(a) Solve x² + 2x - 21 < 13.

The 25° angle the light ray makes with the normal, B which is the angle of incidence, according to the laws of reflection, indicates that the angle of reflection, which is the angle the reflected ray makes with the normal, C, is 25°

The laws of reflection states that (1) The angle of incidence and the angle of reflection have the same value (2) The incident ray, the reflected ray and the normal are coplanar (3) The refracted ray is on the opposite side of the normal with regards to the incident ray.

The law of reflection states that the angle the in of reflection and the angle of incidence are equal; [tex]\theta_i = \theta_r[/tex]

The angle the incident ray makes with the normal of the plain mirror and the angle the reflected ray makes with the normal are the same.

The angle the incident ray in the diagram of the question makes with the normal line segment, B = 25°

The angle the reflected ray makes with the normal line segment = C

Therefore, according the to the laws of reflection, we get;

Angle B = Angle C

Angle B = 25°

Therefore;

Angle C = Angle B = 25° (symmetric property)

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The power series expression for the following integral ∫x[tex]e^{2x^{3} }[/tex]dx is:

∫x[tex]e^{2x^{3} }[/tex]dx=(x²/2)+(2x⁵/5)+(x⁸/4)+(4x¹¹/33)+(x¹³/21)+......

A power series is an infinite series in mathematics where an is the coefficient of the nth component and c is a constant. Power series, which develop as Taylor series of indefinitely differentiable functions, are helpful in mathematical analysis.

For square matrices (for which the function is known as the matrix exponential), as well as more generally in any unital Banach algebra B, the exponential function's power series definition makes sense.

A power series is made up of terms with the generic form. Power series are functions because the selected x-value determines whether the series converges or diverges as well as the value it converges to.

Given integral is,

∫x[tex]e^{2x^{3} }[/tex]dx

We know that the Taylor series expansion for eˣ.

eˣ=1+x+(x₂/!)+(x₃/3!)+(x₄/4!).....∞

[tex]e^{2x^{3} }[/tex]=1+2x₃+((2x³)²/2!)+((2x³)³/3!)+((2x³)⁴/4!)+....

[tex]e^{2x^{3} }[/tex]=1+2x³+2x⁶+(4x⁹/3)+(2x¹²/3)+........

x[tex]e^{2x^{3} }[/tex]=x+2x⁴+2x⁷+(4x¹⁰/3)+(2x¹³/3)+......

Now, integral the power series expansion:

∫x[tex]e^{2x^{3} }[/tex]dx=∫(x+2x⁴+2x⁷+(4x¹⁰/3)+(2x¹³/3)+......)dx

∫x[tex]e^{2x^{3} }[/tex]dx=(x²/2)+(2x⁵/5)+(x⁸/4)+(4x¹¹/33)+(x¹³/21)+......

This is the required solution.

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Answer:

Step-by-step explanation:

A

Answer:

C

Step-by-step explanation:

substitute x = 9 into f(x) and evaluate for output

f(x) = [tex]\frac{1}{2}[/tex] (3x + 7)

f(9) = [tex]\frac{1}{2}[/tex] (3(9) + 7)

     = [tex]\frac{1}{2}[/tex] (27 + 7)

     = [tex]\frac{1}{2}[/tex] × 34

     = 17

x+y=14

What is equation?

A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance. Equations come in a variety of forms, including linear, quadratic, cubic, and others. In this tutorial, let's study more about math equations.

Let x = number of 5 ft long strings and 

y = number of 11 ft long strings

Consequently, 5x+11y=106,

because the whole string is 14,

x+y=14.

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The possible reasons might be different that the shape is not a perfect circle as it is difficult to draw a perfect circle and the Measurements could be rounded, not exact.

A circle is a shape consisting of all points in a plane that are given the same distance from a given point called the center.

Consider the provided information that Rita measured the diameter of a circle in two different directions. Measuring other way, she got 3.5 cm, and measuring horizontally, she got 3.6 cm.

The diameters of a circle must have the same length, If the length are not same then the reasons can be:

The shape is not a perfect circle as well it is difficult to draw a perfect circle.

Measurements could be rounded, not exact which may be possible that the thickness of the circle affected the measurement.

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Answer:

The possible values of the angle \alpha that satisfy the given equation are the angles in the range [0, 60] degrees or the angles in the range [300, 360] degrees. This is because we showed that the value of cos(\alpha) must be less than or equal to 1/\sqrt{3} in order for the equation to hold.

Step-by-step explanation:

It looks like you have entered some LaTeX code in your question, but it is not properly formatted. Here is the correct way to enter the equation:

cos(\alpha) + \frac{1}{\sqrt{3}} \cdot sin(\alpha) = 1

To solve this equation, we can use the identity cos^2(\alpha) + sin^2(\alpha) = 1. We can rewrite the left-hand side of the equation as follows:

cos^2(\alpha) + \frac{1}{3} \cdot sin^2(\alpha) = 1

Then, we can solve for sin^2(\alpha):

sin^2(\alpha) = 3 \cdot (1 - cos^2(\alpha))

Since sin^2(\alpha) is nonnegative, we know that cos^2(\alpha) must be less than or equal to 1/3. This means that cos(\alpha) must be less than or equal to 1/\sqrt{3}. Therefore, the possible values of \alpha that satisfy the given equation are the angles in the range [0, 60] degrees or the angles in the range [300, 360] degrees.

Note that this solution assumes that the angle \alpha is measured in degrees. If the angle is measured in radians, then the range of possible values for \alpha will be different.

Answer:

1

Step-by-step explanation:

7 can only go into 10 once because 2 7's is 14 and that's higher than 10.

If the tip was more than 10 percent but less than 15 percent of the cost of the meal, then the total amount paid must have been between

2- 39$ and 41$.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

Given, A restaurant meal cost $35.50 and there was no tax.

Now the tax is more than 10% and less than 15% which is,

= (110/100)×35.50 and (115/100)×35.50.

= $39.05 and $40.825.

So, from the options, the total amount paid must have been between

39$ and 41$.

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The values of the ED and DC are 12 and 18 respectively.

What is the centroid of the triangle?

The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrencies of a triangle. The medians of a triangle are constructed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle.

a) EC = 18

We have,

Point D is the centroid of ΔABC,

a) EC = 18

b) EC = 27

The centroid theorem states that the centroid of the triangle is at 2/3 of the distance from the vertex to the mid-point of the sides.

Here, ΔABC is a triangle having a centroid D. F, G and E are the midpoints of the sides of the triangle BC, AC, and AB, respectively. Hence as per the theorem;

DE = 2/3 CE,

DG = 2/3 BG and

DF = 2/3 BF.

So,

a) EC = 18

ED = 2/3 EC,

ED = 2/3 * 18,

ED = 2 * 6,

ED = 12,

b) EC = 27

DC = 2/3 EC,

= 2/3 * 27,

= 2 * 9,

DC = 18,

Hence, the values of the ED and DC are 12 and 18 respectively.

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Answer:

10.  ED = 6  and  DC = 12

11.  ED = 9  and  DC = 18

Step-by-step explanation:

The centroid of a triangle is the point of intersection of its medians.

A median is a line that joins the vertex of a triangle to the midpoint of the opposite side.

Centroid Theorem

The centroid of the triangle is two-thirds of the distance from the vertex to the midpoint of the opposite side.

Given:

EC is a median that joins vertex C to midpoint E.

Therefore, DC is two-thirds of EC:

[tex]\implies \sf DC=\dfrac{2}{3}EC[/tex]

[tex]\implies \sf DC=\dfrac{2}{3} \cdot 18[/tex]

[tex]\implies \sf DC=\dfrac{36}{3}[/tex]

[tex]\implies \sf DC=12[/tex]

And ED is one-third of EC:

[tex]\implies \sf ED=\dfrac{1}{3}EC[/tex]

[tex]\implies \sf ED=\dfrac{1}{3}\cdot 18[/tex]

[tex]\implies \sf ED=\dfrac{18}{3}[/tex]

[tex]\implies \sf ED=6[/tex]

Given:

EC is a median that joins vertex C to midpoint E.

Therefore, DC is two-thirds of EC:

[tex]\implies \sf DC=\dfrac{2}{3}EC[/tex]

[tex]\implies \sf DC=\dfrac{2}{3} \cdot 27[/tex]

[tex]\implies \sf DC=\dfrac{54}{3}[/tex]

[tex]\implies \sf DC=18[/tex]

And ED is one-third of EC:

[tex]\implies \sf ED=\dfrac{1}{3}EC[/tex]

[tex]\implies \sf ED=\dfrac{1}{3}\cdot 27[/tex]

[tex]\implies \sf ED=\dfrac{27}{3}[/tex]

[tex]\implies \sf ED=9[/tex]

The remainder of the given polynomials is x-3 is 12x³+x.

A polynomial is a type of algebraic expression in which the exponents of all variables should be a whole number. The exponents of the variables in any polynomial have to be a non-negative integer. A polynomial comprises constants and variables, but we cannot perform division operations by a variable in polynomials.

Given that, the polynomial [tex]x^5+x^3+x[/tex] is divided by x-3.

Here, [tex]x^5+x^3+x[/tex]/(x-3)

ow,

x-3 |[tex]x^5+x^3+x[/tex]| [tex]x^4[/tex][tex]+4x^3[/tex]

    -([tex]x^5-3x^4[/tex])

   __________

             [tex]4x^4[/tex][tex]+x[/tex]

           -([tex]4x^4[/tex][tex]-12x^3[/tex])

         __________

             12x³+x

Therefore, the remainder of the given polynomials is x-3 is 12x³+x.

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Answer:

The correct answer is "The APR on the unpaid balance".

Jerome should give importance to the APR on his unpaid balance, since he just applied for two more credit cards, while he already has three. If he plans to apply for two more, he should give attention to the unpaid balances, because the APR may increase due to his missed payments, which will also result in an increase in payments to his credit cards.

Step-by-step explanation:

The height of the given pentagonal such that its base area is 42 units² will be 7.9 units.

Volume is the scalar quantity of any object that specified occupied space in 3D.

For example, the space in our room is referred to as volume.

The volume of the cuboid = length × height × width.

Suppose that height of the pentagonal is h.

Volume = base area x height h

331.8 = 42 x h

h = 7.9 units.

Hence "The height of the specified pentagonal will be 7.9 units such that its base area is 42 units²".

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Step-by-step explanation:

1 mile = 5280 ft

1 hour = 3600 seconds

65 miles / hour =

= 65 × 5280 ft / 3600 seconds = 65 × 528 ft / 360 s =

= 13 × 264 ft / 36 s = 13 × 66 ft / 9 s = 13 × 22 ft / 3 s =

= 13 × 11 ft / 1.5 s = 143 ft / 1.5 s

so, at 65 mph the driver travels 143 ft before the driver reacts (1.5 s) to an obstacle.

The value of children ticket is $7 and adults tickets is $10.

From the information, Two adult tickets and 4 child tickets cost $48. This will be:

2x + 4y = 48

Five adult tickets and 2 child tickets cost $64. This will be:

5x + 2y = 64

x = adult tickets

y = child tickets.

The two equations will be combined thus:

2x + 4y = 48

5x + 2y = 64

Multiply equation i by 5

Multiply equation ii by 2

10x + 20y = 240

10x + 4y = 128

Subtract the equations

16y = 112

Divide

y = 112/16

y = 7

Children tickets is $7

2x + 4y = 48

2x + 4(7) = 48

2x + 28 = 48

x = (48 - 28)/2

x = 10

Adult tickets cost $10.

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f(g(7)) = 99

g(7) = 7 + 4 = 11

f(11) = 9(11) = 99

g(f(5)) = 49

f(5) = 9(5) = 45

g(45) = 45 + 4 = 49

Hope this helps!

A seismograph station in a particular city might not record any data about some earthquakes because the ground motion it produced might not be strong enough to be detected by the station.

Seismograph stations measure ground motion, and the farther an earthquake is from a station, the weaker the ground motion it produces. If an earthquake is too far away, the ground motion it produces may be too weak to be detected by the seismograph station.

Additionally, if the earthquake occurred in a direction that was not directly aligned with the seismograph station, the ground motion it produced might not be strong enough to be detected by the station.

Thus, a seismograph station in a particular city does not record any data about some earthquakes.

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The value of 45 less than the product of 18 and a number written as an algebraic expression will be B. 45 - 18n

Expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations. As an illustration, the expression x + y has the terms x and y with an addition operator between them.

45 less than the product of 18 and a number written as an algebraic expression will be:

= 45 - (18 × n)

= 45 - 18n.

The correct option is B.

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As BD intersects AC and EC in the same ratio length of AB is 1.5 units.

A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.

As BD is parallel to AD BD intersects AC and EC in the same ratio.

Now, CD is 8 units and DE is 2 units so, CD is (8/2) = 4 times larger than

DE.

Therefore BC is also 4 times larger than AB,

So, 4AB = BC.

4AB = 6.

AB = 1.5

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If a pole has two wires attached to it, one on each side, forming two right triangles as shown, then the height of the pole is 29.6 feet

Consider the triangle on left hand side

The length of the base = 34 feet

The measure of angle = 41 degrees

Opposite side is the height of the pole

Here we have to use the trigonometric functions

sin θ = Opposite side / Hypotenuse

cos θ = Adjacent side / Hypotenuse

tan θ = Opposite side / Adjacent side

Here we have to use the equation of tan θ

Substitute the values in the equation

tan 41 = Opposite side / 34 feet

Opposite side = 34 × tan 41

= 29.6 feet

Therefore, the height of the pole is 29.6 feet

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The required value of c is 16 which is the correct answer that would be an option (C).

In triangle LMN, m∠L = (4c + 47). If the exterior angle to L measures 69°,

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it.

In the case of triangle LMN, we can use the given information to write the following equation:

⇒ 180 - 69 =  (4c + 47)

⇒ 111 = 4c + 47

⇒ 4c = 111- 47

⇒ 4c = 64

⇒ c = 64/4

Apply the division operation, and we get

⇒ c = 16

Thus, the value of c is 16.

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Answer:

Sine:O/H

cos:A/H

Tan:O/A

1-a

2-c

3-b

Answer:

  Ms finch has the higher ratio

Step-by-step explanation:

Given ratios 2 : 6 and 3 : 5, you want to know which is greater.

There are several ways to compare ratios. You can express them over a common denominator, compare them to a common value, convert them to decimal, subtract one from the other.

  2 : 6 vs 3 : 5 = 10 : 30 vs 18 : 30

3 : 5 is greater than 2 : 6, so Ms finch has the higher ratio.

  2 : 6 vs 3 : 5

We can compare each of these to 1/2.

  2 : 6 < 3 : 6 = 1/2

  3 : 5 > 2.5 : 5 = 1/2

3 : 5 is greater than 2 : 6, so Ms finch has the higher ratio.

  2 : 6 ≈ 0.333

  3 : 5 = 0.600

3 : 5 is greater than 2 : 6, so Ms finch has the higher ratio.

  2/6 -3/5 = (10 -18)/(30) = -8/30 so the second ratio is higher

3 : 5 is greater than 2 : 6, so Ms finch has the higher ratio.

Answer:

x = 6

Step-by-step explanation:

∠ 1 and ∠ 2 are alternate exterior angles and are congruent , then

5x - 4 = 3x + 8 ( subtract 3x from both sides )

2x - 4 = 8 ( add 4 to both sides )

2x = 12 ( divide both sides by 2 )

x = 6

Answer:

A. r - s - t

Step-by-step explanation:

Observe the bounds of each integral. (Think of it as a number line if it's confusing.)
r is [1, 4]

s is [1, 2]

t is [3, 4]

Name the integral from [2, 3] = u.

s + t + u = r

u = r - s - t

Answer:

x=2.5

Step-by-step explanation:

Answer:

90

Step-by-step explanation:

Let's call the initial distance from the jet to the airport $x$. We are told that $x = 5$ miles.

Let's call the initial ground distance from the projection of the jet onto the ground to the airport $y$. We are told that $y = 4$ miles.

Let's call the rate at which the distance from the jet to the airport is increasing $v_x$. We are told that $v_x = 180$ miles/hour.

Let's call the rate at which the ground distance from the projection of the jet to the ground to the airport is increasing $v_y$. We are told that $v_y = 150$ miles/hour.

We want to find the vertical speed of the jet at this moment in time. Let's call this quantity $v_z$.

Since the jet is flying in the eastward direction, we can use the Pythagorean theorem to relate the three sides of the right triangle formed by the jet, the projection of the jet onto the ground, and the airport. Specifically, we have

$$x^2 = y^2 + z^2$$

where $z$ is the vertical distance from the jet to the ground (i.e., the height of the jet above the ground).

Since the distance from the jet to the airport and the ground distance from the projection of the jet to the airport are both increasing, we can take the derivative of both sides of the equation above to find the rate of change of $z$ with respect to time. This gives us

$$2x \frac{dx}{dt} = 2y \frac{dy}{dt} + 2z \frac{dz}{dt}$$

where $t$ is time.

We are given the values of $x$, $y$, $v_x$, and $v_y$, so we can plug these values into the equation above to solve for $v_z$. This gives us

$$2 \cdot 5 \cdot 180 = 2 \cdot 4 \cdot 150 + 2z \frac{dz}{dt}$$

Solving for $\frac{dz}{dt}$, we get

$$\frac{dz}{dt} = \frac{2 \cdot 5 \cdot 180 - 2 \cdot 4 \cdot 150}{2z}$$

We are told that $z = x - y$, so we can plug this into the equation above to find $\frac{dz}{dt}$. This gives us

$$\frac{dz}{dt} = \frac{2 \cdot 5 \cdot 180 - 2 \cdot 4 \cdot 150}{2(x-y)}$$

Substituting the values of the known quantities, we get

$$\frac{dz}{dt} = \frac{2 \cdot 5 \cdot 180 - 2 \cdot 4 \cdot 150}{2(5-4)} = \frac{180}{2} = 90$$

Therefore, the vertical speed of the jet at this moment in time is $v_z = 90$ miles/hour.

The vertical speed of the jet is approximately 300 miles/hour.

Velocity is the pace and direction of an item's movement, whereas speed is the time rate at which an object is traveling along a route.

Let's call the distance from the jet to the airport "d" and the height of the jet "h". Then we have two related rates:

d is increasing at a rate of 180 miles/hour.

The horizontal distance is increasing at a rate of 150 miles/hour.

Using the Pythagorean theorem, we can relate these two distances:

d² = h² + 4²

Taking the derivative of both sides with respect to time, we get:

2d(dd/dt) = 2h(dh/dt)

Simplifying, we get:

dd/dt = (h/d)(dh/dt)

We're looking for the vertical speed of the jet, which is dh/dt. We can solve for this by plugging in the given values:

d = 5 miles

dd/dt = 180 miles/hour

h² + 4² = 5² (from the Pythagorean theorem)

dh/dt = ?

Solving for h, we get:

h² = 5² - 4² = 9

h = 3 miles

Now we can plug in all the values and solve for dh/dt:

dh/dt = (dd/dt)(d/h) = (180 miles/hour)(5 miles/3 miles) ≈ 300 miles/hour

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Answer: 4 hours

Step-by-step explanation:

in a bacteria growing experiment, a biologist will observe that the number of bacteria in a certain culture triples every 4 HOURS. After 12 hours, it is.

The discount amount would Chester receive from using his coupon is $17.34

What is the percentage?

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.

The Latin word "per centum," which means "by the hundred," is where the word "percentage" originally came from. With 100 as the denominator, percentages are fractions. To put it another way, it's the relationship between a component and a whole in which the value of the whole is always assumed to be 100.

2/5 ---> .40---> 40%

$28.90(.40)= $11.56

$28.90-$11.56

=$17.34

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