each picnic table seats 6 people how many picnic tables are needed to seat 24 people?

Answer: 6 Drumsticks

Step-by-step explanation:

Final answer:

To find the integral of the vector field over the solid half ball, we calculate two surface integrals, one over the flat disk and one over the hemisphere's curved surface, and then sum their contributions.

Explanation:

The student has asked to calculate the surface integral of the vector field f = (x + 3y^5)i + ( y + 10xz)j + (z - xy)k over the upper half of the sphere [tex]x^2 + y^2 + z^2 = 1[/tex], with z ≥ 0. To solve this, we can use the divergence theorem, noting that the divergence of f is ∇ · f, and then integrating that over the volume of the half sphere. We divide the surface S into two parts: the flat circular disk at the bottom (where z = 0) and the curved surface of the half sphere. The flux through the curved surface can be found by integrating over the sphere's surface, while the disk contributes a separate integral. This requires calculating two different surface integrals, and adding their results.

The surface integral [tex]\(\int_{S} \mathbf{f} \cdot d\mathbf{s}\)[/tex] for the given vector field and half-ball surface is [tex]\(2\pi\)[/tex].

To calculate the surface integral [tex]\(\int_{S} \mathbf{f} \cdot d\mathbf{s}\)[/tex] where [tex]\(S\)[/tex] is the entire surface of the solid half-ball [tex]\(x^2 + y^2 + z^2 \leq 1, z \geq 0\) and \(\mathbf{f} = (x + 3y^5)\mathbf{i} + (y + 10xz)\mathbf{j} + (z - xy)\mathbf{k}\)[/tex], we will use the Divergence Theorem.

The Divergence Theorem states that for a vector field [tex]\(\mathbf{f}\)[/tex],

[tex]\[\int_{S} \mathbf{f} \cdot d\mathbf{s} = \int_{V} (\nabla \cdot \mathbf{f}) \, dV\][/tex]

where [tex]\(S\)[/tex] is the boundary surface of the volume [tex]\(V\),[/tex] oriented by the outward-pointing normal.

1. Calculate the Divergence [tex]\(\nabla \cdot \mathbf{f}\)[/tex]:

[tex]\[\nabla \cdot \mathbf{f} = \frac{\partial}{\partial x}(x + 3y^5) + \frac{\partial}{\partial y}(y + 10xz) + \frac{\partial}{\partial z}(z - xy)\][/tex]

Computing each term:

[tex]\[\frac{\partial}{\partial x}(x + 3y^5) = 1\][/tex]

[tex]\[\frac{\partial}{\partial y}(y + 10xz) = 1\][/tex]

[tex]\[\frac{\partial}{\partial z}(z - xy) = 1\][/tex]

So the divergence is:

[tex]\[\nabla \cdot \mathbf{f} = 1 + 1 + 1 = 3\][/tex]

2. Set up the volume integral:

The volume (V) is the solid half-ball defined by [tex]\(x^2 + y^2 + z^2 \leq 1\)[/tex] and [tex]\(z \geq 0\)[/tex].

We will integrate the constant divergence \(3\) over the volume of the half-ball. The volume of a full ball of radius 1 is \(\frac{4}{3} \pi (1^3) = \frac{4}{3} \pi\). Since we have a half-ball, the volume is:

[tex]\[\text{Volume of half-ball} = \frac{1}{2} \cdot \frac{4}{3} \pi = \frac{2}{3} \pi\][/tex]

3. Evaluate the volume integral:

[tex]\[\int_{V} (\nabla \cdot \mathbf{f}) \, dV = \int_{V} 3 \, dV = 3 \int_{V} dV = 3 \cdot \text{Volume of half-ball} = 3 \cdot \frac{2}{3} \pi = 2 \pi\][/tex]

Therefore, the surface integral is:

[tex]\[\int_{S} \mathbf{f} \cdot d\mathbf{s} = 2\pi\][/tex]

So, the final answer is:

[tex]\[\boxed{2\pi}\][/tex]

Answer:

Volume of each Marble: 0.06 liters

Water required for the marbles and water: 14 Liters

Answer:

C is the correct answer.

Step-by-step explanation:

Took the test.

x(yz)

The decimal value of the 2 in the hexadecimal number f42ac16 is 2.

The hexadecimal number f42ac16 can be split into individual digits, where each digit represents a power of 16. Starting from the rightmost digit, the 2 represents the digit in the 16^0 (1) place, which is simply 2. Therefore, the decimal value of the 2 in the hexadecimal number f42ac16 is 2.

https://brainly.com/question/36345940

#SPJ11

Answer:

C ) 8 sheets of red and 13 sheets of yellow felt

Step-by-step explanation:

Final answer:

Caleb's teacher needs to purchase 7.5 sheets of red felt and 12.5 sheets of yellow felt so that each of the 20 students can make a dog toy.

Explanation:

To calculate the amount of red and yellow felt Caleb's teacher needs to buy for 20 students to each make a dog toy, we can multiply the amount of felt needed for one toy by the total number of students.

Red felt needed: 3/8 sheet per toy

Yellow felt needed: 5/8 sheet per toy

For red felt:

3/8 sheet x 20 students = 60/8 sheets

To simplify this, we divide 60 by 8, which equals 7.5 sheets. So, the teacher needs to buy 7.5 sheets of red felt.

For yellow felt:

5/8 sheet x 20 students = 100/8 sheets

After simplifying, 100 divided by 8 equals 12.5 sheets. Therefore, the teacher needs to acquire 12.5 sheets of yellow felt.

Answer:

Option A is correct

5 feet  is the length of the actual picnic table

Step-by-step explanation:

Proportion states that two ratios or fractions are equal.

As per the statement:

A blueprint for a picnic table has a scale of 1 : 20.

The table in the blueprint is 3 in. long.

Let x be the length of the actual picnic table.

By definition of proportion;

[tex]\frac{1}{20} = \frac{3}{x}[/tex]

By cross multiply we have;

[tex]x = 20 \cdot 3 = 60[/tex]

⇒x = 60 inches

Use conversion:

1 inches = [tex]\frac{1}{12}[/tex] feet

then;

60 inches = [tex]\frac{60}{12} = 5[/tex] feet

Therefore, the length of the actual picnic table is, 5 feet

The correct answer is:

b) his repair and maintenance costs were more than 3% a year

Explanation:

Lewis's house increases in value by 3% each year. However, if he spends more than 3% of the value of his home each year in repairs and maintenance, when he sells the home, he will not be making money; he will in fact lose money, compared to what he's spent the last two decades.

Answer:

b) x=86,y=75,z=94

Step-by-step explanation:

Answer:

There are six trigonometric function, the first three are mainly used and other three are not use often, they are:

They are mainly the ratio of two side of triangle.i.e.

[tex]Sin x = \dfrac{Perpendicular}{Hypotenuse}[/tex]

[tex]Cos x = \dfrac{Base}{Hypotenuse}[/tex]

[tex]Tan x = \dfrac{Perpendicular}{Base}[/tex]

[tex]Cosec x = \dfrac{Hypotenuse}{Perpendicular}[/tex]

[tex]Sec x = \dfrac{Hypotenuse}{Base}[/tex]

[tex]Cot x = \dfrac{Base}{Perpendicular}[/tex]

On the basis of these ratios there are different relations defined in Trigonometry Chapter.

Answer:

length of chords

Explanation:

In the study of the earth and astronomy, mathematicians developed trigonometric functions as lengths of chords in a circle, not as ratios. The mathematician Ptolemy calculated chord lengths by inscribing regular polygons in a circle. Remarkably, he was able to calculate these lengths to an accuracy of about five decimal places. Ptolemy's table of chords is equivalent to our table of sine values. In this lesson, you will begin to see the relationship between the trigonometric functions and the circle.

The solution of the expression are,

⇒ x = 3.5 and x = - 3.5

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ 12 - x² = 0

Now, We can simplify as;

⇒ 12 - x² = 0

⇒ 12 = x²

⇒ x = √12

⇒ x = ± 3.5

⇒ x = 3.5 and x = - 3.5

Thus, The solution of the expression are,

⇒ x = 3.5 and x = - 3.5

Learn more about the mathematical expression visit:

brainly.com/question/1859113

#SPJ3

Answer: finds the tax rate for his income level: 10%

Enters the base amount 29.16

Enters the amount of tax owed: 700

Divides by: 10 =:700

Step-by-step explanation:

I took the test hope this helps