Find the volume of the square pyramid below

It seems that the boundary of [tex]M[/tex] is the circle [tex]x^2+y^2=49[/tex] in the plane [tex]z=0[/tex]. By Stokes' theorem,

[tex]\displaystyle\iint_M(\nabla\times\vec f)\cdot\mathrm d\vec S=\int_{\partial M}\vec f\cdot\mathrm d\vec r[/tex]

Parameterize [tex]\partial M[/tex] by

[tex]\vec r(t)=(7\cos t,7\sin t,0)[/tex]

with [tex]0\le t\le2\pi[/tex]. Then the line integral is

[tex]\displaystyle\int_{\partial M}\vec f(x(t),y(t),z(t))\cdot\mathrm d\vec r=\int_0^{2\pi}(56\sin t,35\cos t,0)\cdot(-7\sin t,7\cos t,0)\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^{2\pi}(245\cos^2t-392\sin^2t)\,\mathrm dt=\boxed{-147\pi}[/tex]

The question involves computing the double surface integral of the curl of the given vector field over a specified surface. The curl of the vector field is found first using a determinantal formula, and the surface integral is then evaluated over the capped cylindrical surface. These concepts are fundamental in vector calculus and have applications in fields such as physics and engineering.

This question requires the application of vector calculus concepts, specifically surface integrals and the divergence theorem. Given the vector field f=(zx+z²y+8y, z³yx+5x, z⁴x²), your task is to compute the double surface integral of the curl of f over a capped cylindrical surface m, which is a union of a cylinder and a hemispherical cap.

To solve it, you will start by finding the curl of the vector field using the determinant of a special kind of 3x3 matrix, called the curl matrix, containing the unit vectors i, j, k, the coefficients of the derivatives in the Cartesian coordinate system, and the components of the vector field. Once you get an expression for the curl of f, you will set up and evaluate a double surface integral over the given region m to find the desired quantity.

The concept of surface integrals is prevalent in many fields including physics and engineering where it is used to calculate quantities like flux. The divergence theorem, also known as Gauss's theorem, connects the flux of a vector field through a closed surface to the divergence of this field in the volume enclosed by the surface.

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Answer:

x =16

Step-by-step explanation:

You can use Theorem Pythagoras method to solve this question.

The formula is a^2 + b^2 =c^2

In this question a=x , b=12, and c= 20

to find a which means x you should use this formula : c^2 - b^2=20^2-12^2

= 400 - 144

=√256

=16

You also can use another method by using the second diagram.

In that diagram you have to know that 12÷3 is 4, 20÷5 is 4, hence x÷4 is 4

so , x=16.

I hope this two ways for answering this question will be helpful for you.

Answer:

x = 16

Step-by-step explanation:

Since the triangles are similar then the ratios of corresponding sides are equal, that is

[tex]\frac{x}{4}[/tex] = [tex]\frac{20}{5}[/tex] ( cross- multiply )

5x = 4 × 20 = 80 ( divide both sides by 5 )

x = 16

Answer:

It is B because this triangle Q and R are congruent

Step-by-step explanation:

Answer:

B. 68

Step-by-step explanation:

Because the side lengths PQ and PR both equal 5 and are congruent, the angles opposite them (angle Q and angle R) are also congruent to each other.

Angle Q is 68, so angle R is also 68.

Congruent means they are the same. Match the letter with the corresponding dimension on the other cone.

A = 6.2/2 = 3.1 units

B = 4.2 units

C = 5.2 units

D = 42 units^3

The unknown measures of the cones are A = 3.1, B = 4.2, C = 5.22 and D = 42 units³

Two figures are said to be congruent if they have the same shape and the their corresponding sides are the same.

Given that both cones are congruent. Hence:

A = 6.2 / 2 = 3.1

B = 4.2

Using Pythagoras:

C² = A² + B²

C² = 3.1² + 4.2²

C = 5.22

D = 42 units³

The unknown measures of the cones are A = 3.1, B = 4.2, C = 5.22 and D = 42 units³

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Answer:

(10 yd) ............=x.....

See attachment for solution steps and answer.

The slope of the line through the points (-1,6) and (7,-2) is -1. Substitute this and one of the points into the point-slope form, 'y - y1 = m (x - x1)', to get the equation of the line. The final equation is y = -x + 5.

To complete the equation of the line through points (-1,6) and (7,-2), we first need to find the slope (m) of the line using the formula m=(y2-y1)/(x2-x1). Here, (-1,6) is (x1,y1) and (7,-2) is (x2,y2). When we substitute these values in, we get m=(-2-6)/(7-(-1)), which simplifies to m=-8/8 or m=-1.

Now, you can use the point-slope form of a line, which is y - y1 = m (x - x1). You can use either of the points given, but let's use (-1,6). The equation then becomes y - 6 = -1(x - (-1)). To simplify, this becomes y - 6 = -x - 1. If you add 6 to both sides, the final equation of the line is y = -x + 5.

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ANSWER

4.1 units to the nearest tenth.

EXPLANATION

The graph of the two functions are shown in the attachment.

The coordinates of point A is (1.6,4.1).

The coordinates of point B is (7.1,6.1)

The x-axis represents the ground level.

The distance of point A from the ground level is how far the y-coordinate of this point is from the x-axis.

Which is |4.1-0|=4.1

Answer:

13

Step-by-step explanation:

You need to use the Pythagorean theorem. Which is a^2+b^2=c^2

so 5^2+12^2

which simplifies to 25+144=c^2

then it simplifies to 169=c^2

Now you take the square root

c=sqrt(169)

c=13

For this case we have that by definition, the Pythagorean theorem states that:

[tex]c = \sqrt {a ^ 2 + b ^ 2}[/tex]

Where:

c: It is the hypotenuse

a, b: Are the legs

Substituting the values of the figure:

[tex]c = \sqrt {5 ^ 2 + 12 ^ 2}\\c = \sqrt {25 + 144}\\c = \sqrt {169}\\c = 13[/tex]

Thus, the value of the hypotenuse is 13.

Answer:

13

Option B

Answer:

  True

Step-by-step explanation:

The first power of the variable is what makes it a linear term.

Answer:

25 ft

Step-by-step explanation:

The tree's shadow is 4 times the length of the man's shadow, so we presume the tree is 4 times the height of the man: 4 × (6 ft 3 in) = (24 ft 12 in) = 25 ft.

Answer: (3b + 5)(3b - 5)

Because both terms, 9b² and 25, are perfect squares, you can factor by taking the square roots of both terms.

The square root of 9b² is 3b (3b × 3b = 9b²).

The square root of 25 is 5 (5 × 5 = 25).

9b² - 25 has a negative, so the factored expression would be

(3b + 5)(3b - 5). The signs (+ and -) alternate in this case because the  expression, 9b² - 25, has no middle term.

You can check your work by using FOIL. See the attachment below.

F irst

O utside

I nside

L ast

Answer:

(3b + 5)(3b - 5)

Step-by-step explanation:

(3b + 5)(3b - 5)

Answer:

The fraction will decrease [tex]33.33\%[/tex]

Step-by-step explanation:

Let

x/y ----> the fraction

we know that

100%-50^=50%=50/100=0.50

100%-25%=75%=75/100=0.75

substitute

[tex]\frac{x}{y}*\frac{0.50}{0.75} =\frac{2}{3}(\frac{x}{y})[/tex]

therefore

The percent that the fraction will decrease is equal to

[tex](1-\frac{2}{3})*100=33.33\%[/tex]

Answer:

The measure of angle JKL is m∠JKL=35°

Step-by-step explanation:

step 1

Find the measure of the arc KL

we know that

arc KL+arc KN+arc NL=360° ----> by complete circle

arc KL+150°+140°=360°

arc KL=360°-290°=70°

step 2

Find the measure of angle JKL

we know that

The inscribed angle is half that of the arc it comprises.

so

m∠JKL=(1/2)[arc KL]

substitute

m∠JKL=(1/2)[70°]=35°

Answer:

y = ln( x + 6 ) - 11

h = 6

k = -11

h + k = -5

Step-by-step explanation:

Horizontal shift changes whats inside the brackets, in other words, changes just the x-value.If the shift is right, the number should be subtracted, if the shift is left, the number should be added.

Vertical shift changes the equation as a whole.if the shift is up, the number of should be added.If the shift is down, the number should be subtracted.

The changes given to us are 3 units left ( horizontal translation ) and 5 units down ( vertical translation )

If we rewrite the equation, we have:

y = ln( x + 3 + 3 ) - 6 - 5

y = ln( x + 6 ) - 11

Answer: B) 0.40 mi

Step-by-step explanation:

Using the Pythagorean Theorem.

A squared + B squared = C squared.

A = leg B = leg C = hypotenuse

A = 0.3 B = ? C = 0.7

0.3 squared x ? squared = 0.7 squared

Answer:

on usatestprep its 0.63

Step-by-step explanation:

Answer:

3a5b

Step-by-step explanation:

Answer:

3a5b

Step-by-step explanation:

Answer:

what

Step-by-step explanation:

Answer:

B. greater mean = class B greater median = class B

Step-by-step explanation:

It will take them 30 minutes

The faster computer will take approximately 22.5 minutes to do the job on its own.

Let's assume that the faster computer can complete the job on its own in x minutes.

If the slower computer takes 45 minutes to complete the job on its own, it means that in 1 minute it completes 1/45th of the job.

Working together, the two computers can complete the entire job in 15 minutes. So in 1 minute, they can complete 1/15th of the job.

Therefore, 1/45 + 1/x = 1/15

Simplifying the equation, we get 1/x = 1/15 - 1/45

Substituting the numerator values with a common denominator, 1/x = (3/45) - (1/45) = 2/45

Now, solving for x, we get x = 45/2 = 22.5

So, it will take the faster computer approximately 22.5 minutes to do the job on its own.

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Answer:

(D)[tex]x=104^{\circ}[/tex]

Step-by-step explanation:

Given: It is given that circle O is circumscribed about quadrilateral ABCD such that ∠ABC=91° and ∠ADC=x-15°.

To find: the value of x.

Solution: It is given that circle O is circumscribed about quadrilateral ABCD such that ∠ABC=91° and ∠ADC=x-15°.

We know that the sum of opposite angles of the cyclic quadrilateral is 180°, therefore

[tex]{\angle}ABC+{\angle}ADC=180^{\circ}[/tex]

Substituting the given values, we have

[tex]91^{\circ}+x-15^{\circ}=180^{\circ}[/tex]

[tex]x+76^{\circ}=180^{\circ}[/tex]

[tex]x=104^{\circ}[/tex]

thus, the value of x is [tex]104^{\circ}[/tex].

Hence, option D is correct.

Answer:

the answer is 104 i just took quiz 6.11.3

Step-by-step explanation:

Answer:

The correct choice is C

Step-by-step explanation:

The given curve is described by the parametric equations:

[tex]x=4-t[/tex]

[tex]y=t^2-2[/tex]

Let us eliminate the parameter by making t the subject in the first equation and substitute into the second equation;

[tex]t=4-x[/tex]

We substitute this into the second equation to get:

[tex]y=(4-x)^2-2[/tex]

This is the equation of a parabola whose vertex is at (4,-2)

The correct choice is C

Final answer:

The slope of a line perpendicular to a line with slope 3/4 is -4/3. This follows from the principle that slopes of perpendicular lines are negative reciprocals of each other.

Explanation:

The question pertains to finding the slope of a line that is perpendicular to another line with a given slope. Given that the slope (m) of line l is 3/4, let's find the slope of line m which is perpendicular to l. In algebra, the slope of perpendicular lines are negative reciprocals of each other. This means that to find the slope of m, you take the negative reciprocal of 3/4, which is -4/3.

Thus, the correct answer is C. - 4 over 3.

Answer:

The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is -80 feet per second.

Step-by-step explanation:

The function that models the height of the ball is:

[tex]f(t)=-16t^2+48t=160[/tex]

At t=3, [tex]f(3)=-16(3)^2+48(3)+160=160[/tex]

At t=5, [tex]f(5)=-16(5)^2+48(5)+160=0[/tex]

The average rate of change is simply the slope of the secant line connecting.

[tex](3,f(3))[/tex] and [tex](5,f(5))[/tex].

The average rate of change

[tex]=\frac{f(3)-f(5)}{3-5}[/tex]

[tex]=\frac{160-0}{-2}[/tex]

[tex]=-80fts^{-1}[/tex]

Answer:

  a) 708.00

  b) 108.00

  c) 11

Step-by-step explanation:

a) The cost is $59 for each of 12 months, so the total cost is ...

  12×$59 = $708

__

b) The "carrying charges" are the difference between what is paid and the cash price:

 $708 -600 = $108

__

c) Saving at the rate of $59 per month, it will take ...

  $600/($59/mo) ≈ 10.17 mo

to save the money. The amount saved will be $590, or $10 short of the required amount after 10 months, so it will take 11 months to save enough for the cash purchase.

Answer:

 a) 708.00

 b) 108.00

 c) 11

Step-by-step explanation:

Answer:

64/3 cc or 64/3 cm³

Step-by-step explanation:

The formula for the volume of a triangular pyramid is

V = (1/3)(area of base)(height)

Here we have a square prism (actually, a cube), whose square base is 4 cm by 4 cm (4 cm is the cube root of 64 cc).  The height of this cube is also 4 cm.

The volume of a triangular pyramid of base area (4 cm)² and height 4 cm is

V = (1/3)(base area)(height)

   = (1/3)(16 cm²)(4 cm) = 64/3 cc

Answer:

64/3 cc or 64/3 cm³

Step-by-step explanation:

Answer:

  [4, 8]

Step-by-step explanation:

You want to find t such that ...

  T ≥ 69

  24t − 2t^2 + 5 ≥ 69

  t^2 -12t +32 ≤ 0 . . . . . subtract the left side, divide by 2

  (t -4)(t -8) ≤ 0 . . . . . . . factor

The factors will have different signs, so their product will be negative, for values of t between 4 and 8. Of course, the equality holds at t=4 and t=8, so the solution interval is ...

  t ∈ [4, 8]

The expression 6(x − 5) means the product of 6 and the difference of x minus 5. If x = 7, the value of the expression is 12.

We can see that the number 6 to the left of the expression 6(x - 5), this means 6 times (x - 5), The reason we write it that way is because the multiplication symbol, ×, can be confused with the variable x, follow by (x - 5) which is the difference of x minus 5. All together is the product of 6 and the difference of x minus 5.

If x = 7, then:

6(7 - 5) = 6(2) = 12

Following are the solution to the given expression:

Given:

[tex]6(x-5) \\\\ x=7 [/tex]

To find:

value=?

Solution:

[tex]\to 6(x-5)\\\\ [/tex]

putting the x value into the above expression:

[tex]\to 6(7-5)\\\\ \to 6(2)\\\\ \to 12[/tex]

Therefore the final answer is "12".

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Answer:

A

Step-by-step explanation:

Answer:

answer is A

Step-by-step explanation:

There are no numbers here

A prism is rectangular shaped,. a pyramid is triangular shaped.

The ratio would be  1 to 3

The formula for the pyramid is V = 1/3Bh