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The table represents a quadratic function C(t).



t C(t)

2 4

3 1

4 0

5 1

6 4


What is the equation of C(t)?

C(t) = −(x − 4)2

C(t) = (x − 4)2

C(t) = −x2 + 4

C(t) = x2 + 4

Evaluation of E(x+y+z)dV where E is the solid in the first octant that lies under the paraboloid  z = 4 - x² - y² is [tex]\frac{128}{15} + \frac{8\pi}{3}[/tex]

Cylindrical coordinates are a set of three coordinates used to locate a point in the cylindrical coordinate system.

When the polar coordinates are extended to a three-dimensional plane, an additional z coordinate is added.

These three measurements combine to form cylindrical coordinates. The coordinates describe two distances and one angle.

According to the question,

We have to find the area under paraboloid

Given : E is a region that lies under paraboloid  z = 4 - x² - y² and in the first octant.

The intersection of this paraboloid with the xy plane will give z = 0,

z = 4 - x² - y² , z = 0, that is a circle  x² +y² = 4 with radius 2

Therefore , 0 ≤ z ≤ 4 - r²

Therefore, the region E in cylindrical coordinates is

[tex]E = ( r , \theta , z ) | 0 \leq \theta \geq \frac{\pi}{2} , 0 \leq 4 \geq , 0 \leq z \geq 4 - r^2[/tex]

f(x , y ,z ) = x + y + z

=> f(x , y ,z ) = rcosθ + rsinθ + z

=> f(x , y ,z ) = r(cosθ + sinθ) + z

Integration ,

[tex]{\int \int \int }\limits_E {z} \,dv[/tex]

=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r ( cos \theta + sin \theta) + z] r \, dzdrd\theta[/tex]

simlifying,

[tex]= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r^2 ( cos \theta + sin \theta) + rz] \, dzdrd\theta\\= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [r^2 ( cos \theta + sin \theta)z + r\frac{z^2}{2}] \, drd\theta[/tex]

=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [(4r^{2} - r^{4})( cos \theta + sin \theta) + \frac{r(4 - r^2)^2}{2}] \, drd\theta[/tex]

=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(\frac{4}{3} r^{3} - \frac{r^{5}}{5})( cos \theta + sin \theta) + \frac{(4 - r^2)^3}{12}]^{2}_{0} \, d\theta[/tex]

Putting the limit and solving,

=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(cos \theta + sin\theta )\frac{64}{15} + \frac{16}{3} \theta ]\, d\theta[/tex]

=> [tex][(sin\theta - cos\theta )\frac{64}{15} + \frac{16}{3} \theta ]^\frac{\pi}{2} _0[/tex]

Evaluating the limits,

[tex]\int\int\int\limits_E {x + y +z } \,dV = \frac{128}{15} + \frac{8\pi}{3}[/tex]

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Answer:

Triangle B

Step-by-step explanation:

The triangles have three pairs of congruent sides.

Amplitude is 5, midline is 3, period is π and the frequency is 2

We can describe the cosine function as y = Acos(B(x + C )+ D

where,

amplitude is A

Frequency is B

period is 2π/B

phase shift is C (positive is to the left)

vertical shift is D

Step 1:

Identify

5 cos(2x) + 3 → Acos(B(x + C )+ D

Hence

A = 5 = Amplitude

B = 2.C = 0. So,

Period = 2π / B

= 2π / 2 = π

Period = π

Frequency = B = 2

Vertical shift = D = 3

Step 2:

Midline

The equation of the midline of periodic function is the average of the maximum and minimum values of the function.

a) we have a maximum when

cos(2x) = 1

x =0, because (cos 0) = 1 Now, replace

y = 5 cos(2x) + 3

y = 5cos(2. 0 ) +3

= 5.1 + 3

= 8

So, the maximum is 8

b) we have a minimum when

cos(2x) = -1

x = π/2 , because

cos(2.π/2) = cos(π) = -1

Now, replace

y = 5cos(2.π/2) + 3

= 5. (-1) + 3

= -5+3

= -2

So, the midline is the average of 8 and -2

Midline = y = [8 + (-2)] / 2

= 6/2

y = 3

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b,d and e option are correct

What is statement?

A statement, like "Pizza is tasty," is a declaration that something is true. In the fields of law, finance, and government, there are more types of statements. Every statement makes a claim or a point.

These statements are true:

The least Marcus would pay for a laptop from electro's is $500

Marcus can accept to pay about $700 for a laptop from electro's and  $1500 for a laptop from bits and bytes.

The middle 50% of laptops from electro's are priced between $500 and $900.

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Initial size of bacteria is 313, population of bacteria after 5 hours is 3.346 ×10¹¹.

here growth means increases in the number of bacterial populations with respect to time. Generally, the rate of growth of bacteria is occurred exponentially. We use the following exponential equation to calculate the number of bacteria.

N= N° e∧(kt)

In the above equation, N and N° denotes the number of populations after t minutes and initial time respectively.

k is a constant of proportion

t is the time in minutes

as per given data, the number is doubled after 10 min

N=2N°, t=10

2N°=N° e∧(k10)

2= e∧(k10)

㏑2 = k10 [we took ㏑ in both side]

k= 0.0693

after t= 80 min, the population N= 80000

now using the equation, 80000=N° e∧(k80)

as k=0.0693, N°= 313

the initial number is 313'

after t=5 hours =300 min, the population will be

N= 313 e∧ (0.0693x300)

N = 3.346×10¹¹

hence, initial population is 313 and the number increase to 3.346×10¹¹ after 5 hours.

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The total ticket sales was $ 2298, and the theater was filled to capacity

Children: 158

Students: 158

Adults: 79

The amount of space occupied by a flat (2-D) surface or an object's shape is known as its area. Draw a square on a piece of paper using a pencil. A 2-D figure, that is. The term "Area" refers to the area that a shape occupies on paper. Now picture that your square is composed of smaller unit squares.

According to Study.com, area is a mathematical concept that is defined as the two-dimensional space occupied by an item. Area is used in many real-world contexts, including building, farming, architecture, science, and even determining how much carpet you'll need to cover your home's rooms.

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LM is translated 5 units right and 2 units up to LM' as the graph shows the move in positive direction.

In elementary mathematics, the term "graph" designates a plot or a function graph. A graph is, in the language of mathematicians, a set of points and the connections between some subset of those points (which may be empty). In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way. The relationships between two or more items are frequently represented by the points on a graph.

Here,

As the graph indicates a move in the right direction, LM is translated 5 units right and 2 units up to LM'.

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Answer: x=27/6

Step-by-step explanation:

-1 2/3 = -5/3

17/6 = -5/3 + x

x = 17/6 + 5/3

x = 17/6 + 10/6

x = 27/6

Answer:  t₁=1.5-0.5√3 c ≈0.634 c       t₂=1.5+0.5√3 c ≈2.366 c

Step-by-step explanation:

h=-8t²+24t+4      h=16 feet     t₂=?

[tex]16=-8t^2+24t+4\\\\16-16=-8t^2+24t+4-16\\\\0=-8t^2+24t-12\\[/tex]

Divide both parts of the equation by -4:

[tex]0=2t^2-6t+3[/tex]

Thus,

[tex]2t^2-6t+3=0[/tex]

a=2     b=-6    c=3

D=b²-4ac

D=(-6)²-4(2)(3)

D=36-24

D=12

√D=√12

√D=√(4*3)

√D=2√3

[tex]t=\frac{-bб\sqrt{D} }{2a} \\\\t=\frac{-(-6)б2\sqrt{3} }{2(2)} \\\\t=\frac{6б2\sqrt{3} }{4} \\\\t_1=1.5-0.5\sqrt{3}\ c \\\\t_2=1.5+0.5\sqrt{3}\ c[/tex]

Answer:

[tex]\huge\boxed{\sf d = 1.25\ m}[/tex]

Step-by-step explanation:

Work = W = 22 J

Force = F = 17.5 N

Distance = d = ?

W = Fd

Put the given data in the above formula.

22 = 17.5 (d)

Divide 17.5 to both sides

22/17.5 = d

1.25 m = d

[tex]\rule[225]{225}{2}[/tex]

Answer:

∠1+∠2≅∠3+∠4 cant be proven because sum of two obtuse angles is sure to be greater than sum of two acute angles of a triangle.

The pairs of triangles that are congruent are;

A. Δ1 and Δ5

B. Δ1 and  Δ2

C. Δ2 and Δ3

D. Δ3 and Δ5

E. Δ1 and Δ3

F. Δ3 and Δ4

G. Δ1 and Δ4

H. Δ4 and Δ5

Congruent triangles are triangles that have all three sides  and all three angles on each triangle having the same length and the same measure.

The congruent angles are indicated by the number of markings

The angles in the figures are;

Angle with  one arc

Angles with two arcs marks

Angles with three arc marks

The congruent sides are indicated by one dash, and two dashes

Therefore;

Δ2 s congruent to Δ1 is congruent to Δ3 by  Angle-Side-Angle rule of congruency

Similarly, we have;

Δ3 is congruent to Δ4 by Angle-Side-Angle congruency postulate

Therefore;

Triangle Δ2 is congruent to Δ3  by Angle-Side-Angle congruency postulate

From the diagram, the one arc, two arcs, and three arc angles form a linear pair, therefore;

Therefore, Δ4 is congruent to Δ5

Δ1 is congruent to Δ4

Δ2 is congruent to Δ5, therefore; Δ3 is congruent to Δ5

Similarly;

Δ1 is congruent to Δ3

Δ1 is congruent to Δ2, therefore , therefore, Δ1 is congruent to Δ5

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Answer:

Step-by-step explanation:

To find 50% of 950, you can multiply 950 by 0.50. 50% is the same as 0.50.

The calculation is: 50% * 950 = 0.50 * 950 = <<50*.01*950=475>>475

So, 50% of 950 is 475.

Answer:

5/8=25/40   7/10=28/40

Step-by-step explanation:

5/8<7/10

Interior angles on the same side are supplementary, which means they add up to 180°. We can write the equation as follows and refer to the two angles as x° and (x + 20)°.

When two rays are linked at their ends, they create an angle in geometry. The sides or arms of the angle are what are known as these rays.

When two lines meet at a point, an angle is created.

An "angle" is the measurement of the "opening" between these two rays. It is symbolized by the character. The circularity or rotation of an angle is often measured in degrees and radians.

Angles are a common occurrence in daily life.

Angles are used by engineers and architects to create highways, structures, and sports venues.

According to our question-

x + x + 20 = 180

2x + 20 = 180

2x = 160

If x = 80, the angles are 80° and 80 + 20 = 100°, respectively.

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Answer:10

Step-by-step explanation:10 times two is twelve and minus 2 is ten.

The number of  chaperones that are needed is 128/15.

What is unitary method?

The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value.

Given that the number of students in 8th grade is 400.

Four fifth of total number of students go for trip.

The number of student who goes for the trip is 400 × 4/5

                                                                                = 320.

To carry 75 students, we need 2 chaperones.

Apply unitary method:

To carry 1 student, we need 2/75 chaperones.

To carry 320 students, we need (2 ×320)/75  chaperones.

                                                  = (2 × 64)/15

                                                  = 128/15

                                                  = 8.533

                                                 = 9

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The three sticks will not form a triangle.

What is the triangle inequality theorem?

The triangle inequality theorem explains the connection between a triangle's three sides. This theorem states that for any triangle, the sum of the lengths of the first two sides is always greater than the length of the third side. In other terms, this theorem states that a straight line is always the shortest distance between any two places.

Given, Mr Hansen has three sticks that measure 5 inches, 10 inches and 17 inches

From the above definition of the Triangle inequality theorem,

Sum of two sides = 5 + 10 = 15 which is less than

the length of the third side = 17

So, the three sticks will not form a triangle.

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The rate of change of temp w.r.t time is -1.875°F

What is rate of change?

The rate of change (ROC) measures how quickly a variable alters over a predetermined amount of time. When discussing momentum, the term "rate of change" (ROC) is frequently used. It is typically defined as the ratio of a change in one variable to a matching change in another; visually, the rate of change is represented by the slope of a line. The Greek letter delta () is frequently used to represent the ROC.

The term "rate of change" (ROC) describes the rate at which something changes over time.

Thus, it is not the amount of individual changes themselves but rather the acceleration or slowdown of changes (i.e., the pace).

Time             Temperature              Difference

4                          89.5                               -

10                        78.25              78.25 - 89.5 = -11.25

16                        67.00              67 - 78.25 = -11.25      

There is a common difference of -11.25 in each successive term.

Therefore, cooling of a chemical represents a linear function.

Rate of change of temp from 4 min to 10 min

=78.25 - 89.5 = -11.25

= -11.25/6

=-1.875

So, the rate of change of temp w.r.t time is -1.875°F

which means temperature is decreasing continuously when the time is increasing

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Answer:

7 5 15 22

step,by step explanation: yes

After using the Pythagorean theorem the missing length of c is √148

What is the Pythagorean theorem in plain English?

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.

How may the Pythagorean Theorem be demonstrated the simplest way?

Angles CAB, AC'B, and AB'C must be equal in order for AC2 + AB2 to equal BC(CB' + BC'). Triangles ABC, AC'B, and AB'C are indeed comparable. Since AB/BC' = BC/AB and AC/CB' = BC/AC, we have the necessary identity right away. Theorem reduces to Pythagorean proposition and proof if angle A is correct.

Pythagorean Theorem: a²+b²=c²

so (12)^2+(2)^2 = 144+4 = 148

c²=148

c=√148

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Answer:

5.46

Step-by-step explanation:

precio original = x

0.7(1.3x) = x - 0.54

0.91x - x = -0.54

x = 6

6 + 30% de 6 = 7.8

7.8 - 30% de 7.8 = 5.46

Arturo compró por 6.

Aumentó 30% para 7.80.

Bajó 30% para 5.46.

6 - 5.46 = 0.54

The required number of candles Marcella has in the starting is 56.

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
As mentioned in the question,
She has some candles in the starting  before she starts collecting the candles, let x be the candle she had in the starting,
Now,
According to the question she is buying 13 candles every week, at the end of 8 weeks she has 160 candles,
So.
x + 13(8) = 160
x + 104 = 160
x = 160 - 104
x = 56

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The length of the line segment GH is 34 units

The complete question is added at the end of this solution

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

GH= 7z - 1

BC = 4z - 3

Also from the question, we understand that the line segment BC is half the line segment GH

This can be represented mathematically as follows

BC = 1/2 * GH

Substitute the known values in the above equation, so, we have the following representation

4z - 3 = 1/2(7z - 1)

Multiply through by 2

8z - 6 = 7z - 1

Collect the like terms

8z - 7z = 6 - 1

Evaluate the like terms

z = 5

Recall that

GH= 7z - 1

Substitute the known values in the above equation, so, we have the following representation

GH= 7 * 5 - 1

Evaluate

GH= 34

Hence, the length is 34 units

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Complete question

The line segment BC is half the line segment GH

If GH = 7z- 1 and BC=4z- 3,

What is GH?

The value of missing coordinate d of the point (d, 7) in the line such that its slope is 1 / 3 is - 9.

In this problem we must determine the value of the missing coordinates of a point in a line such that its slope is 1 / 3. The slope of the line can be determined by means of the secant line formula:

m = Δy / Δx

Where:

If we know that (x₁, y₁) = (d, 7), (x₂, y₂) = (0, 10) and m = 1 / 3, then the missing coordinate is:

1 / 3 = (10 - 7) / (0 - d)

1 / 3 = - 3 / d

d = - 9

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Answer:

(8, 11)

x=8

y=11

Step-by-step explanation:

I solved using elimination.

I set the equations over one another getting x+2y=30 over -2x-2y=-38.

I chose elimination because the y values were already the opposite so it can easily cancel each other out. After it get's canceled out I get x=30 over  -2x=-38. When using elimination we always add so you'd add x+ -2x and 30+ -38. When you add x + -2x you'd get -1x. When you add 30 + -30 you'd get -8. Now you have 1x=-8 divide both sides by -1 and you get x=8. Now you plug in 8 to one of the equations. I chose to plug it into the 1st equation which is x+ 2y=30. But because I plugged in the 8 the equation is now 8+2y=30. I subtracted 8 on both sides canceling the 8 out. So now I have 2y=22. divide both sides by 2 and x=11. So your order pair is (8, 11)
(Hopefully you were able to understand it..)

Answer: -4/5x - 22/5

Step-by-step explanation:

To find an equation of a line that is perpendicular to a given line and passes through a given point, we need to first find the slope of the given line. The slope of a line is the ratio of the change in the y-coordinate to the change in the x-coordinate of any two points on the line. For example, the slope of the line 5/4x + 2 is 5/4.

To find the equation of a line that is perpendicular to a line with slope m, we need to use the equation:

y - y1 = -1/m(x - x1)

where (x1, y1) is the given point that the line passes through. In this case, the given line has slope 5/4 and the given point is (-6,2), so the equation of the line perpendicular to 5/4x + 2 that passes through (-6,2) is:

y - 2 = -1/((5/4))(x + 6)

y - 2 = -4/5(x + 6)

y - 2 = -4/5x - 24/5

y = -4/5x - 22/5

Therefore, the equation of the line perpendicular to 5/4x + 2 that passes through (-6,2) is -4/5x - 22/5 (option "-4/5x - 22/5").

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