If x and y are two nonnegative numbers and the sum of twice the first ( x ) and three times the second ( y ) is 60, find x so that the product of the first and cube of the second is a maximum.

Answers

Answer 1

Final answer:

To find the value of x such that the product of the first number and the cube of the second number is a maximum, we need to solve the equations and find the critical points. From the critical points, we can determine the maximum value of the product.

Explanation:

To find the value of x such that the product of the first number and the cube of the second number is a maximum, we need to use the given condition that the sum of twice the first number and three times the second number is 60. Let's solve this step-by-step.

Let the first number be x and the second number be y.According to the given condition, 2x + 3y = 60.We need to maximize the product xy^3.To maximize the product, we can use the method of substitution.From step 2, we have 2x = 60 - 3y.Substituting the value of 2x from step 5 into the product xy^3, we get (60 - 3y)y^3.To find the maximum value, we need to find the critical points, which are the points where the derivative is equal to zero or does not exist.Find the derivative of (60 - 3y)y^3, which is -9y^2 + 180y - 3y^4.Set the derivative equal to zero and solve for y.From the values of y, find the corresponding values of x using the equation 2x = 60 - 3y.Compare the values of xy^3 for different y values to find the maximum.

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Related Questions

How to find the x intersept

Answers

The x intercept (spelled with a "c", not an "s") is any point where the function curve either touches or crosses the x axis. The x axis is the bold flat horizontal line. This axis is often labeled with an "x" in many diagrams.

For problem 22, the x intercept is the point (-4,0) as this is where the S shaped curve crosses through the horizontal x axis. See the attached diagram for a visual of what I'm talking about. Note the highlighted point in red.

The area of a rectangular plot 24 feet long and 16 feet wide will be doubled by adding an equal distance to each side of the plot. What is the distance added to each side?

Answers

24*16 = 384

384*2 = 768

24+d * 16+d =768

384 + 40d+d^2 = 768

d^2 + 40d-384 =0

(d+48) (d-8) = 0

d=-48, d=8 can't use a negative number so d = 8

check:

24+8=32, 16+8=24, 32x 24 = 768

so 8 feet is added to each side

The best approximation for the square root of 10 is.. A).5 B).100 C).3.1 D).25

Answers

The square root of 10 is 3.16 so the closet is c)3.1

Answer:

It is approximately 3.1

Step-by-step explanation:

Convert this percent into decimal form.

Answers

12 1/4 can be rewritten as 12.25

Then move the decimal place two places to the left to divide by 100 (percent means per 100),

Final answer: 0.1225

The sum of two numbers, x and y, is 12. The difference of x and two times y is 6. What are the values of x and y? x = 8, y = 4 x = 10, y = 2 x = 18, y = -6 x = 20, y = -8

Answers

x+y=12
x-2y=6

hmm
multiply 2nd equation by -1 and add to first

-x+2y=-6
x+y=12 +
0x+3y=6

3y=6
y=2
sub back
x+y=12
x+2=12
x=10

x=10, y=2

X=10 will be a answer

last question
help me pls c:

Answers

The answer is 200 cm squared. You can find this out by know the diameter of the circle which is 20 cm and by the photo the square's corners touch the circle so if a line when through the square diagonally you would get 20cm. Now you know the hypotenuse you can find the side lengths. You can use the 45 45 90 triangle where both side lengths equal x and the hypotenuse equals x* the square root of 2, but instead you would divide 20 by the square root of 2 and you would get your side lengths and then just multiply them

Cost to rent a bicycle is $5 plus $3 per hour for x hours what equation represents the total cost for c hours

Answers

The answer is c = $3x + $5
Hope I helped :)

In the diagram, ∠ABC = 90°. What is the radius of the circle?

A. 5.7 in
B. 16.5 in
C. 24.6 in
D. 12.3 in

Answers

I had that question on my test. its D. 12.3

Answer;

D. 12.3 in

Explanation and solution;We are give that ∠ABC = 90°; therefore a line from point A to point C is the diameter, this is because a diameter subtends an right angle to the circumference of the circle. Therefore; triangle ABC is a right-angled triangle, thus AC is the hypotenuse.

Using the Pythagoras theorem;

AC² = AB² + BC²

= 22.1² + 10.9²

= 607.22

AC = √ 607.22

= 24.64

But, since AC is the diameter and the radius is half of the diameter, then

Radius = 24.64/2

= 12.32

≈ 12.3 (to 1 decimal place)

Find PS if ABC=PQR, AD is an altitude of ABC, PS is an altitude of PQR, AD=12, AC=16 and PR=10

a. 7.5
b. 19.2
c. 4.62
d. 19.5

Answers

If ABC = PQR, then PS will also be similar to AD.

First you need to find how they're similar.

16/10 = 1.6

Then, multiply 12 (AD) by 1.6 to find PS.

12 * 1.6 = 19.2

The answer is B. 19.2

You roll two standard number cubes. What is the probability that the sum is odd, given than one of the number cubes shows a 1? Show your work.

Answers

When you roll 2 standard number cubes there are 36 different sums.
This probability is different because they give us a restriction with the "given that one f the cubes shows a one". The denominator becomes the number total number of rolls with a 1, and the numerator is the sum is odd.
P (odd sum is, given 1 number is a 1) = 6/11

Which theorem could Chelsea use to show the measure of angle KPR is equal to the measure of angle QRL?

Answers

KPS is not an angle, unless we're consented to construct otherwise.

B) Alternate Interior Angle Theorem

Which of the following points lie in the solution set to the following system of inequalities?

y ≤ x − 5
y ≥ −x − 4

(−5, 2)
(5, −2)
(−5, −2)
(5, 2)

Answers

The answer is (5,-2) because when you plug it into the first problem you would get 0 which is greater than or equal to -2 and for the second you would get -9 which is less than or equal to -2

Answer: Second option : (5, −2)

Step-by-step explanation: Given system of inequalities

y ≤ x − 5

y ≥ −x − 4

Plugging x=5 and y=-2 in first inequality

-2 ≤ 5 − 5

-2 ≤ 0 : True.

Plugging x=5 and y=-2 in second inequality

-2 ≥ −5 − 4

-2 ≥ -9 : Also true.

Point (5, −2) satisfied both of the given inequalities in the system.

Therefore, (5,-2) is correct option.

Three cities lie along a perfectly linear route: Springfield, Clarksville, and Allentown. Molly lives in Springfield and works in Allentown. She makes it to work using two gallons of gas in her car. Her friend Edgar lives in Allentown and works in Clarksville. It takes Edgar one gallon of gas to get to work. If Molly's car averages 26 miles per gallon, and Edgar's car averages 17 miles per gallon, about how far apart are Springfield and Clarksville?

Answers

When it says that the cities are routed linearly, it means that these cities are located right next to each other. From the left, the order is Springfield, Clarksville, then Allentown. You have details on the number of gallons and the mileage. To get the distance travelled, just multiply the mileage with the gallons, to cancel out gallons leaving you with the distance in miles.

Now, since Molly travelled from Springfield to Allentown, the total distance she covered is the distance Edward travelled and the unknown distance between Springfield and Clarksville, denoted as x in the picture. So, the equation is then

Molly's distance = x + Edgar's distance
26 miles/gal(2 gal) = x + 17 miles/gal(1 gal)
x = 35 miles

150 centimeters is equivalent to

Answers

150 centimeters is equivalent to 1 1/2 meters

The answer to this question is: 5.9 inches, 150 millimeters, 150,000 micrometers.

John is participating in a marathon that is 26.2 miles. His distance (d, in miles) depends on his time (t, in hours). Which is an appropriate range for this situation?

Answers

The appropriate range for John's distance in miles (d) during the marathon is A. [tex]$0 \leq d \leq 26.2$[/tex].

In a marathon, the distance (d) John covers depends on the time (t) he spends running. The distance is fixed at 26.2 miles, so we need to find the appropriate range for the time (t) he spends running.

Let's calculate John's average speed (v) during the marathon. We know that speed is given by:

[tex]\[ v = \frac{d}{t} \][/tex]

Where:

- v = average speed (miles per hour)

- d = distance covered (miles)

- t = time spent running (hours)

Given that John's distance is 26.2 miles, and the marathon covers this distance, we have:

[tex]\[ 26.2 = \frac{26.2}{t} \][/tex]

Solving for t:

[tex]\[ t = \frac{26.2}{26.2} = 1 \][/tex]

So, John takes 1 hour to cover the 26.2 miles.

Now, let's consider the maximum and minimum possible times for John to complete the marathon:

- Minimum time: John completes the marathon in the fastest time possible. Let's say this is 0. This implies he runs the marathon in 0 hours.

- Maximum time: John takes his time and completes the marathon at the slowest pace possible. Let's use the average time for a marathon, which is around 4.5 hours.

Thus, the appropriate range for the time (t) is:

[tex]\[ 0 \leq t \leq 4.5 \][/tex]

This corresponds to option C: [tex]$0 \leq t \leq 4.5$[/tex].

Complete Question:
John is participating in a marathon that is 26.2 miles. His distance (d, in miles) depends on his time (t, in hours) Which is an appropriate range for this situation?

A. [tex]$0 \leq d \leq 26.2$[/tex]

B. [tex]$0 \leq d \leq 4.5$[/tex]

c. [tex]$0 \leq t \leq 4.5$[/tex]

D. [tex]$0 \leq t \leq 26.2$[/tex]

Which answer is correct

Answers

diagonal is square root of A^2+b^2

so C is the correct answer

The local theater has three types of seats for broadway plays: main floor, balcony, and mezzanine. main floor tickets are $⁢59, balcony tickets are $⁢50, and mezzanine tickets are $⁢40. one particular night, sales totaled $73,785. there were 435 more main floor tickets sold than balcony and mezzanine tickets combined. the number of balcony tickets sold is 78 more than 33 times the number of mezzanine tickets sold. how many of each type of ticket were sold?

Answers

Final answer:

10 mezzanine tickets, 408 balcony tickets, and 853 main floor tickets were sold.

Explanation:

Let's solve this problem step-by-step to find out how many of each type of ticket were sold:

Let's assume that the number of mezzanine tickets sold is x. Therefore, the number of balcony tickets sold is 33x + 78 (since it is 78 more than 33 times the number of mezzanine tickets sold).

The number of main floor tickets sold is 435 + (33x + 78) + x = 435 + 34x + 78 = 34x + 513 (since there were 435 more main floor tickets sold than balcony and mezzanine tickets combined).

The total sales amount is $73,785.

Now, we can set up an equation to solve for x:

$40x + $50(33x + 78) + $59(34x + 513) = $73,785

Simplifying the equation:

40x + 1650x + 3900 + 59(34x + 513) = 73785

40x + 1650x + 3900 + 2006x + 30567 = 73785

3696x + 34467 = 73785

3696x = 39318

x = 39318/3696

x = 10.65

Since we can't have a fraction of a ticket, we can round down to the nearest whole number. So, x = 10.

Therefore, 10 mezzanine tickets were sold, 33x + 78 = 408 balcony tickets were sold, and 34x + 513 = 853 main floor tickets were sold.

Consider the relation y = 4|x + 2| + 7. What are the coordinates of the vertex?

(7, −2)
(2, 7)
(4, −2)
(−2, 7)

Answers

To answer, we set the expression inside the absolute value symbol (II) equal to 0. That is,
x + 2 = 0
Then, we solve for the value of x.
x + 2 - 2 = 0 - 2
Hence, the value of x from the equation above is equal to -2.

Then, substitute the value of x to the equation and drop the absolute value symbol.
y = 4(x + 2) + 7
Substituting,
y = 4(-2 + 2) + 7
y = 4(0) + 7
y = 7

Thus, the vertex of the absolute value equation is equal to (-2,7). The answer is the last choice.

The diffrence between a term and
coefficient

Answers

The difference between a term, and a coefficient is...

- A coefficient tells you how many times to multiply a variable.
· Ex: 4xy = Coefficient: 4

- A term is each of the numbers in an equation, ratio, etc.
· Ex: 5x + 7y + (4x - 4y) + 2 = Terms: 5x, 7y, (4x - 4y), and 2

I am 99.9% sure this is the correct answer, but if it isn't I am truly sorry, and please forgive me.

Conditional probabilities are based on some event occurring given that something else has already occurred?

Answers

The answer is true. A conditional probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A in the condition B", is usually written as P (A|B). The conditional probability of A given B is well-defined as the quotient of the probability of the joint of events A and B, and the probability of B.

The sum of twice a number and a larger number is 145. The difference between the numbers is 55. Let x represent the smaller number and y represent the larger number. Which equations represent the situation? Check all that apply.

A. x-y=55
B. 2(x+y)=145
C. 2x+y=145
D. y-x=55
E. y=x+55

Answers

C, D, and E all represent parts of the situation.

A does not, because x is actually 55 smaller than y, which is why both D and E work.

B does not because it's doubling the sum of x = y, but the description says to double x and then add y (which is why C works).

x and y are the numbers
y>x

sum (addition) of twice a number (2x) and larger number (y) is (=) 145
2x+y=145

difference between them is 55 (y is bigger so it would be x is subtracted from y)
y-x=55

so the equations are

2x+y=145 and
y-x=55

we could add x to both sides in the 2nd equation to obtain
y=x+55, but that doesn't really represent that the difference is 55, though they are the same euation

so I would say C and D only

Which shows 54^2 − 46^2 being evaluated using the difference of squares method?
54^2 − 46^2 = (2916 + 2116)(2916 − 2116) = 4,025,600
54^2 − 46^2 = (54 + 46)(54 − 46) = (100)(8) = 800
54^2 − 46^2 = 2916 − 2116 = 800
54^2 − 46^2 = (54 − 46)^2 = 8^2 = 64

Answers

A difference of squares is of the form (a^2-b^2) and always factors to:

(a^2-b^2)=(a+b)(a-b)

So in this case:

(54^2-46^2)=(54+46)(54-46)=(100)(8)=800

54^2 − 46^2 = (54 + 46)(54 − 46) = (100)(8) = 800

hope it helps

Michelle found a new violin on sale at 30% off. how much would she pay the cashier if it originally sells for $250 in a city that had no sales tax

Answers

100%-30% = 70%

70%=0.70

250*0.70 = 175

she would pay $175

can you help me????????

Answers

check the picture below.

now, if AC is that much, recall, BD is the midsegment, thus AB = BC, so AB is just one of the equal halves of AC, so, AB is AC/2.

[tex] \frac{x}{5}+\frac{3x}{15}=\frac{2x}{3} } [/tex]+2 Answer plz math help

Answers

Multiply all terms by 15
3x+3x= 10x+30

Combine like terms
6x=10x+30

Subtract 10x from both sides
-4x=30

Divide both sides by -4
x= -7.5

Final answer: -7.5

Given f(x) = x2 + 4x − 1 and g(x) = 5x − 7, identify (fg)(x).

Answers

f(x) = x^2 + 4x − 1 and g(x) = 5x − 7
(fg)(x) = (x^2 + 4x − 1)(5x − 7)
(fg)(x) = 5x^3 + 20x^2 - 5x - 7x^2 - 28x + 7
(fg)(x) = 5x^3 + 13x^2 - 33x + 7

answer is C. third choice

(fg)(x) = 5x^3 + 13x^2 - 33x + 7

The product of the functions[tex]$ f(x) = x^2 + 4x - 1 $ and $ g(x) = 5x - 7 $ is $ 5x^3 + 13x^2 - 33x + 7 $.[/tex]

The correct answer is indeed [tex]{C} \),[/tex] which matches [tex]$ 5x^3 + 13x^2 - 33x + 7 $.[/tex]

To find the product[tex]$ (f \cdot g)(x) $[/tex], where [tex]$ f(x) = x^2 + 4x - 1 $[/tex] and [tex]$ g(x) = 5x - 7 $,[/tex]we need to perform the multiplication of these two functions.

Start by expanding [tex]$ f(x) \cdot g(x) $:[/tex]

1. Write down ( f(x) ):

[tex]\[ f(x) = x^2 + 4x - 1 \][/tex]

2. Write down ( g(x) ):

[tex]\[ g(x) = 5x - 7 \][/tex]

3. Perform the multiplication [tex]$ f(x) \cdot g(x) $[/tex]:

[tex]\[ f(x) \cdot g(x) = (x^2 + 4x - 1)(5x - 7) \][/tex]

4. Distribute [tex]$ x^2 + 4x - 1 $[/tex] across ( 5x - 7 ):

[tex]\[ f(x) \cdot g(x) = x^2 \cdot (5x - 7) + 4x \cdot (5x - 7) - 1 \cdot (5x - 7) \][/tex]

5. Perform the multiplications:

[tex]\[ x^2 \cdot (5x - 7) = 5x^3 - 7x^2 \][/tex]

[tex]\[ 4x \cdot (5x - 7) = 20x^2 - 28x \][/tex]

[tex]\[ -1 \cdot (5x - 7) = -5x + 7 \][/tex]

6. Combine all the terms:

[tex]\[ f(x) \cdot g(x) = 5x^3 - 7x^2 + 20x^2 - 28x - 5x + 7 \][/tex]

7. Simplify by combining like terms:

[tex]\[ f(x) \cdot g(x) = 5x^3 + (20x^2 - 7x^2) + (-28x - 5x) + 7 \][/tex]

[tex]\[ f(x) \cdot g(x) = 5x^3 + 13x^2 - 33x + 7 \][/tex]

Therefore, the product [tex]$ (f \cdot g)(x) $ is $ 5x^3 + 13x^2 - 33x + 7 $.[/tex]

The correct answer is indeed [tex]{C} \),[/tex] which matches [tex]$ 5x^3 + 13x^2 - 33x + 7 $.[/tex]

What is the solution of sqrt 2x + 4 = 16 ? x = 6 x = 72 x = 126 no solution

Answers

Answer: Third option is correct.

Step-by-step explanation:

Since we have given that

[tex]\sqrt{2x+4}=16[/tex]

We need to find the value of 'x'.

First we squaring the both sides:

[tex](\sqrt{2x+4})^2=16^2\\\\2x+4=256\\\\2x=256-4\\\\2x=252\\\\x=\dfrac{252}{2}\\\\x=126[/tex]

Hence, the value of x is 126.

Therefore, Third option is correct.

Answer:

C on Edge

Step-by-step explanation:

Received a 100% on the quiz.

Find the surface area of a sphere with a volume of 36π in3.
SHOW WORK
will give medals and mark brainliest

its 113.10 inches squared

Answers

First use known volume to solve for radius:
[tex]V=\frac{4}{3}\pi r^{3}[/tex]
[tex]r=\sqrt[3]{\frac{3V}{4\pi}}[/tex]
[tex]r=\sqrt[3]{\frac{3(36\pi)}{4\pi}}[/tex]
[tex]r=\sqrt[3]{27}[/tex]
[tex]r=3[/tex]

Then use the radius to get the surface area:
[tex]A_{surf}=4 \pi r^{2}[/tex]
[tex]A_{surf}=4 \pi 3^{2}[/tex]
[tex]A_{surf}=4 \pi 9[/tex]
[tex]A_{surf}=36\pi\approx113.09[/tex]

Final answer:

To find the surface area of a sphere given the volume, first solve for the radius using the volume formula (4/3πr³). In this case, the radius is 3. Then, use the surface area formula (4πr²), which gives the surface area as 36π square inches, or an approximate value of 113.10 square inches.

Explanation:

The volume of a sphere is given by the formula V = 4/3 * π * r³. First, we need to find the radius of the sphere. Set the volume of the sphere (36π in³) equal to the volume formula and solve for r:

36π = 4/3πr³

From this, we find that r = 3. Now, we use the radius to find the surface area with the formula A = 4πr²:

A = 4π * 3² = 36π in²

So, the surface area of a sphere with a volume of 36π in³ is 36π square inches or approximately 113.10 square inches.

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(05.02)
What is the y-intercept of the line shown?

−1
0
0.5
1

Answers

The y-intercept of the line is -1.

The y-intercept is -1!

What is the answer? (Tip- to undo multiply both sides by 4/7)

x|4/7 = 28

Answers

x / |4/7| = 28

Multiply by |4/7|

x = |4/7| x 28

Ignore the absolute for a second and note 4/7 x 28 is 16 because...
28 / 7 = 4
4 x 4 = 16

x = |16|

[tex]\frac{x}{\frac{4}{7}}=28\\\\(\frac{x}{\frac{4}{7}})*\frac{4}{7}=28*\frac{4}{7}\\\\x=\frac{28*4}{7}=\frac{4*7*4}{7}=\frac{16}{1}\\\\\\x=16[/tex]

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