MATHEMATICS COLLEGE

Find the volume of the solid cut from the thick-walled cylinder 1 less than or equals x squared plus y squared less than or equals 2 by the cones z equals plus or minus StartRoot 4 x squared plus 4 y squared EndRoot.

Answers

Answer 1

Answer:

Answer:

$4/3(2√(2) -1)*\pi$ this is the answer

Step-by-step explanation:

$1<= x^2+y^2 <= 2 by the cons z= +- √((x^2 + y^2))$

Let suppose

x = r∙cos(θ)

y = r∙cos(θ)

z = z

The differential volume element changes to

dV = dxdydz = r drdθdz

The limits of integration in cylindrical coordinates are:

(i)

$1 \leq x^2 + y^2\leq 2$

$1 \leq r^2 \leq 2$

since r is always positive

$1 \leq r \leq √(2)$

(ii)

$- √((x^2+ y^2)) \leq z \leq +√((x^2+ y^2))$

$- r \leq z \leq r$

(iii)

$0 \leq Theta \leq 2∙π$

we have no restrictions in radial direction.

$V = \int\limits^a_b { dV} \,$

Remaining derivation has been explained in the atatchment where we get the volume of the cylinder

Answer 2

Answer:

The volume of the solid cut from the thick-walled cylinder and cones is 4π/3 (2√2 - 1) using cylindrical coordinates.

To determine the volume of the solid cut from the given cylinder and cones, we'll utilize cylindrical coordinates. The cylinder is represented by 1 ≤ r² ≤ 2, or 1 ≤ r ≤ √2, in cylindrical coordinates, where x² + y² = r². Similarly, the cones are represented by z = ±2r. The volume differential in cylindrical coordinates is dV = rdzdrdθ.

Integrating over the appropriate limits, we obtain the volume of the solid:

V = ∫₀²π ∫₁√₂ -2r²r²dzdrθ

Evaluating the integral, we get:

V = 4π/3 (2√2 - 1)

Learn more about Volume calculation in cylindrical coordinates here:

brainly.com/question/33911093

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A line with a slope of -2 passes through the point (4, 7). Write an equation for this line in point-slope form.

Answers

Answer:

The equation in point-slope is $y-7=-2(x-4)$ .

Step-by-step explanation:

Point-slope is a specific form of linear equations in two variables:

$y-b=m(x-a)$

When an equation is written in this form, m gives the slope of the line and (a, b) is a point the line passes through.

We want to find the equation of the line that passes through (4, 7) and whose slope is -2. Well, we simply plug m = -2, a = 4, and b = 7 into point-slope form.

$y-7=-2(x-4)$

$\bf (\stackrel{x_1}{4}~,~\stackrel{y_1}{7})~\hspace{10em} slope = m\implies -2 \n\n\n \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\n \cline{1-1} \n y-y_1=m(x-x_1) \n\n \cline{1-1} \end{array}\implies y-7=-2(x-4)$

What is a rational number

Answers

A rational number is any number, positive or negative. This can be numbers including fractions, decimals, whole numbers, and so on.

As long as it is a real number it's a rational number.

In addition, any number that can be turned into a fraction is rational.

Answer:

A Rational Number can be made by dividing two integers.

Step-by-step explanation:

Gas prices went up this week from $4.00 a gallon to $4.20. What is the percent increase of the price?

Answers

Answer:5% increase

Step-by-step explanation:

What is the exact quotient of 434 divided by 7

Answers

434 divided by 7 equals 62.

The admission fee at an amusement park is $1.50 for children and $4 for adults. On a certain day,276 people entered the park, and the admission fees collected totaled 664.00 dollars. How many
children and how many adults were admitted?
Your answer is
number of children equals____
number of adults equals____

Answers

Is there answer choices?

Analyze the biconditional statement below and complete the instructions that follow.A polygon is a pentagon if and only if it has five sides.

Express the given biconditional as the conjunction of two conditionals.

A.
If it is a pentagon, then it has five sides. If it is a polygon, then it is a pentagon.
B.
If it is a pentagon, then it is a polygon. If it is a polygon, then it is a pentagon.
C.
If it is a pentagon, then it is a polygon. If it is a polygon, then it has five sides.
D.
If it is a pentagon, then it has five sides. If the polygon has five sides, then it is a pentagon

Answers

Answer:

D. If it is a pentagon, then it has five sides. If the polygon has five sides, then it is a pentagon

Step-by-step explanation:

Given statements;

A polygon is a pentagon if and only if it has five sides.

From the statement we see that a pentagon has 5 sides.
A body of this structure with 5 sides can be classified as a polygon.

Therefore, this bi-conditional statement is premised on two "if"s,

If it is a pentagon then we are dealing with a body with 5 sides.

And if such body has five sides, it is a polygon called pentagon

The correct option is D

Choice A, B and C are logically flawed.

A. If it is a pentagon, then it has five sides. If it is a polygon, then it is a pentagon.

The emboldened part we do not know.

B. If it is a pentagon, then it is a polygon. If it is a polygon, then it is a pentagon

The emboldened part is flawed.

C. If it is a pentagon, then it is a polygon. If it is a polygon, then it has five sides.

We do not know if all pentagons have 5 sides.

Final answer:

The conjunction of two conditional statement for the biconditional 'A polygon is a pentagon if and only if it has five sides' is 'If it is a pentagon, then it has five sides' and 'If a polygon has five sides, then it is a pentagon'.

Explanation:

The given biconditional statement 'A polygon is a pentagon if and only if it has five sides' can be split into two conditional statements as follows: 'If it is a pentagon, then it has five sides' and 'If a polygon has five sides, then it is a pentagon'.

These two separate conditional statements show both possible directions of the biconditional statement. Option D correctly represents these two conditions as the conjunction of two conditional statements.