(4u)(5v^2)-2u+uv^2-4uv^2

Consider the ratio written like a fraction:

Count of sandwichescount of people→46

But we need the count of sandwiches for 1 person. So we need to change the 6 people into 1 person. To change 6 into 1 divide it by 6

6÷6=1

For ratios, when multiplying or dividing, what we do to the bottomwe do to the top.

So to change 46 into what we want we apply

4÷66÷6=231← Count of sandwichescount of people

So each person ends up with 23 of a sandwich

Now see where the shortcut comes from 46=23 of a sandwich

The y-value increases by 4 when the x-value increases by 2, so the slope is

... 4/2 = 2

Your equation for the line with slope m through point (h, k) is

... y - k = m(x - h)

and you have m = 2, (h, k) = (-6, 6)

Filling in those values gives

... y - 6 = 2(x - (-6))

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

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Answer:

He has surveyed the students who are already participating in the choir. So naturally the number of students who likes the choir class will be higher than in other classes. You can see it reflected in the data.

Step-by-step explanation:

He has surveyed the students who are already participating in the choir. So naturally the number of students who likes the choir class will be higher than in other classes. You can see it reflected in the data.

If he needs to make an inference on the whole school he needs to randomly pick people from the school and ask the same question. Then he can make a better inference about the favorite  class of students.

Answer:

His sample group does not represent the entire school body.

Step-by-step explanation:

Answer:

Cost of 200 gram chess  = $0.88.

Step-by-step explanation:

Given  : If cheese is $4.40 per kilogram.

To find : What should I pay for 200 gram.

Solution : We have given

1 kilogram = 1000 gram.

Cost of 1000 gram chess =  $4.40.

Cost of 1 gram chess = [tex]\frac{4.40}{1000}[/tex].

Cost of 200 gram chess =  [tex]\frac{4.40}{1000}* 200[/tex].

                                        = $0.88.

Therefore, Cost of 200 gram chess  = $0.88.

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

Answer:

Volume of each Marble: 0.06 liters

Water required for the marbles and water: 14 Liters

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]

Final answer:

To find the distance the flying squirrel must glide to reach the acorn, the Pythagorean theorem is used. The tree's height (15 feet) and the horizontal distance to the acorn (8 feet) provide the two sides of a right triangle. The squirrel's glide distance is the hypotenuse, calculated to be 17 feet.

Explanation:

To calculate the distance the flying squirrel needs to glide to reach the acorn, we can apply the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this scenario, the tree height forms one side of a right triangle, the distance from the base of the tree to the acorn forms the second side, and the squirrel's glide path represents the hypotenuse.

Given:

Tree height (side a): 15 feet

Distance from the tree to the acorn (side b): 8 feet

Using the Pythagorean theorem:

c² = a² + b²

c² = 15² + 8²

c² = 225 + 64

c² = 289

c = √289

c = 17 feet

Therefore, the flying squirrel needs to glide 17 feet to reach the acorn.

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 side length of the square window with a perimeter of (8x+12) feet is found by dividing the perimeter by 4, resulting in an expression for the side length as 2x + 3 feet.

To find the side length of a square window with a given perimeter, you can use the fact that the perimeter (P) of a square is equal to four times the side length (s). In this case, the perimeter is expressed as (8x+12) feet. Since there are four sides of equal length in a square, you divide the perimeter by 4 to find the length of one side.

The formula for the perimeter of a square is:

P = 4s

Given the perimeter P = (8x+12), we can solve for the side length s by dividing both sides of the equation by 4:

s = (8x+12) ÷ 4

Simplifying the equation, we get:

s = 2x + 3

Therefore, the expression for the side length of the square window in feet is 2x + 3.

The dimension of the square is 1.125 by 1.125 units and the area of the square is 1.265625 square units.

It is the method to separate the polynomial into parts and the parts will be in multiplication. And the value of the polynomial at this point will be zero.

The given expression represents the area of a square is 64r² – 144r + 81.

The dimension of the square will be

Using the formula method, we have

64r² – 144r + 81

Then

[tex]\rm r = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

Then we have

a = 64

b = –144

c = 81

Then

[tex]r = \dfrac{-(-144) \pm \sqrt{(-144)^2-4*64*81}}{2*64}\\\\\\r = \dfrac{144 \pm \sqrt{20736- 20736}}{128}\\\\\\r = \dfrac{144}{128}\\\\\\r = 1.125[/tex]

The dimension of the square is 1.125 by 1.125 units and the area of the square is 1.265625 square units.

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