During a fireworks display, a shell is shot into the air with a velocity of 58 m/s, its height in meters after t seconds is given by y=58t-20t^2
Find velocity when t=0.5

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: A

Step-by-step explanation:

From the statement.

Times = Multiplication

The sum of a number and 18.3 = (r + 18.3)

Equals = 27.9

Therefore,

-6 x (r + 18.3) = 27.9

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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Answer:10

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

If log₁₀2=0.3010, the value of  log₂10 will be 3.3×10⁻⁴.Logarithm identity is used to find the value of the log₂10.

It is another way to represent the power of numbers, and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.

[tex]\rm a^b = c\\log_ac =b[/tex]

It is given that, log₁₀2=0.3010

We have to find the log₂10

The following logarithm identity is used as,

logₐn=logₙa

log₂10 =1/ log₁₀2

log₂10 =1/ 0.3010

log₂10 =3.3×10⁻⁴

Thus, if log₁₀2=0.3010, the value of  log₂10 will be 3.3×10⁻⁴. Logarithm identity is used to find the value of the log₂10.

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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:

x + y = 87

1.5x + 0.5y = 78.50

Step-by-step explanation:

Let x be the number of nachos sold and y be the number of sodas sold

x nachos will cost 1.50x dollars
y sodas will cost 0.50y dollars

So total sales = 1.5x + 0.5y and we know he made $78.50

so one equation is
1.5x + 0.5y = 78.50

Total sold = x + y and we know that this number is 87

So the second equation is

x + y = 87

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..)

In linear equation, 96 is the quotient.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.

                             Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

f(x) =  6x^3-+18x^2-18x+12

x - 2 = 0

x = 2

f( x) = 6 * 8 + 18 * 4 - 18 * 2 + 12

     =  48 + 72 - 36 + 12

      = 132 - 36

      = 96

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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:

-9

Step-by-step explanation:

-3+-6=-9

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]

For the first graph, slope = 1/5.

For the second graph, slope = -1/2.

For the third graph, slope = 2.

For the fourth graph, slope = 1.42.

What is the slope of the line?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

                  [tex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

For the first graph, the coordinates will be (-15, 0) and (0, 3)

          Slope = 3 - 0/0-(-15)

                     = 3/15

            Slope = 1/5

For the second graph, the coordinates will be (0, 5) and (10, 0)

          Slope = 0 - 5/10- 0

                     = -5/10

            Slope = -1/2

For the third graph, the coordinates will be (2.5, 0) and (0, -5)

          Slope = -5-0/0 - 2.5

                     = -5/-2.5

            Slope = 2

For the fourth graph, the coordinates will be (-2.8, 0) and (0, 4)

          Slope = 4 - 0/0 - (-2.8)

                     = 4/2.8

            Slope = 1.42

Hence,

For the first graph, slope = 1/5.

For the second graph, slope = -1/2.

For the third graph, slope = 2.

For the fourth graph, slope = 1.42.

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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:155m

Step-by-step explanation:GK is congruent to HI

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]

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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A circle is a two-dimensional figure with a radius and circumference of

2πr.

The arc length of AB is 10.05.

A circle is a two-dimensional figure with a radius and circumference of

2πr.

We have,

The radius of the circle = 12

The circumference of the given circle.

= 2πr

This means,

The whole circle is 360°.

= 360/360 x 2πr

= 2πr

Now,

We need to find 48°

So,

The arc length of AB.

= 48/360 x 2πr

= 48/360 x 2 x 3.14 x 12

= 48/30 x 2 x 3.14

= 10.048

Rounding to the nearest hundredths.

= 10.05

Thus,

The arc length of AB is 10.05.

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Answer: 10.05

Step-by-step explanation:

The arc length of AB.

= 48/360 x 2πr

= 48/360 x 2 x 3.14 x 12

= 48/30 x 2 x 3.14

= 10.048

Rounding to the nearest hundredths.

= 10.05

The arc length of AB is 10.05.

Ty's visibility if out the upper window of a building with an altitude of 1,602 feet is 49.

What is altitude?

The height of an object in relation to a planet's or a natural satellite's surface, like sea level or land. the degree to which a celestial body is elevated above the horizon.

Here, we have

Given

The formula V = 1.225 √a is used to estimate how far a person can see on a clear day, where V is the visibility in miles and A is the altitude in feet.

We have to find Ty's visibility out the upper window of a building with an altitude of 1,602 feet.

We know that

V = 1.225 √a

a = 1,602 feet.

V = 1.225 √1602

V = 49

Hence, Ty's visibility out the upper window of a building with an altitude of 1,602 feet is 49.

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The quadratic equation of the function C(t) is; C(t)= (x - 4)²

We are given the coordinates as;

When t = 2, C(t) = 4

When t = 3, C(t) = 1

When t = 4, C(t) = 0

When t = 5, C(t) = 1

When t = 6, C(t) = 4

The general formula formula for quadratic equation is;

y = ax² + bx + c

where c is y-intercept

We are given that x = 4, when y = 0. Thus, x = 4 is a root of the equation.

Put x = 2and y = 4  into the equation to get;

4a + 2b + c = 4    -----(1)

Put x = 3 and y = 1  into the equation to get;

9a + 3b + c = 1    -----(2)

Put x = 4 and y = 0  into the equation to get;

16a + 4b + c = 0    -----(3)

Using simultaneous equation solver gives;

a = 1, b = -8 and c = 16. Thus, our equation is;

y = x² - 8x + 16

By factorization;

y = x² - 4x - 4x + 16

y = x(x - 4) - 4(x - 4)

y = (x - 4)²

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Answer:

  a.  $3155.19

  b.  $1,135,868.40

  c.  $705,868.40

  d.  interest

Step-by-step explanation:

You want to find the monthly payment, total repaid, and finance charge on a 30-year mortgage for $430,000 at the rate of 8%.

You can use a spreadsheet, financial calculator, app, or the amortization formula to find the payment amount.

  A = P(r/12)/(1 -(1 +r/12)^(-12t))

where P is the principal amount of the loan, r is the annual interest rate, and t is the number of years.

  A = 430,000(0.08/12)/(1 -(1 +0.08/12)^(-12·30)) ≈ 3155.19

The monthly payment is $3,155.19.

The total of the 12·30 = 360 payments is ...

  360 · $3155.19 = $1,135,868.40

It will take about $1,135,868.40 to pay off the loan.

The finance charge is the difference between the amount repaid and the original loan amount:

  $1135868.40 -430000 = $705,868.40

The finance charge is $705,868.40.

The finance charge at $706k is considerably greater than the original cost of the home, $430k.

The finance charge is greater.

If a graph G has an Euler path, then it must include precisely two odd vertices. Or, to put it another way, G cannot have an Euler path if the number of odd vertices is less than 2.

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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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B. There is no function rule

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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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: -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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The vertex labels of Δ QRT are : ∠2 label is Q, ∠8 label is T and the remaining vertex label is R. The vertex labels for ΔSTR are: ∠5 label is S, ∠9 label is R and the remaining label is T.

The polygon having three sides and three vertices are known as triangle.

If we separate the two triangles as shown in the figure. We can find the vertex labels with the help of the angles marked in the figure.

For Δ QRT, the vertex labels are:

∠2 is Q, ∠1 is R and ∠8 is T

Similarly. for ΔSTR, the vertex labels are:

∠5 is S, ∠6 is T, and ∠9 is R

Hence,  For Δ QRT, we can label as ∠2 is Q, ∠1 is R and ∠8 is T and for ΔSTR we can label as  ∠5 is S, ∠6 is T, and ∠9 is R.

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If the midpoint of AB is at (0,4). and A = (3,0) then B is at point (-3, 8)

The formula to determine the center point of a straight line using the coordinates of its endpoints is known as the midpoint formula in coordinate geometry.

The formula is of the form

X= (x₂ - x₁)/2

Y = (y₂ - y₁)/2

solving for B (x₂, y₂)

0 = (x₂ + 3)/2

0 = x₂ + 3

x₂ = -3

4 = (y₂ + 0)/2

8 = y₂ + 0

y₂ = 8

point B = (-3, 8)

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