer:
\(\huge\boxed{\bf{(2x + 4)(x - 1) = 2x^2 + 8x - 4}} \)
Step-by-step explanation:
\(\sf \to (2x + 4)(x - 1) \)
\(\sf \to x(2x + 4) + 4(x - 1) \)
\(\sf \to 2x^2 + 4x + 4x - 4 \)
\(\sf \to 2x^2 + 8x - 4 \)
\(\: \)
\(\huge\colorbox{red}{BlackPain} \)
Solve both equations, and determine if the equations are sometimes, always, or never true?
a) 3(x+2)+ x = 2(2x-1)+8
b) 4 + 2(x + 5) = 3 -(x - 2)
D) a. always true
b. sometimes true
A) a. never true
b. sometimes true
B) a. always true
b. never true
C) a. sometimes true
b. never true
Answers
Ab= AD and BC=DC
Y=?
Answers
Answer:
y = 63°
Step-by-step explanation:
ABCD is a kite.
The main diagonal BD bisects the opposite angles, thus
∠ DBC = ∠ DBA = 35°
The sum of the 3 angles in Δ BCD = 180° , thus
y = 180° - (35 + 82)° = 180° - 117° = 63°
Gerald earns $100.00 per week selling magazine subscriptions. For each subscription he sells.
he earns 25% on his total sales (S). If he wants to earn at least $600.00 next week, how much
does he need to sell?
Answers
Answer:
d. is the correct answer in my opinion
Easy math question please answer ASAP!!!
Q. The sum of 9/26 and -4/13
The options are attached in the photo
A.
Answers
9/26+ (-8/26)
answer 1/26
A multiple choice test has 10 questions with 3 choices each. Only one choice per question is correct. What is the probability to ace the test (answer correctly all 10 questions)
Answers
Answer:
Since each question has 3 choices and only one is correct, the probability of answering a single question correctly is 1/3.
To find the probability of answering all 10 questions correctly, we need to multiply the probabilities of each individual question together, since the events are independent.
Therefore, the probability of acing the test is:
(1/3)^10 = 0.000005904
Or approximately 0.0006%, which is a very small probability.
Step-by-step explanation:
example of definition of derivative:lim as h approaches 0f(a+h) - f(a) / hThe slope of f(x) = -3x^2 +11 at any value of x
Answers
The slope of the function f(x) = -3x^2 +11 at any value of x is given by 6x.
The definition of derivative is given by the limit as h approaches 0 of [f(a+h) - f(a)] / h,
where f(x) is a function and a is a fixed value of x.
To find the slope of the function\(f(x) = -3x^2 +11\) at any value of x, we need to use this definition of derivative.
So, let's plug in f(x) = -3x^2 +11 and a = x into the formula.
This gives us:
lim as h approaches 0 of \([-3(x+h)^2 + 11 - (-3x^2 + 11)] / h\)
Simplifying this expression, we get:
lim as h approaches 0 of \([-3x^2 - 6hx - 3h^2 + 11 + 3x^2 - 11] / h\)
Combining like terms, we get:
lim as h approaches 0 of \([-6hx - 3h^2] / h]\)
Factoring out an h from the numerator, we get:
lim as h approaches 0 of [-h(6x + 3h)] / h
Canceling out the h in the numerator and denominator, we get:
lim as h approaches 0 of 6x + 3h
Now, we can plug in h = 0 to find the slope of the function at any value of x:
6x + 3(0) = 6x.
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a rectangle has area 81 m2. express the perimeter of the rectangle as a function of the length l of one of its sides.
Answers
Let l be the length of the rectangle and w be the width of the rectangle. Therefore, the area of the rectangle is given by the formula:
We know that the area of the rectangle is given as 81m².
So, 81 = lw
Let's solve for w: w = 81/l
The perimeter of the rectangle is given by the formula: Perimeter of Rectangle = 2(Length + Width)P
= 2(l + w)
Substituting the value of w from the above equation: P = 2(l + 81/l) This is the required expression to calculate the perimeter of the rectangle in terms of length. In order to find the perimeter of a rectangle, we need to know the length and width of the rectangle. We can then use the formula for the perimeter of a rectangle, P = 2(l + w), and substitute the value of w that we just found: P = 2(l + 81/l) This is the required expression to calculate the perimeter of the rectangle in terms of length l.
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Find the solution set of the inequality
- 4x +35> 2
Answers
Answer:
There is a answer to the question
Answer:
Inequality Form:
x
<
33
4
Interval Notation:
(
−
∞
,
33
4
)
Step-by-step explanation:
Complete the missing parts of the table for the following function y=1/2 x
Answers
The table of values for the function y = (1/2)^x with x = -2 to 3 x is as shown below
Completing the table of values for the functionGiven that
y= (1/2)^x
To complete the table of values for the function y = (1/2)^x, we can substitute the given values of x into the function and evaluate for y. The table is as follows:
x y
-2 4
-1 2
0 1
1 1/2
2 1/4
3 1/8
Here's how we get each value of y:
When x = -2: y = (1/2)^(-2) = 4When x = -1: y = (1/2)^(-1) = 2When x = 0: y = (1/2)^(0) = 1When x = 1: y = (1/2)^(1) = 1/2When x = 2: y = (1/2)^(2) = 1/4When x = 3: y = (1/2)^(3) = 1/8Therefore, the table of values for the function y = (1/2)^x with x = -2 to 3 and an increment of 1 in x is as shown above.
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If the area of a triangle is 30 in.? and the base is 5 in., what is the height?
Answers
Therefore 12 x 5 = 60 and 60/2 = 30 :)
Which describes the slope of this line?
O positive slope
O negative slope
• zero slope
O undefined slope
Answers
Explanation:
Zero slopes are slopes that don’t increase on the Y axis at all, therefore this isn’t a zero slope. Undefined slopes are like elevators, they can only move straight up and straight down. It got its name “undefined” from the fact that it is impossible to divide by zero.
Remember:
Slopes that only have a run are zero slopes
Slopes that only have a rise are undefined slopes
Anything else is either a positive or a negative slope.
Which simplified expression represents the area of the parallelogram? –4x3 14x – 24 square centimeters 2x3 – 6x2 – 14x 24 square centimeters –4x3 – 14x 24 square centimeters 2x3 6x2 14x 24 square centimeters.
Answers
The simplified expression that represents the area of the parallelogram is 2x^3 - 6x^2 -14x + 24
How to calculate the area of a parallelogramArea of a parallelogram = base * height
Given the following
Base = 2x^2 + 2x - 6
Height = x - 4
Substitute into the formula to have:
Area of a parallelogram= (x-4)(2x^2 + 2x - 6)
Area of a parallelogram= 2x^3 + 2x^2 - 6x - 8x^2 - 8x + 24
Area of a parallelogram= 2x^3 - 6x^2 -14x + 24
Hence the simplified expression that represents the area of the parallelogram is 2x^3 - 6x^2 -14x + 24
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y5+7=17
Responses
y = 50
y = 50
y = 2
y = 2
y = 24
y = 24
y = 120
Answers
Answer:
y=2
Step-by-step explanation:
Lacey earn $680 each week she pays 91.80 in federal income tax the other texas is $17 her pay after taxes is 547.40 how much does lacey paying state income tax
Answers
Lacey's pay in state income tax is $132.60.
To determine Lacey's pay in state income tax, we need to calculate the difference between her gross pay and her pay after taxes.
Lacey's gross pay is $680 per week. She pays $91.80 in federal taxes and $17.00 in other taxes, so her total tax amount is $91.80 + $17.00 = $108.80.
Lacey's pay after taxes is $547.40.
To find the state income tax, we subtract the pay after taxes from the gross pay:
State income tax = Gross pay - Pay after taxes = $680 - $547.40 = $132.60.
Therefore, Lacey's pay in state income tax is $132.60.
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Lacey earns $680 each week. She pays $91.80 in fed other taxes are $17.00. Her pay after taxes is $547.40 Lacey pay in state income tax? $132.60 $38.60 $23.80 $115,60
Input value for the function machine that gives an output value of 3.
Answers
Answer:
x = –1
Step-by-step explanation:
From the question given above, the following data were obtained:
y = √(–2x + 7)
y = 3
x =?
The value of x can be obtained as follow:
y = √(–2x + 7)
3 = √(–2x + 7)
Take the square of both side
3² = –2x + 7
9 = –2x + 7
Collect like terms
9 – 7 = –2x
2 = –2x
Divide both side by –2
x = 2 / –2
x = –1
40 POINTS!! PLEASE!! I AM SOOO LOST!!
A horizontal line passes through $(1,1)$. A vertical line passes through $(2,2)$. What is the point of intersection of these two lines? (No image)
Write an equation for the graph shown in the form y = (The image is below)
In which quadrant(s) do the graphs of $y = x^2$ and $y = 3x +10$ intersect? (No image)
The graphs of these two equations intersect in how many points?
\begin{align*}
x^2 + y^2 &= 25,
x^2 - y^2 &= 3.
(No image)
Answers
Answer:
(2,1)
Step-by-step explanation:
since the equations are y=1 and x=2, the answer is (2,1)
In the diagram below, m A = 59°, m B = 55° and m C = 78°. Find m D.
Answers
Answer:
D=36
Step-by-step explanation:
All angles of the triangle add up to 180. Subtract 180-59-55=66
Then the missing measurement is a vertical which means that E=66.
So then 180-78-66=36
evaluate dy for the given values of x and dx. (a) y = e x/10 , x = 0, dx = 0.1.
Answers
The value of dy for y = \(e^{(x/10)}\), x = 0, and dx = 0.1 is 0.01.
How evaluate dy for the given values of x?To evaluate the value of dy for the given values of x and dx, we first need to find the derivative of y with respect to x, which can be computed as follows:
\(y = e^{(x/10)}\)
Differentiating both sides with respect to x using the chain rule, we get:
dy/dx = d/dx [\(y = e^{(x/10)}\\\)]
=\(y = e^{x/10}\) * d/dx [x/10]
= \(y = e^{(x/10)}\) * (1/10) * d/dx [x]
=\(y = e^{(x/10)}\) * (1/10)
Now, we substitute the values x = 0 and dx = 0.1 in the above expression to get the value of dy:
dy = (1/10) *\(e^{(0/10)}\) * dx
= (1/10) * (1) * (0.1)
= 0.01
Therefore, the value of dy for y = \(e^{(x/10)}\), x = 0, and dx = 0.1 is 0.01.
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