The largest rectangular box volume in the first octant, with one vertex on [tex]\(x + 2y + 3z = 9\),[/tex] is [tex]\(\frac{486}{125}\).[/tex]
To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex on the plane [tex]\(x + 2y + 3z = 9\),[/tex]we can set up the problem using optimization techniques.
Let the coordinates of the vertex of the box that lies on the plane [tex]\(x + 2y + 3z = 9\)[/tex] be[tex]\((x, y, z)\).[/tex] Since the other vertices are on the coordinate planes, the dimensions of the box are [tex]\(x\), \(y\), and \(z\).[/tex]
The volume [tex]\(V\)[/tex]of the rectangular box is given by:
[tex]\[V = x \cdot y \cdot z\][/tex]
Given that this vertex lies on the plane [tex]\(x + 2y + 3z = 9\),[/tex] we have the constraint:
[tex]\[x + 2y + 3z = 9\][/tex]
We need to maximize [tex]\(V\)[/tex] subject to this constraint. To do this, we can use the method of Lagrange multipliers. We introduce a Lagrange multiplier [tex]\(\lambda\)[/tex] and define the Lagrangian function:
[tex]\[\mathcal{L}(x, y, z, \lambda) = x y z + \lambda (9 - x - 2y - 3z)\][/tex]
To find the critical points, we take the partial derivatives of [tex]\(\mathcal{L}\)[/tex] with respect to [tex]\(x\), \(y\), \(z\), and \(\lambda\)[/tex] and set them to zero:
[tex]\[\frac{\partial \mathcal{L}}{\partial x} = yz - \lambda = 0\]\[\frac{\partial \mathcal{L}}{\partial y} = xz - 2\lambda = 0\]\[\frac{\partial \mathcal{L}}{\partial z} = xy - 3\lambda = 0\]\[\frac{\partial \mathcal{L}}{\partial \lambda} = 9 - x - 2y - 3z = 0\][/tex]
From the first three equations, we can express [tex]\(\lambda\)[/tex] as follows:
[tex]\[\lambda = yz\]\[\lambda = \frac{xz}{2}\]\[\lambda = \frac{xy}{3}\][/tex]
Equating these expressions for [tex]\(\lambda\):[/tex]
[tex]\[yz = \frac{xz}{2} \implies 2yz = xz \implies x = 2y \quad \text{(if \(z \neq 0\))}\]\[yz = \frac{xy}{3} \implies 3yz = xy \implies y = 3z \quad \text{(if \(x \neq 0\))}\][/tex]
Substituting [tex]\(y = 3z\) and \(x = 2y = 2(3z) = 6z\)[/tex] into the constraint [tex]\(x + 2y + 3z = 9\):[/tex]
Now, using [tex]\(z = \frac{3}{5}\):[/tex]
[tex]\[y = 3z = 3 \left(\frac{3}{5}\right) = \frac{9}{5}\]\[x = 6z = 6 \left(\frac{3}{5}\right) = \frac{18}{5}\][/tex]
The dimensions of the box are:
[tex]\[x = \frac{18}{5}, \quad y = \frac{9}{5}, \quad z = \frac{3}{5}\][/tex]
The volume[tex]\(V\)[/tex] is:
[tex]\[V = x \cdot y \cdot z = \left(\frac{18}{5}\right) \left(\frac{9}{5}\right) \left(\frac{3}{5}\right) = \frac{18 \cdot 9 \cdot 3}{5^3} = \frac{486}{125} = 3.888\][/tex]
Therefore, the volume of the largest rectangular box is:
[tex]\[\boxed{\frac{486}{125}}\][/tex]
The volume of the largest rectangular box with one vertex on the plane x + 2y + 3z = 9 is found using Lagrange multipliers. The maximum volume is 4.5 cubic units. The calculations involve the gradient method and substitution.
To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 9, we need to maximize V = xyz subject to the constraint x + 2y + 3z = 9.
We can use the method of Lagrange multipliers for this problem:
Define the function we want to maximize, f(x, y, z) = xyz.Introduce the constraint as a new function, g(x, y, z) = x + 2y + 3z - 9 = 0.Set up the system of equations using the gradient of the function and the constraint: ∇f = λ∇g.This gives us the following system of equations:
yz = λxz = 2λxy = 3λx + 2y + 3z = 9From these equations, we can solve for x, y, z, and λ:
λ = yzλ = xz / 2λ = xy / 3Equating and solving, we obtain x = 1.5, y = 1.5, and z = 2.
Finally, substituting these values into V = xyz gives the volume V = (1.5) imes (1.5) imes 2 = 4.5.
2. Quadrilateral ABCD is inscribed in a circle. Find the measure of each of the angles of the quadrilateral. Show your work.
A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle.
Opposite angles in a cyclic quadrilateral add to 180 degrees. Quadrilateral ABCD is a cyclic quadrilateral.
< A+<C=180
Substituting <A and <B values given in figure:
x+2+x-2=180
Adding like terms:
2x=180
Dividing both sides by 2
x=90°
<A=x+2=90+2=92°
<C=x-2=90-2=88°
<D=x-10=90-10=80°
<B+<D=180
<B=180-80=100°
The angles of the quadrilateral ABCD are
<A=92°
<B=100°
<C=88°
<D=80°
Mah problem, in class no substitute beed help
Last summer we took an auto trip of 3427 miles . After we had driven 1578 miles , how many are left ?
What is the solution set to the inequality-3x 5.92-15.4 for x in the set (-10,-5, 0, 5, 10)?
Final answer:
The solution set to the given inequality '-3x < 5.92 - 15.4' is {5, 10}.
Explanation:
The student's question seems to be about solving an inequality to find which values from a given set satisfy it. While the inequality itself appears to be incorrectly written or has a typo, I'll assume it's meant to be '-3x < 5.92 - 15.4'.
We can begin by simplifying the right side of the inequality:
5.92 - 15.4 = -9.48
So the inequality becomes:
-3x < -9.48
To solve for x, we divide both sides by -3, remembering to reverse the inequality sign because we are dividing by a negative number:
x > 9.48 / 3
x > 3.16
Now we can compare this result to the set of numbers given: (-10, -5, 0, 5, 10).
The only numbers greater than 3.16 are 5 and 10.
Therefore, the solution set to the inequality is {5, 10}.
What is the solution set for 3x>6?
Mrs. Bell has a rain gauge outside. On a very rainy day, the gauge measured 2.5 inches. The actual amount of rain that fell was 2.3 inches. To the nearest tenth, what is the percent error in Mrs. Bell's measurement?
The Philips family had 3 cats. They needed to split 20 ounces of cat food among the three cats. How much cat food would each cat get? between what two whole numbers does your answer lie?
Each cat would get approximately 6.67 ounces of cat food, which falls between 6 and 7 ounces. The calculation is done by dividing the total amount of food (20 ounces) by the number of cats (3).
Explanation:The Philips family had to split 20 ounces of cat food among their 3 cats. To find out how much food each cat gets, we need to divide the total amount of food by the number of cats. Calculating this, 20 ounces ÷ 3 cats equals approximately 6.67 ounces per cat. Even though this is a decimal, if we consider whole numbers, the amount each cat gets falls between 6 and 7 ounces, since 6.67 is more than 6 and less than 7.
When dealing with fractions and division, it's essential to understand the concept of division and be able to find averages. The task at hand may often find itself related to dividing a quantity evenly among a number of recipients, much like in scenarios where we have to split a pie or determine individual portions.
To divide 20 ounces of cat food equally among three cats, each cat gets approximately 6.67 ounces. The amount of food per cat lies between the whole numbers 6 and 7.
Explanation:The question is asking how to divide 20 ounces of cat food equally among three cats. To find out how much each cat would get, you divide the total amount of cat food by the number of cats:
Determine the total amount of food: 20 ounces.Count the number of cats: 3 cats.Divide the total amount of food by the number of cats: 20 ounces ÷ 3 cats = approximately 6.67 ounces per cat.This calculation shows that each cat would get approximately 6.67 ounces of food. This value lies between the whole numbers 6 and 7. Therefore, when the Philips family divides the cat food, each cat gets a little more than 6 ounces but less than 7 ounces of food.
Write 4/10 as a fraction with a denominator of 100. Then write 4/10 as a decimal.
Would it be 4/100 or 40/100? The decimal would be 0.4 ?
Find the fuction h (x) =f(x) - g (x) if f (x)=3^x and g (x)= 3^2x - 3^x
To find h(x) = f(x) - g(x), we substitute and simplify the given functions. The result is h(x) = 2 × 3ˣ - 3²ˣ. This solution involves basic algebraic manipulation and exponent rules.
Let's start by defining the given functions:
f(x) = 3ˣg(x) = 3²ˣ - 3ˣWe need to find h(x) = f(x) - g(x). Substitute the expressions for f(x) and g(x):
h(x) = 3ˣ - (3²ˣ - 3ˣ)Now simplify the expression inside the parentheses:
h(x) = 3ˣ - 3²ˣ + 3ˣCombine like terms:
h(x) = 3ˣ+ 3ˣ - 3²ˣh(x) = 2 × 3ˣ - 3²ˣThus, the function h(x) = 2 × 3ˣ - 3²ˣ is found by subtracting g(x) from f(x).
Determine the taylor series about the point x0=2 for the function 11−x. determine the radius of convergence of the series.
A Bird Is flying 5 m/s. How far will it travel if it flies for 3minutes
A. 15m
B. 5m
C. 900m
D. 36m
the area of this circle is 72π m^2 what is the area of a 45° sector of this circle
You have a $50 gift card to the same site. You want to buy an album with 16 tracks for $12.99 and then use the rest of the gift card for single tracks. How many songs can you buy with the gift card?
Final answer:
After purchasing an album for $12.99 from the $50 gift card, the remaining balance is $37.01, which allows the purchase of 37 more single tracks, assuming each track costs $1.
Explanation:
To determine how many songs can be bought with a $50 gift card after purchasing an album for $12.99, we need to subtract the cost of the album from the total amount on the gift card. So, we have:
Total amount on gift card: $50Cost of album: $12.99Subtracting the cost of the album from the gift card amount gives us:
$50 - $12.99 = $37.01
Assuming that each single track costs $1 as per the scenario provided, we can divide the remaining balance by the cost per single track:
$37.01 ÷ $1 per track = 37 tracks
This means that you can purchase 37 more single tracks with the remaining balance on the gift card.
The left and right page numbers of an open book are two consecutive integers whose sum is 423.
Lorne subtracted 6x3 – 2x + 3 from –3x3 + 5x2 + 4x – 7. Use the drop-down menus to identify the steps Lorne used to find the difference. 1. (–3x3 + 5x2 + 4x – 7) + (–6x3 + 2x – 3) 2. (–3x3) + 5x2 + 4x + (–7) + (–6x3) + 2x + (–3) 3. [(–3x3) + (–6x3)] + [4x + 2x] + [(–7) + (–3)] + [5x2] 4. –9x3 + 6x + (–10) + 5x2 5. –9x3 + 5x2 + 6x – 10
To find the polynomial difference, Lorne transformed the subtraction into addition of the opposite, combined like terms, and simplified the expression to get the result of -9x³ + 5x² + 6x - 10.
To find the difference between the two polynomials, Lorne correctly followed the steps of subtraction by changing the subtraction to the addition of the opposite. The process is outlined below:
Add the opposite of the second polynomial to the first polynomial by changing the sign of each term in the second polynomial.
Combine like terms by adding the coefficients of terms with the same degree of x.
Rearrange the terms in descending order of their degrees.
Write down the final simplified form of the polynomial.
By following these steps, the result would be:
Step 1:
(-3x³ + 5x² + 4x - 7) + (-6x³ + 2x - 3)
Step 2:
(-3x³ + 5x² + 4x - 7) + (-6x³) + 2x + (-3)
Step 3:
[(-3x³) + (-6x³)] + [4x + 2x] + [(-7) + (-3)] + [5x²]
Step 4:
-9x³ + 6x + (-10) + 5x2
Step 5:
-9x³ + 5x² + 6x - 10
Lorne subtracted [tex]6x^{3} - 2x + 3[/tex] from [tex]-3x^{3} + 5x^{2} + 4x - 7[/tex]. So, the main answer is 5. [tex]-9x^{3} +5x^{2} +6x-10.[/tex]
The steps Lorne used to find the difference between [tex]-3x^{3} +5x^{2} +4x-7[/tex] and [tex]6x^{3} -2x+3[/tex] are:
[tex]1. (-3x^{3} +5x^{2} +4x-7)+(-6x^{3} +2x-3)\\2. (-3x^{3} )+5x^{2} +4x+(-7)+(-6x^{3} )+2x+(-3)\\3. [(-3x^{3} )+(-6x^{3} )]+[4x+2x]+[(-7)+(-3)]+[5x^{2} ]\\4. -9x^{3} +6x+(-10)+5x^{2} \\5. -9x^{3} +5x^{2} +6x-10[/tex]
So, the main answer is 5. [tex]-9x^{3} +5x^{2} +6x-10.[/tex]
Lorne first added the two polynomials by combining like terms, then simplified the result by collecting like terms. The final expression represents the difference between the two polynomials after combining and simplifying.
COMPLETE QUESTION:
Lorne subtracted [tex]6x^{3} - 2x + 3[/tex] from [tex]-3x^{3} + 5x^{2} + 4x - 7[/tex]. Use the drop-down menus to identify the steps Lorne used to find the difference.
[tex]1. (-3x^{3} +5x^{2} +4x-7)+(-6x^{3} +2x-3)\\2. (-3x^{3} )+5x^{2} +4x+(-7)+(-6x^{3} )+2x+(-3)\\3. [(-3x^{3} )+(-6x^{3} )]+[4x+2x]+[(-7)+(-3)]+[5x^{2} ]\\4. -9x^{3} +6x+(-10)+5x^{2} \\5. -9x^{3} +5x^{2} +6x-10[/tex]
Philip is going on a 400040004000-kilometer road trip with three friends. The car consumes 666 liters of gas per 100100100 kilometers, and gas costs \$1.50$1.50dollar sign, 1, point, 50 per liter.
Miguel’s teacher asks him to color for 4/8 of his grid. He must use three colors: red blue and green. There must be more green sections then red sections. How can Miguel color the section service grid to follow all the rules?
7b divided by 12=4.2 what is b
To solve for b in the equation 7b divided by 12 equals 4.2, multiply both sides by 12 and then divide by 7, resulting in b being equal to 7.2.
Explanation:To solve the equation 7b ÷ 12 = 4.2 for b, we must isolate b on one side of the equation. We can begin by multiplying both sides of the equation by 12 to cancel out the division by 12 on the left side. This results in the equation 7b = 4.2 × 12. To find the value of b, we then divide both sides of the equation by 7:
Multiply both sides by 12: 7b ÷ 12 × 12 = 4.2 × 12Simplify: 7b = 50.4Divide both sides by 7: 7b ÷ 7 = 50.4 ÷ 7b = 7.2Therefore, the value of b is 7.2.
To find the value of b in the equation 7b/12 = 4.2, multiply both sides by 12 and then divide by 7 to get b = 7.2. The value of b is 7.2.
Explanation:The student is asking how to find the value of b in the equation 7b divided by 12 equals 4.2. To solve for b, you can follow these steps:
Write down the original equation: 7b / 12 = 4.2.Multiply both sides of the equation by 12 to eliminate the denominator: 12 * (7b / 12) = 12 * 4.2, which simplifies to 7b = 50.4.Divide both sides by 7 to solve for b: 7b / 7 = 50.4 / 7, which gives b = 7.2.Therefore, the value of b is 7.2.
Use Cavalieri’s Principle to calculate the exact volume of an oblique cylinder with a height of 10 meters and a circular base with a radius of 8 meters
Answer: Volume of an oblique cylinder is 2011.43 sq.m.
Step-by-step explanation:
Since we have given that
Radius of a cylinder = 8 meters
Height of a cylinder = 10 meters
Cavelieri's Principle states that volume of an oblique cylinder is same as the volume of right circular cylinder with equal radius and height.
As we know the formula for "Volume of cylinder":
[tex]Volume=\pi r^2h\\\\V=\frac{22}{7}\times 8\times 8\times10\\\\V=\frac{14080}{7}\\\\V=2011.43[/tex]
Hence, Volume of an oblique cylinder is 2011.43 sq.m.
I need some help what 9 + 10?
The system of equations 4x-y=4(x+1) , y = 6 has:
A. one solution
B. infinitely many solutions
C. no solution
The system of equations 4x-y=4(x+1) and y=6 has no solution.
To determine the solution(s) of the system of equations, let's analyze each equation:
Equation 1: 4x - y = 4(x + 1)
Equation 2: y = 6
To simplify Equation 1, we can distribute 4 on the right side:
4x - y = 4x + 4
By rearranging terms, we have:
- y = 4
Comparing Equation 2 with this result, we see that the two equations are contradictory. Equation 2 states that y = 6, while the modified Equation 1 states that y = -4. This means there is no value of y that can simultaneously satisfy both equations.
Thus, the system of equations has no solution.
Therefore, the correct answer is C. no solution.
To know more about system of equations, refer here:
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What is the area of the figure?
Express your answer as a mixed number in simplest form.
? m2
Mel Sturbridge needs $24,700 to remodel his home.
Find the face value of a simple discount note that will provide the $24,700 in proceeds if he plans to repay the note in 180 days and the bank charges an 8% discount rate.
Round to the nearest cent.
To find out the face value of a simple discount note that will provide a proceed of $24,700 with an 8% discount rate for 180 days, use the formula FV = P / (1 - (r * t)). Plug in the given values and compute the answer. The result is the required face value and should be rounded to the nearest cent.
Explanation:To find out the face value of a simple discount note, we can use the formula FV = P / (1 - (r * t)) where FV represents the face value, P is the proceeds, r is the discount rate, and t is the time in years.
In the problem, P = $24,700, r = 8% or 0.08 (expressed as a decimal), and t = 180/365 because there are 365 days in a year and the note is for 180 days. Let's substitute these values into the formula:
FV = 24700 / (1 - (0.08 * (180/365))
When you calculate the expression on the right, the answer will be the face value required to get a proceed of $24,700 with an 8% discount rate for 180 days. Round to the nearest cent for your final answer.
Learn more about Simple Discount Note here:https://brainly.com/question/37160732
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three friends are in the same algebra class. Their scores on a recent test are three consecutive odd intergers whose sum is 279. find each score
Make a line graph of the data in the table and conjecture on the minimum degree of a polynomial model...
sin(76)cos(31)-cos(76)sin(31)
What is the completely factored form of x2y – 2xy – 24y?
Answer:
C: y(x + 4)(x – 6)
Step-by-step explanation:
I took the test on Edge -2022
Sam is training for an upcoming wrestling match. He trains for 3.5 hours each day Monday through Wednesday. He trains 2.5 hours on Thursday and Friday. How many hours will he train if he trains for six weeks
Help with this question
You spin the spinner, flip a coin, then spin the spinner again. Find the probability of the compound event. Write your answer as a fraction or percent. If necessary round your answer to the nearest hundredth.
The probability of spinning an odd number, flipping heads, then spinning a yellow is ????
probability = number of favourable outcomes / total number of outcomes
probability of spinning any number = 1/3
probability of tossing a coin = 1/2
probability of compound event = 1/3 * 1/2 * 1/3 = 1/18
probability of spinning an odd number, flipping heads, then spinning a yellow is = 2/3*1/2*1/3 = 1/9