(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 solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]

To solve the system of equations (3x - 4y = 35) and (3x + 4y = 5), we'll use the method of elimination.

Add the two equations together to eliminate the variable (y).

(3x - 4y) + (3x + 4y) = 35 + 5

3x - 4y + 3x + 4y = 40

6x = 40

[tex]\[ x = \frac{40}{6} \][/tex]

[tex]\[ x = \frac{20}{3} \][/tex]

Substitute [tex]\(x = \frac{20}{3}\)[/tex] into one of the original equations. Let's use the first equation:

[tex]\[ 3\left(\frac{20}{3}\right) - 4y = 35 \][/tex]

20 - 4y = 35

-4y = 35 - 20

-4y = 15

[tex]\[ y = \frac{15}{-4} \][/tex]

[tex]\[ y = -\frac{15}{4} \][/tex]

So, the solution to the system of equations is [tex]\(x = \frac{20}{3}\)[/tex] and [tex]\(y = -\frac{15}{4}\).[/tex]

Answer:

T = 13.9 + 12.85 + 14.82 + 12.74

Step-by-step explanation:

Great question, it is always good to ask away and get rid of any doubts that you may be having.

Based on the information in the question, the total time for the relay would be calculated by simply adding each of the four-relay members time during the 100 meter run. Therefore the expression would be

T = 13.9 + 12.85 + 14.82 + 12.74

T = 54.31 seconds.

If we decided to solve for the Total time (T) it would add up to 54.31 seconds for the relay time. But since the question did not ask for the total but just the expression. Then the answer is the expression above.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Answer:

Option 1 is correct.

Step-by-step explanation:

The given polynomial is

[tex]f(x)=20x^4+x^3+8x^2+x-12[/tex]

we have to find  all the rational roots of the polynomial f(x)

The Rational Root Theorem states that the all possible roots of a polynomial are in the form of a rational number i.e in the form of [tex]\frac{p}{q}[/tex]

where p is a factor of constant term and q is the factor of coefficient of leading term.

In the given polynomial the constant is -12 and the leading coefficient is 20.

[tex]\text{All possible factor of -12 are }\pm1,\pm2, \pm3, \pm4,\pm6,\pm12[/tex]

[tex]\text{All possible factor of 20 are }\pm1,\pm2,\pm4,\pm5,\pm10,\pm20[/tex]

So, the all possible rational roots of the given polynomial are,

[tex]\pm1,\pm2, \pm3, \pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\pm\frac{1}{5},\pm\frac{3}{5},\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}[/tex]

Now, the rational roots of polynomial satisfy the given polynomial

[tex]f(-\frac{4}{5})=20(-\frac{4}{5})^4+(-\frac{4}{5})^3+8(-\frac{4}{5})^2-\frac{4}{5}-12=\frac{256}{625}\times 20-\frac{64}{125}+\frac{128}{125}-\frac{4}{5}-12[/tex]

[tex]=\frac{1024}{125}-\frac{64}{125}+\frac{128}{25}-\frac{4}{5}-12[/tex]

[tex]=\frac{960}{125}+\frac{128}{25}-\frac{4}{5}-12=12-12=0[/tex]

Hence, rational root.

[tex]f(\frac{3}{4})=20(\frac{3}{4})^4+(\frac{3}{4})^3+8(\frac{3}{4})^2+\frac{3}{4}-12=\frac{405}{64}+\frac{27}{64}+\frac{9}{2}+\frac{3}{4}-12=0[/tex]

rational root

[tex]f(1)=20(1)^4+(1)^3+8(1)^2+1-12=20+1+8-11=18\neq 0[/tex]

not a rational root.

hence, option 1 is correct

Answer:

The first option

Step-by-step explanation:

Right on edg:)

The car can drive 216 miles on 12 gallons.

The rate of change of position is called as displacement.

Now it is given that,

Distance traveled by car = 630 miles

Gas required to travel 630 miles = 35 gallons

Therefore, Distance travel n 1 gallon = Distance traveled by car / Gas required to travel 630 miles

⇒ Distance travel n 1 gallon = 630 / 35 miles

⇒ Distance travel n 1 gallon = 18 miles

Therefore, Distance travel n 12 gallon = 12 * Distance travel n 1 gallon

⇒ Distance travel n 12 gallon = 12 * 18 miles

⇒ Distance travel n 12 gallon = 216 miles.

Thus, the car can drive 216 miles on 12 gallons.

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Answer:

Parameter

Step-by-step explanation:

To factor -1/4 out of -1/2-5/4y, you can use the distributive property and simplify the expression to 1/8 + 5/16y.

To factor -1/4 out of -1/2-5/4y, you can use the distributive property. First, rewrite the expression as (-1/2) + (-5/4)y. Then, factor out -1/4: (-1/4)(-1/2) + (-1/4)(-5/4)y. This simplifies to 1/8 + 5/16y.

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In this exercise we have to perform the parameterization of the given function and we will have:

[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]

In there is some scalar function [tex]f(x, y)[/tex] such that:

[tex]\nabla f(x,y) = f(x, y) = x^2i+y^2j[/tex]

Then we want to find [tex]f[/tex] such that:

[tex]f'(x)= x^2= f(x,y) = \frac{x^3}{3} + g(y)\\f'(y)= y^2= f(x,y)= \frac{y^3}{3} + C\\f(x, y)= \frac{x^3}{3} +\frac{y^3}{3} + C[/tex]

So the vector filed [tex]f(x, y)[/tex] is conservative, which means the fundamental theorem applies; the line integral of [tex]f[/tex] along any path [tex]C[/tex] parameterized by some vector-valued function [tex]r(t)[/tex] over [tex]a\leq t\leq b[/tex] is given by:

[tex]\int\limits_C {f} \, dr\\\int\limits^t=a_t=b {f(r(t))} \, dt= f(r(b))-f(r(a))[/tex]

In the case:

[tex]\int\limits_C {f} \, dr = f(1, 2)-f(-1, 2)= 2/3[/tex]

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This problem requires calculation in multivariable calculus. Function f = ∇f when f(x, y) = x³/3 + y³/3. To calculate the line integral of ∇f · dr over the curve c, take the difference between the end and start points of the scalar function.

The subject for this calculation is multivariable calculus, dealing with operations such as the gradient (∇) and line integrals.

According to the given function f(x, y) in part (a), it can be seen that f = ∇f for f(x, y) = x³/3 + y³/3. The vector field of this function is conservative since it corresponds to the gradient of a scalar function.

For part (b), you will need to calculate the line integral of ∇f · dr over the curve c, which is y = 2x² from (-1, 2) to (1, 2). The result of this will be the difference between the end and start points of the scalar function: f(1, 2) - f(-1, 2).

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I believe your answer is B)160. But i'm not 100%. <3

The approximate cost per square foot for the tiles is $3.22 when rounded to the nearest cent.

To calculate the price per square foot of tiles, divide the total cost of the tiles by the total number of square feet. The total cost given is $40.21, and the total area is 12 1/2 square feet.

First, convert the mixed number to an improper fraction. 12 1/2 is the same as 25/2 square feet. Now, to find the cost per square foot, perform the following division:

Price per square foot = Total cost / Total area in square feet

Price per square foot = $40.21 / (25/2 square feet)

Multiply by the reciprocal:

Price per square foot = $40.21 / 1 x 2 / 25

Price per square foot = $40.21 x 2 / 25

Price per square foot = $80.42 / 25

Price per square foot = $3.2168

Therefore, the approximate cost per square foot for the tiles is $3.22 when rounded to the nearest cent.

The approximate price per square foot is $3.22.

To find the price per square foot, divide the total cost by the total square footage. In this case, the cost is $40.21 and the square footage is 12 1/2 square feet.

1. Cost per square foot = Total cost / Total square footage

2. Cost per square foot = $40.21 / 12.5 sq ft ≈ $3.22/sq ft

1. Convert 12 1/2 square feet to a decimal: 12.5 sq ft.

2. Divide the total cost by the total square footage: $40.21 / 12.5 sq ft ≈ $3.22/sq ft.

To calculate the cost per square foot, you need to divide the total cost by the total square footage. In this case, the total cost is $40.21, and the total square footage is 12 1/2 square feet. First, convert 12 1/2 square feet to a decimal, which is 12.5 sq ft. Then, divide the total cost by the total square footage:

Cost per square foot = $40.21 / 12.5 sq ft ≈ $3.22/sq ft.

Therefore, the approximate price per square foot is $3.22.

complete question

A box contains 12 1/2 square feet of tiles and costs $40.21 what is the approximate price per square foot

a, b, d, e true: empty set fits in its own container.

c, f false: bigger container doesn't hold smaller container.

Here are the truths and falsehoods of the statements:

a) ∅ ∈ {∅} - True. The empty set is an element of the set containing only the empty set.

b) ∅ ∈ {∅, {∅}} - True. The empty set is also an element of the set containing both the empty set and the set containing the empty set.

c) {∅} ∈ {∅} - False. The set containing the empty set is not an element of the set containing only the empty set.

d) {∅} ∈ {{∅}} - True. The set containing the empty set is an element of the set containing only the set containing the empty set.

e) {∅} ⊂ {∅, {∅}} - True. The set containing the empty set is a subset of the set containing both the empty set and the set containing the empty set.

f) {{∅}} ⊂ {∅, {∅}} - False. The set containing the set containing the empty set is not a subset of the set containing only the empty set and the set containing the empty set.




The probable question can be: Determine whether these statements are true or false.

a) ∅ ∈ {∅}

b) ∅ ∈ {∅, {∅}}

c) {∅} ∈ {∅}

d) {∅} ∈ {{∅}}

e) {∅} ⊂ {∅, {∅}}

f) {{∅}} ⊂ {∅, {∅}}

Answer:

paralell lines, i think.

Step-by-step explanation:

The only value of x where f(x) = 0 can be determined by looking at the function's behavior near the x-axis and using the Intermediate Value Theorem. In this case, the potential x value is -9.

The one and only value of x where f(x) = 0 is the point of the function that intersects the x-axis. Since the function is continuous, it must cross the x-axis at this point due to the Intermediate Value Theorem.

To determine the possible value of x, we look at the given function values at x = -10 and x = -8 and identify where the function changes sign from negative to positive. The value of x where this sign change happens is the possible x value.

From the provided choices, the only value that would make this sign change possible is -9. Therefore, -9 could be the value of x where f(x) = 0.

The expression that represents "the difference between -17 and -8" is the second option:

-17 - (-8)

This expression translates to "negative seventeen minus negative eight," which yields the difference between the two numbers.

This expression represents "negative eight minus negative seventeen." When you subtract a negative number, it's equivalent to adding its positive counterpart. So, this expression can be simplified to:

-8 + 17 = 9

However, this result does not represent the difference between -17 and -8; it gives the difference between 17 and 8.

This expression represents "negative seventeen minus eight." This directly calculates the difference between -17 and -8. It simplifies to:

-17 - 8 = -25

This indeed represents the difference between -17 and -8.

This expression represents "negative seventeen minus negative eight." Similar to the first expression, subtracting a negative number is the same as adding its positive counterpart. So, this simplifies to:

-17 + 8 = -9

This result does not represent the difference between -17 and -8; it gives the difference between -17 and 8.

This expression represents "negative eight plus negative seventeen." It simplifies to -25, which again represents the difference between -17 and -8.

Among these expressions, only the second expression, -17 - 8, correctly represents the difference between -17 and -8.

The decimal 0.8333 is equivalent to the fraction 2/3.

To convert the decimal 0.8333 to a fraction, follow these steps:

Step 1: Let x be the decimal value:

x = 0.8333

Step 2: Multiply both sides of the equation by 10000 to remove the decimal:

10000x = 8333.3

Step 3: Subtract x from both sides of the equation:

10000x - x = 8333.3 - 0.8333

9999x = 8332.4667

Step 4: Divide both sides of the equation by 9999 to isolate x:

x = 8332.4667 / 9999

Divide both the numerator and the denominator by their greatest common divisor, which is 3333:

x = (8332.4667 ÷ 3333) / (9999 ÷ 3333)

x = (2.49995) / 3

Now, the fraction as a whole number over the denominator:

x = 2 / 3

Therefore, the fraction is 2/3.

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