All categories

2 months ago

Maximum value of x3y3 + 3 x*y when x+y = 8.

Mathematics

2 answers:

Inessa [12.5K]2 months ago

8 0

Response:

The outcome is 4144.

Detailed explanation:

We need to determine the peak value of f(x)= $x^{3}y^{3}+3xy$ when $x+y=8$

We can represent $x+y=8$ as $y=8-x$

Plugging the value of y

$f(x)=x^{3}(8-x)^{3} +3x(8-x)$

= $3x^{2}(8-x)^{2}[-x+8-x]+3[-x+8-x]$

= $3(8-2x)[x^{2}(8-x)^{2}+1]$

To find the maximum, we'll set the equation to 0.

Thus, we find:

$8-2x=0$ => x = 4

And since $y=8-x$

$y =8-4= 4$ > y = 4

Hence, we will substitute these values into the equation to ascertain the maximum value.

= $x^{3}y^{3}+3xy$

= $xy(x^{2}y^{2}+3)$

= $4\times4((16\times16) +3)$

= $16(259)$ = 4144

Svet_ta [12.7K]2 months ago

6 0

The highest value of x³y³ + 3 x.y, given x + y = 8, is 4144. For clarity, let's revisit foundational concepts related to derivatives:

Let's get started on solving this!

First, we have the equation: with x + y set to 8, which gives us y = 8 - x. Now, we can proceed to analyze the equation further.

You might be interested in

Given that Ray E B bisects ∠CEA, which statements must be true? Select three options. m∠CEA = 90° m∠CEF = m∠CEA + m∠BEF m∠CEB =

tester [12383]

Answer:

Attached is the question in consideration.

$m\angle CEA =90 \ (deg)$

$m\angle BEF=135\ (deg)$

$\angle CEF$ forms a straight line.

$\angle AEF$ depicts a right triangle.

The options $1,4,5,6$ represent the correct answers.

Step-by-step explanation:

⇒ Given that $\ ray\ AE$ is ⊥ $FEC$ it constitutes a right triangle, leading to $m\angle CEA =90\ (deg)$ .

⇒ The measure for $\angle BEF =135\ (deg)$ equals $\angle BEF =\angle AEB +\angle AEF = (45+90)=135\ (deg)$ as $\angle AEB$ bisects $\angle AEC$ , implying that $\angle AEB$ is half of $\angle AEC$ , thus $\angle AEB = 45\ (deg)$ .

⇒ $\angle CEF$ represents a straight line, as the angle measures across it yield $180\ (deg)$ .

⇒ The angle measure for $\angle AEF = 90\ (deg)$ is derived from the linear pair concept.

Since $\angle CEA + \angle AEF = 180\ (deg)$ , inserting the values of $m\angle CEA =90\ (deg)$ leads to $\angle AEF = 90\ (deg)$ .

The other two options are incorrect as:

$m\angle CEF=m\angle CEA + m\angle BEF = (90+135)=225$

it surpasses $180\ (deg)$ while $\angle CEF$ is a

straight line.

Also, $m\angle CEB=2(m\angle CEA)$ is inaccurate.

As $\angle CEA = 90\ (deg)$ and $\angle CEB=45\ (deg)$

Thus, we have a total $4$ valid answers.

The confirmed options are $1,4,5,6$ .

5 0

2 months ago

Read 2 more answers

Two birds sit at the top of two different trees. The distance between the first bird and a birdwatcher on the ground is 32 feet.

Svet_ta [12734]

Response:

Step-by-step breakdown:

When you sketch that diagram (great description, by the way!), what you essentially have is a right triangle with a base of 32 and a hypotenuse of 45. The right angle resides at one of the base's ends, and x represents the vertex angle. We must find this vertex angle first to determine the angle of depression from the second bird to the watcher. The side measuring 32 is opposite to angle x, with 45 being the hypotenuse; hence, the trigonometric relation we need is sine:

$sin(x)=\frac{32}{45}$ and

sin(x) =.711111111

Go to your calculator, press the 2nd key followed by the sin key, and your display will show:

$sin^{-1}($

then, enter in your decimal.711111111 and hit equals. You should arrive at an angle of 45.325. That angle is x. However, that's not the angle of depression. The angle of depression is the complementary angle to x.

Angle of depression = 90 - angle x and

Angle of depression = 90 - 45.325, resulting in

Angle of depression = 44.67 or 44.7 degrees.

8 0

1 month ago

Find a solution to the linear equation 13x + y = 26 by filling in the boxes with a valid value of x and y.

Inessa [12570]

Answer:

MOM

Step-by-step explanation:

6 0

2 months ago

Ice Cream Scoops - In shops with lots of ice-cream flavors there are many different flavor combinations, even with only a 2-scoo

PIT_PIT [12445]

Answer:

The universal formula applicable for determining the necessary combinations of n ice cream flavors.

$N_n = \frac{n(n+1)}{2}$

For 10 flavors of ice cream

$N_{10} = \frac{10(10+1)}{2}$

$N_{10} = 55$

Step-by-step explanation:

Let's denote V = Vanilla, C = Chocolate, S = Strawberry, P = Pineapple, B = Berry

For 2 flavors of ice cream:

V, C

{V/V, V/C, C/C}

3 possible combinations

For 3 flavors of ice cream:

V, C, S

{V/V, V/C, V/S, C/S, C/C, S/S}

6 possible combinations

For 4 flavors of ice cream:

V, C, S, P

{V/V, V/C, V/S, V/P, C/S, C/P, S/P, C/C, S/S, P/P}

10 possible combinations

For 5 flavors of ice cream:

V, C, S, P, B

{V/V, V/C, V/S, V/P, V/B, C/S, C/P, C/B, S/P, S/B, P/B, C/C, S/S, P/P, B/B }

15 possible combinations

Thus, we derive a sequence of

3, 6, 10, 15...

This sequence is recognized as the Triangular number series

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15

The general formula for this sequence is

Utilizing this formula allows us to predict the number of 2-scoop combinations from 10 flavors.

$N_n = \frac{n(n+1)}{2}$

$N_{10} = \frac{10(10+1)}{2}$

$N_{10} = 55$

Consequently, there are 55 varying combinations for a 2-scoop cone when selecting from 10 flavors.

5 0

1 month ago