Question 3 B0/5 pts 100 Details If the eighth term of a geometric sequence is 81920, and the eleventh term of an geometric sequence is 5242880 its first term a and its common ratio r = Question Help:

Therefore, the maximum value of f(x,y) inside the triangle is 80/9, which occurs along the line y = (-1/2)x + 4 at the point (8/3, 10/3), and the minimum value is -32, which occurs at the critical point (-8,4).

To find the maximum and minimum values of f(x,y) = xy on the region inside the triangle whose vertices are (6,2), (0,3), and (6,0), we use the method of Lagrange multipliers.

First, we need to find the critical points of f(x,y) subject to the constraint that (x,y) lies inside the triangle. We can express this constraint using the equations of the lines that form the sides of the triangle:

y = (-1/2)x + 4

y = (3/2)x

y = 0

Next, we set up the Lagrange multiplier equation:

∇f = λ∇g

where g(x,y) is the equation of the constraint, i.e., the triangle.

We have:

f(x,y) = xy

∇f = <y, x>

g(x,y) = y - (-1/2)x - 4 = 0

∇g = <-1/2, 1>

Setting ∇f = λ∇g, we get:

y = (-1/2)λ

x = λ

Substituting these into the constraint equation, we get:

(-1/2)λ - 4 = 0

Solving for λ, we get:

λ = -8

Substituting this into y = (-1/2)λ and x = λ, we get:

x = -8 and y = 4

Therefore, the only critical point of f(x,y) inside the triangle is (-8,4).

Next, we need to check the values of f(x,y) at the vertices and along the sides of the triangle.

At the vertices:

f(6,2) = 12

f(0,3) = 0

f(6,0) = 0

Along the line y = (3/2)x:

f(x, (3/2)x) = (3/2)x^2

Using the vertex (6,2) and the x-intercept (4/3, 2), we can see that the maximum value of (3/2)x^2 on this line occurs at x = 4. Therefore, the maximum value of f(x,y) along this line is:

f(4,6) = 24

Along the line y = (-1/2)x + 4:

f(x, (-1/2)x + 4) = (-1/2)x^2 + 4x

Using the vertex (6,2) and the x-intercept (8,0), we can see that the maximum value of (-1/2)x^2 + 4x on this line occurs at x = 8/3. Therefore, the maximum value of f(x,y) along this line is:

f(8/3,10/3) = 80/9

Finally, we need to check the values of f(x,y) at the critical point (-8,4). We have:

f(-8,4) = -32

For more similar questions on minimum function

brainly.com/question/10439235

#SPJ11

A function is a rule that assingns each value of independent variable to exactly one value of the dependent variable.

A function is a mathematical concept that relates two sets of values, known as the domain and the range. The domain is the set of independent variables, while the range is the set of dependent variables. A function is a rule that assigns to each value in the domain exactly one value in the range.

For example, if we have a function f(x) = 2x + 3, the domain would be any possible value of x, and the range would be any possible value of 2x + 3. So if we put x = 2, then f(x) = 2(2) + 3 = 7. Therefore, the function assigns the value of 7 to the value of 2 in the domain.

Functions are used in various branches of mathematics, science, and engineering to model and analyze relationships between two or more variables. They are an important concept in calculus, where they are used to study rates of change and optimization problems.

Learn more about function at https://brainly.com/question/21084589

#SPJ11

Step-by-step explanation:

use Pythagorean theorem to find the height

c = 38 ft

a = 14 ft

a² + b² = c²

(14)² + b² = (38)²

b² = 1444 - 196

b² = 1248

b = √1248

b = √16 × 78

b = 4√78 feet

The red car is actually traveling along the road at a speed of approximately 26.67 feet per second.

We can start by drawing a diagram of the situation:

     P (police car)

      |

      |

      |

40    |    S (stop sign)

-------|--------------------

      | 160

      |    R (red car)

Let's use the Pythagorean theorem to find the distance between the police car and the red car at any time t:

d(t)² = 40² + (160 - v*t)²

Where v is the speed of the red car in feet per second, and d(t) is the distance between the police car and the red car at time t.

We want to find how fast the red car is actually traveling along the road, so we need to find v when the distance between the police car and the red car is decreasing at a rate of 100 feet per second:

d'(t) = -100

We can take the derivative of the equation for d(t) with respect to time:

2d(t)d'(t) = 0 + 2(160 - v*t)(-v)

Simplifying and plugging in d'(t) = -100, we get:

-4000 + 2v²t = -100(160 - vt)

Solving for v, we get:

v = 80/3 ≈ 26.67 feet per second

Learn more about Pythagorean theorem

https://brainly.com/question/14930619

#SPJ4

Determine whether the two figures are similar: D. No, the two figures are not similar.

In Geometry, two (2) quadrilaterals are similar when the ratio of their corresponding sides are equal in magnitude and their corresponding angles are congruent.

Additionally, two (2) geometric figures such as quadrilaterals are considered to be congruent only when their corresponding side lengths are congruent (proportional) and the magnitude of their angles are congruent;

Ratio = 12/8 = 10/6 = 10/6

Ratio = 3/2 ≠ 5/3 = 5/3

In conclusion, the two figures are not similar because the ratio of their corresponding sides is not proportional.

Read more on quadrilateral here: brainly.com/question/6561516

#SPJ1

Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

? equals 80

Step-by-step explanation:

The value of y that satisfies the equation [tex]\int x^5 dx = \int x^y dx[/tex] is y = -1.

We know that the indefinite integral of x^5 dx is (1/6) x^6 + C, where C is

the constant of integration. Therefore:

[tex]\int x^5 dx = (1/6) x^6 + C[/tex]

We want to find y such that [tex]\int x^5 dx = \int x^y dx[/tex]. Using the power rule of integration, the indefinite integral of [tex]x^y[/tex] dx is [tex](1/(y+1)) x^{(y+1)} + C[/tex], where C is the constant of integration. Therefore:

[tex]\int x^y dx = (1/(y+1)) x^{(y+1)} + C[/tex]

For these two integrals to be equal, we need:

[tex](1/6) x^6 + C = (1/(y+1)) x^{(y+1) } + C[/tex]

Subtracting C from both sides, we get:

[tex](1/6) x^6 = (1/(y+1)) x^{(y+1)}[/tex]

Multiplying both sides by (y+1), we get:

[tex](1/6) x^6 (y+1) = x^{(y+1)}[/tex]

Now, we can equate the powers of x on both sides:

[tex]x^6 (y+1) = x^{(y+1)}[/tex]

Using the fact that[tex]x^a \times x^b = x^{(a+b)}[/tex], we can simplify the left-hand side:

[tex]x^(6(y+1)) = x^{(y+1)}[/tex]

Now, we can equate the exponents on both sides:

6(y+1) = y+1

Simplifying, we get:

6y + 6 = y + 1

5y = -5

y = -1

for such more question on integral

https://brainly.com/question/22008756

#SPJ11

Line x is perpendicular to line y. Line z crosses lines x and y. Only statements 3 and 4 are true.

∠6 = ∠8 is not true because they both lie on the same plane and makes an angle of 180° and can never be true. ∠6 = ∠1 is also not true because ∠1 is clearly obtuse angle and ∠6 is clearly acute angle so they cannot be equal. Hence, statement a and b are false.

∠7 = ∠3 is always true because they are corresponding angles and corresponding angles are always equal. m∠2 + m∠4 = 180° is also true because they lie on same plane and have common vertex and hence, they are supplementary angles and make a sum of 180°.  Hence, statement 3 and 4 is always true.

To know more about corresponding angles:

https://brainly.com/question/12521848

#SPJ1

The relative frequency for selecting a teen out of all the people who like chocolate is calculated by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

To find the relative frequency for selecting a teen out of all the people who like chocolate, you need to follow these steps:

Step 1: Determine the total number of people who like chocolate (let's call this T).
Step 2: Determine the number of teens who like chocolate (let's call this N).
Step 3: Calculate the relative frequency by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

To learn more about relative frequency

https://brainly.com/question/18834138

#SPJ11

Answer:

1st choice:  1/4(y - 10) = 2/3

Step-by-step explanation:

the "variable" is y

"is" means "=" (equals sign)

one fourth =  1/4

"difference of" means subtract

Answer:  1/4(y - 10) = 2/3

The sample space for the number solid is {1, 2, 3, 4, 5, 6} and the probability of rolling 1 is 1/6.

The sample space refers to the set of all possible outcomes of an experiment, while a sample value is a specific outcome in the sample space.

For the given 8-sided solid, the sample space would be {1, 2, 3, 4, 5, 6}, as the faces labeled "skip" are not counted as sample values.

Now, let's calculate the probability of rolling a 1. Probability is the likelihood of a particular outcome occurring, which can be calculated by dividing the number of successful outcomes (in this case, rolling a 1) by the total number of possible outcomes.

The total number of possible outcomes is 6 (1, 2, 3, 4, 5, and 6). There is only one successful outcome: rolling a 1.

So, the probability of rolling a 1 is:

P(1) = (Number of successful outcomes) / (Total number of possible outcomes)
P(1) = 1 / 6

Thus, the probability of rolling a 1 on this 8-sided solid is 1/6 or approximately 0.1667, or 16.67%.\

To learn more about sample space: https://brainly.com/question/2117233

#SPJ11

To find the total volume of the silo, we need to add the volume of the hemisphere on top to the volume of the cylinder at the bottom.

The volume of a hemisphere is given by:

V_hemi = (2/3)πr^3

where r is the radius of the hemisphere.

Substituting r = 9ft, we get:

V_hemi = (2/3)π(9ft)^3

= 1521π ft^3

The volume of a cylinder is given by:

V_cyl = πr^2h

where r is the radius of the cylinder and h is its height.

Substituting r = 9ft and h = 35ft, we get:

V_cyl = π(9ft)^2(35ft)

= 2673π ft^3

Therefore, the total volume of the silo is:

V_silo = V_hemi + V_cyl

= 1521π + 2673π

= 4194π ft^3

≈ 13160 ft^3

Rounding to the nearest cubic foot, the silo can hold approximately 13160 cubic feet of grain.

When the points are rotated 90 degrees clockwise about the origin, the result is:

To rotate a point 90 degrees clockwise about the origin, you can use the following rule: (x, y) becomes (y, -x). Let's apply this rule to the given points:

I - (3, 1)

Rotated I: (1, -3)

J - (5, -1)

Rotated J: (-1, -5)

H - (3, -3)

Rotated H: (-3, -3)

So, after a 90-degree clockwise rotation about the origin, the new coordinates of the points are:

I: (1, -3)

J: (-1, -5)

H: (-3, -3)

Find out more on rotation at https://brainly.com/question/30316631

#SPJ1

The functions that represents the cost are

(a) y = 8800, (b) y = 1900 + 4/7x and (c) y = 4800, x ≤ 150;  y = 1200 + 24x x > 150

From the question, we have the following parameters that can be used in our computation:

The graph

The function (a) is a horizontal line that passes through y = 8800

So, the function is

y = 8800

The function (b) is a linear function that passes through

(0, 1900) and (175, 2000)

So, the function is

y = 1900 + 4/7x

The function c is a piecewise function with the following properties

So, the function is

y = 4800, x ≤ 150

y = 1200 + 24x x > 150

Read more about functions at

https://brainly.com/question/27915724

#SPJ1

After 2 hours, approximately 4.38 grams of Francium remain from the 480-gram sample.

What is Algebraic expression ?

An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. It may contain one or more terms, with each term separated by a plus or minus sign. Algebraic expressions are used in algebra to represent mathematical relationships and formulas.

a) To write an exponential function that models the mass of Francium remaining after t minutes, we can use the formula:

N = N0 * [tex](1/2)^{(t / t1/2)}[/tex]

where N is the amount remaining after time t, N0 is the initial amount, t1/2 is the half-life, and (t/t1/2) means raised to the power of t/t1/2.

In this case, the initial amount is 480 grams, the half-life is 22 minutes, and we want to find the amount remaining after t minutes. Therefore, the exponential function that models the mass of Francium remaining after t minutes is:

N = 480 * [tex](1/2)^{t/22}[/tex]

b) 2 hours is equal to 120 minutes. To find how many grams of Francium remain after 2 hours, we can substitute t = 120 into the exponential function we found in part a):

N = 480 *[tex](1/2)^{ (120 / 22) }[/tex] ≈ 4.38 grams

Therefore, after 2 hours, approximately 4.38 grams of Francium remain from the 480-gram sample.

To learn more about Algebraic expression from given link.

brainly.com/question/31238826

#SPJ1

The wavelength of the light is approximately 0.687 μm.

Using the single slit diffraction formula, we have:

sin θ = (mλ) / w

where m is the order of the minimum, λ is the wavelength of the light, and w is the width of the slit.

We can rearrange the formula to solve for the wavelength of the light:

λ = (w sin θ) / m

Plugging in the given values, we get:

λ = (1.2 μm)(sin 32.3°) / 1 = 0.687 μm

Therefore, the wavelength of the light is approximately 0.687 μm.

The wavelength is the distance between two consecutive peaks or troughs in a wave. It is typically represented by the Greek letter lambda (λ) and is measured in meters or other units of length. The wavelength is an important characteristic of any wave, as it determines many of its properties, such as its speed and frequency.

Learn more about wavelength

https://brainly.com/question/31143857

#SPJ4

Sea "l" la medida del largo del terreno y "a" la medida del ancho del terreno.

De acuerdo con el problema, el ancho es la mitad del largo, es decir, a = l/2.

El perímetro de un rectángulo se calcula sumando las longitudes de sus cuatro lados, por lo que en este caso:

Perímetro = 2l + 2a = 2l + 2(l/2) = 3l

Sabemos que el perímetro es de 105 metros, entonces:

3l = 105

l = 105/3 = 35

Por lo tanto, el largo del terreno es 35 metros. Y, como el ancho es la mitad del largo, entonces:

a = l/2 = 35/2 = 17.5

Por lo tanto, el ancho del terreno es de 17.5 metros.

En resumen, las medidas del terreno son 35 metros de largo y 17.5 metros de ancho.

To know more about del terreno refer here:

https://brainly.com/question/31136130

#SPJ11

The point with the least absolute value on the number line is always the point zero.

The absolute value of a number is the distance that number is from zero on the number line. Therefore, the point on the number line with the least absolute value is the point closest to zero. This point is located at zero itself, as it is the point on the number line that is equidistant from both the positive and negative numbers.

To further explain, consider the following examples:

- The point 3 is 3 units away from zero, but the point -3 is also 3 units away from zero.
- The point 5 is 5 units away from zero, but the point -5 is also 5 units away from zero.
- The point 0 is 0 units away from zero, making it the point with the least absolute value on the number line.

In conclusion, the point with the least absolute value on the number line is always the point zero.

To know more about absolute value, visit:

https://brainly.com/question/1301718#

#SPJ11

The critical points and their classifications are:

(0, 1) - saddle point

(0, -1) - saddle point

(6, 1) - local minimum

(6, -1) - local minimum

To find the critical points of the function f(x, y) = x³ + y³ – 9x² – 3y - 6, we need to find the points where the partial derivatives of f with respect to x and y are zero.

∂f/∂x = 3x² - 18x = 3x(x - 6)

∂f/∂y = 3y² - 3 = 3(y² - 1)

Setting these partial derivatives equal to zero and solving for x and y, we get:

x = 0 or x = 6

y = ±1

So the critical points are (0, 1), (0, -1), (6, 1), and (6, -1).

To classify each critical point, we need to compute the second partial derivatives of f:

∂²f/∂x² = 6x - 18

∂²f/∂y² = 6y

∂²f/∂x∂y = 0

At (0, 1):

∂²f/∂x² = -18 < 0 (concave down)

∂²f/∂y² = 6 > 0 (concave up)

So (0, 1) is a saddle point.

At (0, -1):

∂²f/∂x² = -18 < 0 (concave down)

∂²f/∂y² = 6 > 0 (concave up)

So (0, -1) is a saddle point.

At (6, 1):

∂²f/∂x² = 18 > 0 (concave up)

∂²f/∂y² = 6 > 0 (concave up)

So (6, 1) is a local minimum.

At (6, -1):

∂²f/∂x² = 18 > 0 (concave up)

∂²f/∂y² = 6 > 0 (concave up)

So (6, -1) is a local minimum.

For more such questions on Critical points.

https://brainly.com/question/30889798#

#SPJ11

Step-by-step explanation:

Pick any of the two points...I'll use the first two

calculate slope:     m  = ( y1-y2) / (x1-x2) =  (-14 - -5) / (-2 -1) = -9/-3 = 3

   equation of a line in slope intercept form is y = mx+ b

      so now you have    y = 3x + b  

             sub in any of the x,y  points given (8,16)  to calculate 'b'

                             16 = 3 (8) + b

                                  b = -8

so your first line is     y = 3x - 8

In a similar fashion, for the second one   m = - 5/8    and b = 2

           y = -5/8 x + 2

Answer: Nori

Step-by-step explanation:

2% of 5000 = 100

1% of 1000 = 10

If c represents the cost per pound for the plums, the cost per pound for the plums is $0.90.

First, we need to determine the total cost of the plums before the coupon was applied.

4 1/5 pounds can be written as a mixed number:

4 1/5 = 21/5

So, the total cost of the plums without the coupon can be found by multiplying the cost per pound (c) by 21/5:

Total cost = c * 21/5

Now we can create an equation to represent the total cost after the coupon was applied:

Total cost - coupon = $3.23

Substituting the expression for total cost:

c * 21/5 - 0.55 = 3.23

To solve for c, we can start by adding 0.55 to both sides:

c * 21/5 = 3.78

Then, we can isolate c by multiplying both sides by the reciprocal of 21/5:

c = 3.78 / (21/5)

c = 0.90

Therefore, the cost per pound for the plums is $0.90.

More on cost: https://brainly.com/question/13001095

#SPJ11

Answer:

we 1st can get the weight of rat by

1 rat and 1 cat + 1 dog and rat = 30

2 rat + 1 cat + 1 dog = 30

Then 1 rat and cat measure 24 so

2 rat + 24 =30

2 rat + 24 =30 1 rat = 3 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 10

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 10

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg 1 dog + 3kg = 20 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg 1 dog + 3kg = 20 kg 1 dog = 17kg

so we get the weight of each now we r going to sum them 1 rat + 1 cat + 1 dog = x

1 rat + 1 cat + 1 dog = x 3 kg + 7 kg + 17 kg = x

1 rat + 1 cat + 1 dog = x 3 kg + 7 kg + 17 kg = x 27 kg = x ..... is the mass of 3 of them

There are 42 yellow skittles in the box.

Based on the given information, we know that the ratio of green skittles to yellow skittles is 5:7. This means that for every 5 green skittles, there are 7 yellow skittles.

To find out how many yellow skittles are in the box, we need to know how many sets of 5 green skittles there are. We can do this by dividing the total number of skittles in the box (30) by 5 (since there are 5 green skittles for every set).
30 ÷ 5 = 6

This means there are 6 sets of 5 green skittles in the box.

Now we can use the ratio of 5:7 to find out how many yellow skittles there are in each set:
5 green skittles : 7 yellow skittles

Since there are 7 yellow skittles in each set, we can find the total number of yellow skittles by multiplying 7 by the number of sets (6):
7 x 6 = 42

There are 42 yellow skittles in the box.

To know more about "Ratio" refer here:

https://brainly.com/question/17223098#

#SPJ11

The derivative of the vector function is: r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k

We are given a vector function r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k, and we need to find its derivative r'(t).

The derivative of a vector function is obtained by differentiating each component of the vector function separately.

So, let's differentiate each component:

r(t) = ln(7-t^2)i + sqrt(13+t)j – 4e^(9t)k

r'(t) = (d/dt) ln(7-t^2) i + (d/dt) sqrt(13+t) j - (d/dt) 4e^(9t) k

Using the chain rule of differentiation, we have:

r'(t) = -2t/(7-t^2) i + 1/(2sqrt(13+t)) j - 36e^(9t) k

Therefore, the derivative of the vector function is:

r'(t) = (-2t/(7-t^2)) i + (1/(2sqrt(13+t))) j - 36e^(9t) k

To learn more about vector function visit: https://brainly.com/question/8005711

#SPJ11

I have gotten 113

Step-by-step explanation:

The right angle triangle has three angles and the value of its two angles are 90°and30°.We need to find the third angle which is the sum of 90°and 30°subtract from 180°=60°.60° is vertically opposite to angle C. We'll be having a quadrilateral whose angles add up to 360° .Subtract the sum of the three angles from 360° and you'll get 113°

The correct option is option 3.

Based on the given integral, region D is in the first quadrant, and its boundaries are not explicitly given. However, we can deduce the boundaries of D by looking at the integrand. Since the integrand is f(x,y), we can see that we are integrating over the entire region D, which means that D must be the rectangle that contains all the other regions mentioned in the options.

Therefore, option 1 is not correct, as D is not bounded from the right by x=2, but rather extends indefinitely to the right. Option 2 is also not correct, as D extends beyond the line y=x. Option 4 is not correct either, as D is not bounded above by y=x^2, but rather extends beyond it. Options 5 and 6 are also not correct, as D extends beyond the lines y=0 and y=2.

Therefore, the correct option is option 3, which states that D is the region in the first quadrant that is bounded from the left by the line x=0. This is correct, as D extends indefinitely to the right, and is bounded from the left by x=0.

Learn more about the boundaries of the region.

brainly.com/question/29674302

#SPJ11

To estimate your paycheck, a good percentage to multiply your earning by would be 0.75 or 75%. When calculating your paycheck, it's important to account for taxes, deductions, and other withholdings that may be taken out of your gross pay.  

This accounts for taxes, deductions, and other withholdings that are typically taken out of your paycheck before you receive your net pay. For example, if you earn $1,000 per pay period, multiplying by 0.75 would give you an estimated net pay of $750. However, keep in mind that this is just an estimate and your actual net pay may vary depending on your specific tax situation and other factors.

Learn more about paycheck,

https://brainly.com/question/31171390

#SPJ4