Final answer:
To maximize the area with 160 feet of fencing and one side provided by a barn, the fencing should create a rectangle with equal width for the two sides perpendicular to the barn. By expressing length in terms of width and using calculus to find the maximum area, we find a width of 40 feet and a length of 80 feet maximizes the area.
Explanation:
The farmer is using the barn wall as one side of the rectangular area he wants to fence. The remaining three sides require 160 feet of fencing. To maximize the area of the rectangle with a given perimeter, the shape should be a square; however, since one side is already provided by the barn, the best we can achieve is to have the two sides perpendicular to the barn be equal in length.
Let's use width w to denote the length of the two sides that are perpendicular to the barn and length l to denote the side parallel to the barn but not including the barn's wall itself. The total amount of fencing the farmer has is 160 feet, which we can express using the equation:
2w + l = 160
Since we're optimizing for area A, and A = w x l, we want to find the values of w and l that give us the maximum A. We can express l in terms of w using the perimeter equation:
l = 160 - 2w
Substituting this into the area equation gives us:
A = w x (160 - 2w) = 160w - 2w²
To find the maximum area, we take the derivative of A with respect to w and set it to zero to find critical points.
dA/dw = 160 - 4w = 0
Solving for w gives us w = 40 feet. Therefore, the dimensions that give the maximum area when one side of a rectangle is fixed are a width of 40 feet and a length of 80 feet.
Which reflection rule, if any, can be used to prove that rectangle A(-8, -3), B(-2, -3), C(-2, -6), D(-8, -6) and rectangle A'(8, -3), B'(2, -3), C'(2, -6), D'(8, -6) are congruent?
A) (x, y) → (-x, y)
B) (x, y) → (x, -y)
C) (x, y) → (-x, -y)
D) The rectangles are not congruent.
The school janitor is going to purchase a vacuum cleaner to use in the classrooms. The regular price of a vacuum cleaner is $70. The janitor can buy a used one for $28. How much money will he save if he buys the used vacuum cleaner?
How fast (in rpm) must a centrifuge rotate if a particle 6.00 cm from the axis of rotation is to experience an acceleration of 125,000 g's?
The speed of the particle in rpm is 43170.32 rpm.
Given:
The distance of particle (radius) from the axis is [tex]r=6\rm \: cm=0.06\; m[/tex].
The acceleration (centripetal) experienced by the particle is,
[tex]a=125000\rm \; g's\\a=125000\times 9.81 \rm \; m/s^2\\a=1226250\rm \; m/s^2[/tex]
Let ω be the angular velocity of the particle and N be the speed in rpm.
So, ω will be [tex]\omega=\dfrac{2\pi N}{60}[/tex]
Now, the acceleration of the particle can be written as,
[tex]a=\omega^2r\\a=\left(\dfrac{2\pi N}{60}\right)^2\times 0.06[/tex]
So, the speed of the particle in rpm will be,
[tex]a=\left(\dfrac{2\pi N}{60}\right)^2\times 0.06\\12262550=0.01095N^2\times 0.06\\N=43170.32\rm \; rpm[/tex]
Therefore, the speed of the particle in rpm is 43170.32 rpm.
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(4) use the method of lagrange multipliers to determine the maximum value of f(x, y) = x a y b (the a and b are two fixed positive constants) subject to the constraint x + y = 1. (assume x, y > 0.)
The maximum value of [tex]\(f(x, y)\)[/tex] is given by:
[tex]\(f\left(\frac{b}{a+b}, \frac{a}{a+b}\right) = \left(\frac{b}{a+b}\right)^a \left(\frac{a}{a+b}\right)^b\)[/tex]
To find the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint (x + y = 1, use the method of Lagrange multipliers.
Let's set up the Lagrangian function:
[tex]\(L(x, y, \lambda) = x^a y^b + \lambda(x + y - 1)\)[/tex]
Taking the partial derivatives and setting them to zero:
Equation 1: [tex]\(\frac{\partial L}{\partial x} = a x^{a-1} y^b + \lambda = 0\)[/tex]
Equation 2: [tex]\(\frac{\partial L}{\partial y} = b x^a y^{b-1} + \lambda = 0\)[/tex]
Equation 3: [tex]\(\frac{\partial L}{\partial \lambda} = x + y - 1 = 0\)[/tex]
From equations (1) and (2) it can be written as
Equation 4: [tex]\(a x^{a-1} y^b = -\lambda\)[/tex] ... (4)
Equation 5: [tex]\(b x^a y^{b-1} = -\lambda\)[/tex] ... (5)
Dividing equation (4) by equation (5) gives
[tex]\(\frac{a}{b} \frac{x^{a-1}}{x^a} \frac{y^b}{y^{b-1}} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
From equation (3) [tex]\(y = 1 - x\)[/tex].
Substituting [tex]\(y = 1 - x\)[/tex] into the equation [tex]\(\frac{a}{b} \frac{y}{x} = 1\)[/tex]
[tex]\(\frac{a}{b} \frac{1-x}{x} = 1\)[/tex]
[tex]\(a x = b (1-x)\)[/tex]
[tex]\((a + b) x = b\)[/tex]
[tex]\(x = \frac{b}{a+b}\)[/tex]
Substituting [tex]\(x = \frac{b}{a+b}\)[/tex] into the constraint equation (3):
[tex]\(y = 1 - x[/tex]
[tex]= 1 - \frac{b}{a+b} = \frac{a}{a+b}\)[/tex]
Therefore, the critical point (x, y) that satisfies the constraint equation is:
[tex]\(x = \frac{b}{a+b}\) and \(y = \frac{a}{a+b}\)[/tex]
Therefore, the critical point gives the maximum value of the function [tex]\(f(x, y) = x^a y^b\)[/tex] subject to the constraint x + y = 1.
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Combine like terms: –10.9p + 3.9 = –9.18 Apply the next steps to solve the equation. What is the solution? p =
Answer:
P=1.2
Step-by-step explanation:
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what is 50 divided by 2.5 divided by 0.04
The value of the 50 divided by 2.5 divided by 0.04 is equal to 500.
To calculate 50 divided by 2.5 divided by 0.04, should follow the order of operations,
which is parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
However, there are no parentheses or exponents in this expression, so we can proceed directly to multiplication and division:
50 ÷ 2.5 ÷ 0.04
First, divide 50 by 2.5:
50 ÷ 2.5 = 20
Next, divide the result (20) by 0.04:
20 ÷ 0.04 = 500
Therefore, 50 divided by 2.5 divided by 0.04 equals 500.
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In a chemical compound there are three parts zinc for every 16 parts copper by mass. A piece of the compound contains 320 g of copper. Write and solve an equation to determine the amount of zinc in the chemical compound.
The volume of 10000 drops of liquid is 10 fluid ounces what iscthe volume of 10 drops
The volume of 10 drops of liquid is approximately 0.0001 fluid ounces.
To find the volume of 10 drops of liquid, we can use the given information that the volume of 10000 drops is 10 fluid ounces. Let's set up a proportion to solve for the volume of 10 drops.
Let V be the volume of 10 drops in fluid ounces.
We know that the volume of 10000 drops is 10 fluid ounces, so we can write:
10,000 drops / V = 10 fluid ounces / 10 drops
Now, cross-multiply and solve for V:
10,000 drops * 10 drops = 10 fluid ounces * V
100,000 drops² = 10 fluid ounces * V
Now, divide both sides by 100,000 drops^2 to solve for V:
V = 10 fluid ounces / 100,000 drops^2
V ≈ 0.0001 fluid ounces
Therefore, the volume of 10 drops of liquid is approximately 0.0001 fluid ounces.
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Is the relationship between the temperature scales proportional
Write and solve an equation for the situation. Eight and one-half years ago, Steven was 7 years old. How old is he now? Make sure you explain and know your answer. I am confused and frustrated, also, I have been working of this for a while and I do not fully understand. Thanks, :). Answer Fast. This means a lot.
Eight and one-half years ago the age of Steven was 7 years, then the present age of Steven is 15.5 years.
What is an equation?Mathematical equations with two algebraic symbols on either side of the equal (=) sign are called equation. This connection is illustrated by the left and right expression being equal to one another. The left-hand side equals the right-hand side is a basic, straightforward equation.
Assume the current age of Steven is x years.
From the given information in the question,
8.5 years ago from now,
x - 8.5 = 7 years
x = 15.5 years.
So, the current age is 15.5 years.
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Which of the following sets shows all the numbers from the set {2, 3, 4, 5} that are part of the solution to the inequality 4n + 6 > 22?
How many pounds-mass per cubic foot is 40 grams per cubic centimeter?
Question 1 options:
500 lbm/ft^3
3,200 lbm/ft^3.
2,500 lbm/ft^3
1,200 lbm/ft^3
To convert grams per cubic centimeter to pounds-mass per cubic foot, you can use the conversion factors. By applying the conversion factors of 1 lb = 454 g and 1 ft³ = (30.5 cm)³, you can calculate that 40 grams per cubic centimeter is equal to 500 pounds-mass per cubic foot.
Explanation:To convert grams per cubic centimeter to pounds-mass per cubic foot, we need to use conversion factors. Firstly, we need to convert grams to pounds by using the conversion factor 1 lb = 454 g. Then, we convert cubic centimeters to cubic feet by using the conversion factor 1 ft³ = (30.5 cm)³. Finally, we combine the two conversion factors to find the desired unit: pounds-mass per cubic foot.
Given:
40 grams per cubic centimeterConversion factor:
1 lb = 454 g1 ft³ = (30.5 cm)³Calculation:
40 g/cm³ * (1 lb/454 g) * [(30.5 cm)³/1 ft³] = 40 * 1/(454 * (30.5)³) lb/ft³Simplifying the calculation gives:
40/(454 * (30.5)³) lb/ft³ = 500 lbm/ft³Therefore, 40 grams per cubic centimeter is equal to 500 pounds-mass per cubic foot.
Raphael graphed the functions g(x) = x + 2 and f(x) = x – 1. How many units below the y-intercept of g(x) is the y-intercept of f(x)? The y-intercept of f(x) is units below the y-intercept of g(x).Raphael graphed the functions g(x) = x + 2 and f(x) = x – 1. How many units below the y-intercept of g(x) is the y-intercept of f(x)? The y-intercept of f(x) is units below the y-intercept of g(
Answer:
The y-intercept of f(x) is 3 units below the y-intercept of g(x).
Step-by-step explanation:
We are given,
The functions [tex]f(x)=x-1[/tex] and [tex]g(x)=x+2[/tex].
Now, y-intercept is the point where the graph of the function crosses y-axis.
So, y-intercept is obtained when x= 0.Substituting x= 0 in the functions, we get,
[tex]f(0)=0-1\\f(0)=-1[/tex] and [tex]g(0)=0+2\\g(0)=2[/tex].
Thus, we get that,
The y-intercept of the the function f(x) is -1.
The y-intercept of the function g(x) is at 2.
Thus, the y-intercept of f(x) is 2-(-1)= 3 units below the y-intercept of g(x) as seen below.
How Much Water Can Be Held By A Cylindrical Tank With A Radius Of 12 Feet And A Height Of 30 Feet
Bennett tried to solve exercise 1 a few different ways. Which of his methods are correct? Of the other methods, which makes the most sense to you? Explain
A. 5% sales tax means that for every dollar you spend, you need to pay a nickel in tax. If you buy a nickel tax. If you buy something for $21, you need to pay 21 nickels in tax.
B. You can set up a promotion and solve for the missing value.
$.05 = x <-- (These are fractions)
___ __
$1.00 $21.00
C. If you know that 10% of $21 is $2.10, so 5% should be half of $2.10
D. 5% is equal to 1/20. To find the amount of tax on $21, find $21 ÷ 20
E. 1% of $21.00 is $0.21 so 5% of $21.00 is 5 x $ 0.2
Writing a check on an account with insufficient funds is allowed under certain conditions.
True
False
Answer:
This is FALSE.
Step-by-step explanation:
Writing a check on an account with insufficient funds is never allowed by any bank. This will only lead to check bounce and sometimes can land you up in legal trouble. When a check bounces, you will have to pay your bank the bounce fee also. Hence, we should be careful while writing any check.
The reported unemployment is 5.5% of the population. what measurement scale is used to measure unemployment? nominal ordinal interval or ratio descriptive
The answer is interval or ratio. In interval level of measurement, the distances between have attributes does have a meaning. The best example to be used here is temperature. The distance from 40-50 is like the distance from 70-80. While in ratio level of measurement, there is constantly an absolute zero that is significant. Meaning, that you can build a meaningful fraction (or ratio) with a ratio variable.
A building casts a 250 meter long shadow at the same time a nearby post 6 meters high casts a shadow 30 meters long what is the height of the building
The knife shown below is 14 inches long. write an algebraic expression that represents the length of the blade (in inches).
The required algebraic expression for the length of the blade is x = 14 - h.
Given that,
The knife shown below is 14 inches long.
To write an algebraic expression that represents the length of the blade (in inches).
In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication, and division,
Here let the length of the blade of the knife be x,
length of the knife handle be h
Here the total length of the knife is 14 inches
So,
The total length of the knife = length of the blade + length of the handle
14 = x + h
x = 14 - h
Thus, the required algebraic expression for the length of the blade is x = 14 - h.
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What is the product
3/8 * (-3/6)
Enter your answer as a fraction, in simplified form, please
A combination lock uses three distinctive numbers between 0 and 49 inclusive. how many different ways can a sequence of three numbers be selected? (show work)
Let x have a poisson distribution with a variance of 3. find p(x = 2).
The probability that a Poisson random variable X equals 2, given that its variance is 3, is approximately 0.2240.
Explanation:The student asks about finding the probability that a Poisson random variable X takes on a value exactly equal to 2, given that the variance of X is 3. We know for a Poisson distribution, the variance is equal to its mean, which is denoted by λ (lambda). Therefore, λ = 3.
Using the formula for the Poisson probability mass function (PMF), P(X = k) = (e^(-λ) * λ^k) / k!, we can calculate P(X = 2). Substituting λ = 3 and k = 2, we have:
P(X = 2) = (e^(-3) * 3^2) / 2! = (0.0497871 * 9) / 2 = 0.224041.
We can round this probability to four decimal places:
P(X = 2) ≈ 0.2240.
120t-0.4t^4+1000=0
How can i solve this?
Find three consecutive odd integers whose sum is 87
Step-by-step explanation: Three consecutive odd integers are represents as follows.
X ⇒ first integer
X + 2 ⇒ second integer
X + 4 ⇒ third integer
Since the sum of these is 87, our equation reads X + X + 2 + X + 4 = 87.
Simplifying on the left, we get 3x + 6 = 87.
Subtracting 6 from both sides, 3x = 81.
Now divide both sides by 3 and x = 27.
Since x, our first odd integer is 27, our next consecutive odd integer will be X + 2 or 27 + 2 which is 29 and X + 4 would be 31.
____ as 9 hundreds is 9 thousands
Ten as 9 hundreds is 9 thousands.
What is place value?
Place value is the value of each digit in a number.
For example 5 in 350 is 5 tens or 50.
The given digit is 9 thousand which can be written as 9,000
It can be written as,
9,000 = 9 x 1000
or,9,000 = 9 x 10 x 100
So, for require 9 hundreds multiplying 9 by 100 in the above expression
9,000 = 10 x 900
which can be described as Ten as 9 hundreds is 9 thousand.
Hence, ten as 9 hundreds is 9 thousands
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10 as 9 hundreds is 9 thousands.
How to Obtain place values from a given number ?Suppose a number 12345 is given this can be written as
10000 + 2000 + 300 + 40 + 5 if we sum up all these we will get back our given number.
Now if we are asked what is the place value of 2 & 4
So, the place value of 2 in the given number is 2000 & the place value of 4 is 40.
Place value is defined as the value based on the position of that number whether it's in thousand's place or ten's or in 1's place.
In the given question
___ as 9 hundreds is 9 thousand
⇒ ____ as 900 is 9000
if we divide 9000/900 = 10 we will get our answer.
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The points at x equals=_______ and xequals=_______ are the inflection points on the normal curve
What is the inverse of the function f(x) = 2x – 10?
Answer:
f⁻¹(x) = (1/2)x +5
Step-by-step explanation:
In y = f(x), swap the variables, then solve for y. The expression you get is f⁻¹(x).
... y = 2x -10
... x = 2y -10 . . . . . . swapped variables
... x +10 = 2y . . . . . add 10
... (1/2)x + 5 = y . . . . divide by 2
... f⁻¹(x) = (1/2)x + 5 . . . . . . rewrite using function notation
Graph the line with slope -2 passing through the point (3,-4)
STVU is an isosceles trapezoid. If SV = 6x - 5 and TU = 2x + 7, find the value of x.
x=3
Fifty eight thousand kilograms of solid waste is equivalent to how many metric tons
Answer:58 metric ton
Step-by-step explanation:
Given
Fifty eight thousand kilogram of solid waste
We need to convert it into metric tons
and we know 1 metric ton is equivalent to one thousand kilogram i.e. 1000 kg
so 58,000 kg is equivalent to 58 metric ton