What are some researchable areas of Mathematics
Teaching? Answer briefly in 5 sentences. Thank you!

The correct option is:

c. y¹ = 3 + √(x - 7)

To find the inverse function of y = (x - 3)^2 + 7 for x > 3, we can follow these steps:

Step 1: Replace y with x and x with y in the given equation:

x = (y - 3)^2 + 7

Step 2: Solve the equation for y:

x - 7 = (y - 3)^2

√(x - 7) = y - 3

y - 3 = √(x - 7)

Step 3: Solve for y by adding 3 to both sides:

y = √(x - 7) + 3

So, the inverse function of y = (x - 3)^2 + 7 for x > 3 is y¹ = √(x - 7) + 3.

Therefore, the correct option is:

c. y¹ = 3 + √(x - 7)

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The explicit formula for the nth term (an) of the sequence 27, 9, 3, ... can be expressed as an = 27 / 3^(n-1), where n represents the position of the term in the sequence.

To find the explicit formula for the nth term of the sequence 27, 9, 3, ..., we need to identify the pattern or rule governing the sequence.

From the given sequence, we can observe that each term is obtained by dividing the previous term by 3. Specifically, the first term is 27, the second term is obtained by dividing 27 by 3, giving 9, and the third term is obtained by dividing 9 by 3, giving 3. This pattern continues as we divide each term by 3 to get the subsequent term.

Therefore, we can express the nth term, denoted as aₙ, as:

aₙ = 27 / 3^(n-1)

This formula states that to obtain the nth term, we start with 27 and divide it by 3 raised to the power of (n-1), where n represents the position of the term in the sequence.

For example:

When n = 1, the first term is a₁ = 27 / 3^(1-1) = 27 / 3^0 = 27.

When n = 2, the second term is a₂ = 27 / 3^(2-1) = 27 / 3^1 = 9.

When n = 3, the third term is a₃ = 27 / 3^(3-1) = 27 / 3^2 = 3.

Using this explicit formula, you can calculate any term of the sequence by plugging in the value of n into the formula.

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If a media planner wishes to run 120 adult 18-34 GRPS per week, the frequency of the advertisement needs to be 3 times per week.

The Gross Rating Point (GRP) is a metric that is used in advertising to measure the size of an advertiser's audience reach. It is measured by multiplying the percentage of the target audience reached by the number of impressions delivered. In other words, it is a calculation of how many people in a specific demographic will be exposed to an advertisement. For instance, if the GRP of a particular ad is 100, it means that the ad was seen by 100% of the target audience.

Frequency is the number of times an ad is aired on television or radio, and it is an essential aspect of media planning. A frequency of three times per week is ideal for an advertisement to have a significant impact on the audience. However, it is worth noting that the actual frequency needed to reach a specific audience may differ based on the demographic and the product or service being advertised.

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To find the area of triangle AEB, we use base AB (6 cm) and height 6 cm. Applying the formula (1/2) * base * height, the area is 18 cm².

To find the area of triangle AEB, we need to determine the length of the base AB and the height of the triangle. Since both square ABCD and triangle AABE is standing on the same base AB, the length of AB remains the same for both.

We are given that BD = 6 cm, which means that the length of AB is also 6 cm. Now, to find the height of the triangle, we can consider the height of the square. Since AB is the base of both the square and the triangle, the height of the square is equal to AB.

Therefore, the height of triangle AEB is also 6 cm. Now we can calculate the area of the triangle using the formula: Area = (1/2) * base * height. Plugging in the values, we get Area = (1/2) * 6 cm * 6 cm = 18 cm².

Thus, the area of triangle AEB is 18 square centimeters.

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The sequences are:1. Divergent2. Convergent (limit = 4/9)3. Convergent (limit = 1/4)

The following sequences are:

Aₙ = 9 + 4n³/n + 3n²  

Nₙ = Aₙ / N = (9 + 4n³/n + 3n²) / n³/9n+4  

Xₖ = Xₙ = n³ + 3n/Aₙ + n⁴

Let us determine whether each of the given sequences converges or diverges:

1. The first sequence is given by Aₙ = 9 + 4n³/n + 3n²Aₙ = 4n³/n + 3n² + 9 / 1

We can say that 4n³/n + 3n² → ∞ as n → ∞

So, the sequence diverges.

2. The second sequence is  

Nₙ = Aₙ / N = (9 + 4n³/n + 3n²) / n³/9n+4

Nₙ = (4/9)(n⁴)/(n⁴) + 4/3n → 4/9 as n → ∞

So, the sequence converges and its limit is 4/9.3. The third sequence is  

Xₖ = Xₙ = n³ + 3n/Aₙ + n⁴Xₖ = Xₙ = (n³/n³)(1 + 3/n²) / (4n³/n³ + 3n²/n³ + 9/n³) + n⁴/n³

The first term converges to 1 and the third term converges to 0. So, the given sequence converges and its limit is 1 / 4.

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185 said they like dogs, 170 said they like cats, 86 said they liked both cats and dogs, and 74 said they don't like cats or dogs. The number of people who were surveyed is 515.

The number of people who were surveyed can be found by adding the number of people who liked dogs, the number of people who liked cats, the number of people who liked both, and the number of people who did not like either. So, the total number of people surveyed can be found as follows:

Total number of people who like dogs = 185

Total number of people who like cats = 170

Total number of people who like both = 86

Total number of people who do not like cats or dogs = 74

The total number of people surveyed = Number of people who like dogs + Number of people who like cats + Number of people who like both + Number of people who do not like cats or dogs

= 185 + 170 + 86 + 74= 515

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we have shown that the expression ρ(x,t) = [1/(1 + t^2)] ρ0 [(x/(1 + t^2)) - t] satisfies the advection equation ∂ρ/∂t + v ∂ρ/∂x = -ρ ∂v/∂x.

The density of radioactive material, denoted by ρ(x,t), evolves according to the equation:

∂ρ/∂t + v ∂ρ/∂x = -ρ ∂v/∂x

This equation describes the transport of a substance by a moving medium, where the rate of movement of the radioactive material is influenced by the velocity of the wind, determined by the function v(x,t).

To solve the equation, we use the method of characteristics. We define the characteristic equation as:

x = ξ(t)

and

ρ(x,t) = f(ξ)

where f is a function of ξ.

Using the method of characteristics, we find that:

∂ρ/∂t = (∂f/∂t)ξ'

∂ρ/∂x = (∂f/∂ξ)ξ'

where ξ' = dξ/dt.

Substituting these derivatives into the original equation, we have:

(∂f/∂t)ξ' + v(∂f/∂ξ)ξ' = -ρ ∂v/∂x

Dividing by ξ', we get:

(∂f/∂t)/(∂f/∂ξ) = -ρ ∂v/∂x / v

Letting k(x,t) = -ρ ∂v/∂x / v, we can integrate the above equation to obtain f(ξ,t). Since f(ξ,t) = ρ(x,t), we can express the solution ρ(x,t) in terms of the initial value of ρ and the function k(x,t).

Now, let's solve the advection equation using the method of characteristics. We define the characteristic equation as:

x = x(t)

Then, we have:

dx/dt = v(x,t)

ρ(x,t) = f(x,t)

We need to find the function k(x,t) such that:

(∂f/∂t)/(∂f/∂x) = k(x,t)

Differentiating dx/dt = v(x,t) with respect to t, we have:

dx/dt = (2xt)/(1 + t^2) + 1 + t^2

Integrating this equation with respect to t, we obtain:

x = (x(0) + 1)t + x(0)t^2 + (1/3)t^3

where x(0) is the initial value of x at t = 0.

To determine the function C(x), we use the initial condition ρ(x,0) = ρ0(x).

Then, we have:

ρ(x,0) = f(x,0) = F[x - C(x), 0]

where F(ξ,0) = ρ0(ξ).

Integrating dx/dt = (2xt)/(1 + t^2) + 1 + t^2 with respect to x, we get:

t = (2/3) ln|2xt + (1 + t^2)x| + C(x)

where C(x) is the constant of integration.

Using the initial condition, we can express the solution f(x,t) as:

f(x,t) = F[x - C(x),t] = ρ0 [(x - C(x))/(1 + t^2)]

To simplify this expression, we introduce A(x,t) = (2/3) ln|2xt + (1 + t^2)x|/(1 + t^2). Then, we have:

f(x,t) = [1/(1 +

t^2)] ρ0 [(x - C(x))/(1 + t^2)] = [1/(1 + t^2)] ρ0 [(x/(1 + t^2)) - A(x,t)]

Finally, we can write the solution to the advection equation as:

ρ(x,t) = [1/(1 + t^2)] ρ0 [(x/(1 + t^2)) - A(x,t)]

where A(x,t) = (2/3) ln|2xt + (1 + t^2)x|/(1 + t^2).

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Answer:

the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.

Step-by-step explanation:

In a right triangle, one of the angles is always 90 degrees, which is the right angle. The acute angle in a right triangle is the angle that is smaller than 90 degrees.

To find the value of θ for the acute angle in a right triangle, given that sin(θ) = cos(53°), we can use the trigonometric identity:

sin(θ) = cos(90° - θ)

Since sin(θ) = cos(53°), we can equate them:

cos(90° - θ) = cos(53°)

To find the acute angle θ, we solve for θ by equating the angles inside the cosine function:

90° - θ = 53°

Subtracting 53° from both sides:

90° - 53° = θ

θ= 37°

Therefore, the value of θ for the acute angle in a right triangle, where sin(θ) = cos(53°), is 37 degrees.

The values of x, y, and z in the triangle are x = 4, y = 11, and z = 180 - (3x + 4) - (3x - 4).

In the given problem, we are asked to find the values of x, y, and z in a triangle. The information provided states that angle X is equal to 4 degrees and angle N is equal to 11 degrees. Additionally, we have two expressions involving x: (3x + 4) degrees and (3x - 4) degrees.

To find the value of y, we can use the fact that the sum of the interior angles in a triangle is always 180 degrees. In this case, we have x + y + z = 180. Plugging in the given values, we get 4 + 11 + z = 180. Solving for z, we find that z = 180 - 4 - 11 = 165 degrees.

To find the values of x and y, we can use the fact that the sum of the angles in a triangle is always 180 degrees. In this case, we have angle X + angle N + angle K = 180. Plugging in the given values, we get 4 + 11 + K = 180. Solving for K, we find that K = 180 - 4 - 11 = 165 degrees.

Therefore, the values of x, y, and z in the triangle are x = 4, y = 11, and z = 165 degrees.

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The correct answer is B. y=3x-2.

The slope of a line determines its steepness and direction. Parallel lines have the same slope, so for a line to be parallel to 3x+6y=5, it should have a slope of -1/2. Since none of the given options have this slope, none of them are parallel to the line 3x+6y=5. This line has the same slope of 3 as the given line, which makes them parallel.

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Answer:

Two triangles are said to be congruent if they are exactly identical. We know that a triangle has three angles and three sides. So, two triangles have six angles and six sides. If we can prove the any corresponding three of them of both triangles equal under certain rules, the triangles are congruent to each other. These rules are called axioms.

The method you will use depends on the information you are given about the triangles.

--> SSS(Side-Side-Side): If you know that all three sides of a triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.

--> SAS(Side-Angle-Side): If you know that two sides and the angle between those sides are equal to the another corresponding two sides and the angle between the two sides of another triangle, then you say that the triangles are congruent by SAS axiom.

--> ASA(Angle-Side-Angle): If you know that the two angles and the side between them are equal to the two corresponding angles and the side between those angles of another triangle are equal, you may say that the triangles are congruent by ASA axiom.
--> AAS(Angle-Angle-Side): This method is similar to the ASA axiom, but they are not same. In AAS axiom also you need to have two corresponding angles and a side of a triangle equal, but they should be in angle-angle-side order.

--> RHS(Right-Hypotenuse-Side) or HL(Hypotenuse-Leg): If hypotenuses and any two sides of two right triangles are equal, the triangles are said to be congruent by RHS axiom. You can only test this rule for the right triangles.

Answer:

So, there are four ways to figure out if two triangles are the same shape and size. One way is called SSS, which means all three sides of one triangle match up with the corresponding sides on the other triangle. Another way is called AAS, where two angles and one side of one triangle match two angles and one side of the other triangle. Then there's SAS, where two sides and the angle between them match up with the same parts on the other triangle. Finally, there's ASA, where two angles and a side in between them match up with the same parts on the other triangle.

(a) Without knowing the effect size, it is not possible to calculate the type II error for the given hypothesis test. (b) To detect a true mean diameter of 1.55 inches with a power of at least 0.9, approximately 65 bearings would be needed.

(a) If the true mean diameter is 1.55 inches, the probability of not rejecting the null hypothesis when it is false (i.e., the type II error) depends on the chosen significance level, sample size, and effect size. Without knowing the effect size, it is not possible to calculate the type II error.

(b) To calculate the required sample size to detect a true mean diameter of 1.55 inches with a power of at least 0.9, we need to know the chosen significance level, the standard deviation of the population, and the effect size.

Using a statistical power calculator or a sample size formula, we can determine that a sample size of approximately 65 bearings is needed.

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The general solution of the differential equation is: y(t) = c1e^t + c2te^t + c3t²e^t + c4e^(2t) + c5e^(3t)

Thus, c1, c2, c3, c4, and c5 are arbitrary constants.

To find the general solution of the differential equation y⁵ − 8y⁴ + 16y′′′ − 8y′′ + 15y′ = 0, we follow these steps:

Step 1: Substituting y = e^(rt) into the differential equation, we obtain the characteristic equation:

r⁵ − 8r⁴ + 16r³ − 8r² + 15r = 0

Step 2: Solving the characteristic equation, we factor it as follows:

r(r⁴ − 8r³ + 16r² − 8r + 15) = 0

Using the Rational Root Theorem, we find that the roots are:

r = 1 (with a multiplicity of 3)

r = 2

r = 3

Step 3: Finding the solution to the differential equation using the roots obtained in step 2 and the formula y = c1e^(r1t) + c2e^(r2t) + c3e^(r3t) + c4e^(r4t) + c5e^(r5t).

Therefore, the general solution of the differential equation is:

y(t) = c1e^t + c2te^t + c3t²e^t + c4e^(2t) + c5e^(3t)

Thus, c1, c2, c3, c4, and c5 are arbitrary constants.

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a) The work done by the force F = 5i + 3j + 2k on a body moving from the origin to the point (3, -1, 2) is 13 units.

b) A vector that is perpendicular to both u = -i + 2j - k and v = 2i - j + 3k is -6i - 7j - 3k.

a) The work done by a force F = 5i + 3j + 2k acting on a body that moves from the origin to the point (3, -1, 2) can be determined using the formula:

Work done = ∫F · ds

Where F is the force and ds is the displacement of the body. Displacement is defined as the change in the position vector of the body, which is given by the difference in the position vectors of the final point and the initial point:

s = rf - ri

In this case, s = (3i - j + 2k) - (0i + 0j + 0k) = 3i - j + 2k

Therefore, the work done is:

Work done = ∫F · ds = ∫₀ˢ (5i + 3j + 2k) · (ds)

Simplifying further:

Work done = ∫₀ˢ (5dx + 3dy + 2dz)

Evaluating the integral:

Work done = [5x + 3y + 2z]₀ˢ

Substituting the values:

Work done = [5(3) + 3(-1) + 2(2)] - [5(0) + 3(0) + 2(0)]

Therefore, the work done = 13 units.

b) To find a vector that is perpendicular to both u = -i + 2j - k and v = 2i - j + 3k, we can use the cross product of the two vectors:

u × v = |i j k|

|-1 2 -1|

|2 -1 3|

Expanding the determinant:

u × v = (-6)i - 7j - 3k

Therefore, a vector that is perpendicular to both u and v is given by:

u × v = -6i - 7j - 3k.

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The probability of both days being dry is 0.48 (48%), the probability of both days being wet is 0.08 (8%), and the probability of exactly one dry day is 0.44 (44%).

In the given scenario, we have two events: Monday being dry or wet, and Tuesday being dry or wet. We can represent this situation using a tree diagram:

```

         Dry (0.6)

       /         \

  Dry (0.8)    Wet (0.2)

    /               \

Dry (0.8)       Wet (0.4)

```

The branches represent the probabilities of each event occurring. Now we can answer the questions:

1. The probability of both days being dry is the product of the probabilities along the path: 0.6 ˣ 0.8 = 0.48 (or 48%).

2. The probability of both days being wet is the product of the probabilities along the path: 0.4ˣ  0.2 = 0.08 (or 8%).

3. The probability of exactly one dry day is the sum of the probabilities of the two mutually exclusive paths: 0.6 ˣ  0.2 + 0.4 ˣ  0.8 = 0.12 + 0.32 = 0.44 (or 44%).

By using the tree diagram and calculating the appropriate probabilities, we can determine the likelihood of different outcomes based on the given conditional and independent probabilities.

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The boat's coordinates are (10, 0).

A coordinate plane is a grid made up of vertical and horizontal lines that intersect at a point known as the origin. The origin is typically marked as point (0, 0). The horizontal line is known as the x-axis, while the vertical line is known as the y-axis.

The x-axis and y-axis split the plane into four quadrants, numbered I to IV counterclockwise starting at the upper-right quadrant. Points on the plane are described by an ordered pair of numbers, (x, y), where x represents the horizontal distance from the origin, and y represents the vertical distance from the origin, in that order.

The distance between any two points on the coordinate plane can be calculated using the distance formula. When it comes to the given question, we are given that Each unit on the coordinate plane represents 1 NM.

Since the boat is 10 NM east of the y-axis, the x-coordinate of the boat's position is 10. Since the boat is not on the y-axis, its y-coordinate is 0. Therefore, the boat's coordinates are (10, 0).

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The area of the given figure, we can divide it into two separate shapes: a rectangle and a right triangle. The area of the given figure is 30 cm².

First, let's calculate the area of the rectangle. The width of the rectangle is 5 cm, and the height is 4 cm. The area of a rectangle is given by the formula: A = length × width. Therefore, the area of the rectangle is:

Area of rectangle = 5 cm × 4 cm = 20 cm².

Next, let's calculate the area of the right triangle. The base of the triangle is 5 cm, and the height is 4 cm. The area of a triangle is given by the formula: A = 0.5 × base × height. Therefore, the area of the right triangle is: Area of triangle = 0.5 × 5 cm × 4 cm = 10 cm².

To find the total area of the figure, we add the area of the rectangle and the area of the triangle:

Total area = Area of rectangle + Area of triangle = 20 cm² + 10 cm² = 30 cm².

Therefore, the area of the given figure is 30 cm².

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he probability of getting one tail when a coin is tossed four times is A.

1/4

When a coin is tossed, there are two possible outcomes: heads (H) or tails (T). Since we are interested in getting exactly one tail, we can calculate the probability by considering the different combinations.

Out of the four tosses, there are four possible positions where the tail can occur: T _ _ _, _ T _ _, _ _ T _, _ _ _ T. The probability of getting one tail is the sum of the probabilities of these four cases.

Each individual toss has a probability of 1/2 of landing tails (T) since there are two equally likely outcomes (heads or tails) for a fair coin. Therefore, the probability of getting exactly one tail is:

P(one tail) = P(T _ _ _) + P(_ T _ _) + P(_ _ T _) + P(_ _ _ T) = (1/2) * (1/2) * (1/2) * (1/2) + (1/2) * (1/2) * (1/2) * (1/2) + (1/2) * (1/2) * (1/2) * (1/2) + (1/2) * (1/2) * (1/2) * (1/2) = 4 * (1/16) = 1/4.

Therefore, the probability of getting one tail when a coin is tossed four times is 1/4, which corresponds to option A.

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The speed of light in diamond is approximately 1.24 x 10⁸ meters per second.

The index of refraction (n) of a given media affects how fast light travels through it. The refractive is given as the speed of light divided by the speed of light in the medium.

n = c / v

Rearranging the equation, we can solve for the speed of light in the medium,

v = c / n

The refractive index of the diamond is given to e 2.42 so we can now replace the values,

v = c / 2.42

Thus, the speed of light in diamond is approximately 1.24 x 10⁸ meters per second.

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If the coefficient of x² in the equation f(x) = 3x² is changed to 3, the graph will be affected if the coefficient of x² is changed to the parabola will be narrower. Thus, option A is correct.

A. The parabola will be narrower.

The coefficient of x² determines the "steepness" or "narrowness" of the parabola. When the coefficient is increased, the parabola becomes narrower because it grows faster in the upward direction.

B. The parabola will not be wider.

Increasing the coefficient of x² does not result in a wider parabola. Instead, it makes the parabola narrower.

C. The parabola will not be translated down.

Changing the coefficient of x² does not affect the vertical translation (up or down) of the parabola. The translation is determined by the constant term or any term that adds or subtracts a value from the function.

D. The parabola will not be translated up.

Similarly, changing the coefficient of x² does not impact the vertical translation of the parabola. Any translation up or down is determined by other terms in the function.

In conclusion, if the coefficient of x² in the equation f(x) = x² is changed to 3, the parabola will become narrower, but there will be no translation in the vertical direction. Thus, option A is correct.

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Complete Question:

If the graph of f(x) = x², how will the graph be affected if the coefficient of x² is changed to 3?

A. The parabola will be narrower.

B. The parabola will be wider.

C. The parabola will be translated down.

D. The parabola will be translated up.

There are 19,600 ways to select three different tickets from the given pool of fifty tickets, the correct option is: c. 19,600

To determine the number of ways three different tickets can be selected from a pool of fifty tickets, we can use the concept of combinations. The number of combinations of selecting r items from a set of n items is given by the formula nCr = n! / (r!(n-r)!), where n! represents the factorial of n.

In this case, we need to calculate the number of ways to select 3 tickets from a pool of 50 tickets. Applying the formula, we have:

50C3 = 50! / (3!(50-3)!)

= 50! / (3!47!)

Simplifying further:

50C3 = (50 * 49 * 48 * 47!) / (3 * 2 * 1 * 47!)

= (50 * 49 * 48) / (3 * 2 * 1)

= 19600

Therefore, the correct answer is: c. 19,600

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Answer: SSS

Step-by-step explanation:

The lines on the triangles say that 2 of the sides are equal. Th triangles also share a 3rd side that is equal.

So, a side, a side and a side proves the triangles are congruent through, SSS

The three consecutive even integers are -38, -36, and -34.

Given f(x) = 2x-3 and g(x) = 5x + 4, the composite of f° g(x) = f(g(x)) can be calculated as follows:

Solution: A. Composite (f° g)(x):f(x) = 2x - 3 and g(x) = 5x + 4

Let's substitute the value of g(x) in f(x) to obtain the composite of f° g(x) = f(g(x))f(g(x))

= f(5x + 4)

= 2(5x + 4) - 3

= 10x + 5

B. Composite (g° f)(x):f(x)

= 2x - 3 and g(x)

= 5x + 4

Let's substitute the value of f(x) in g(x) to obtain the composite of g° f(x) = g(f(x))g(f(x))

= g(2x - 3)

= 5(2x - 3) + 4

= 10x - 11

C. Composite (f° g)(-3):

Let's calculate composite of f° g(-3)

= f(g(-3))f(g(-3))

= f(5(-3) + 4)

= -10 - 3

= -13

Given f(x) = x² - 8x - 9 and

g(x) = x²+ 6x + 5,

the composite of f° g(x) = f(g(x)) can be calculated as follows:

Solution: A. Composite (fog)(0):f(x) = x² - 8x - 9 and g(x)

= x² + 6x + 5

Let's substitute the value of g(x) in f(x) to obtain the composite of f° g(x) = f(g(x))f(g(x))

= f(x² + 6x + 5)

= (x² + 6x + 5)² - 8(x² + 6x + 5) - 9

= x⁴ + 12x³ - 31x² - 182x - 184

B. Composite (fog)(1):

Let's calculate composite of f° g(1) = f(g(1))f(g(1))

= f(1² + 6(1) + 5)= f(12)

= 12² - 8(12) - 9

= 111

C. Composite (g° f)(1):

Let's calculate composite of g° f(1) = g(f(1))g(f(1))

= g(2 - 3)

= g(-1)

= (-1)² + 6(-1) + 5= 0

The length and width of an envelope can be calculated as follows:

Solution: Let's assume the width of the envelope to be x.

The length of the envelope will be (x + 4) cm, as per the given conditions.

The area of the envelope is given as 96 cm².

So, the equation for the area of the envelope can be written as: x(x + 4) = 96x² + 4x - 96

= 0(x + 12)(x - 8) = 0

Thus, the width of the envelope is 8 cm and the length of the envelope is (8 + 4) = 12 cm.

Three consecutive even integers whose square difference is 76 can be calculated as follows:

Solution: Let's assume the three consecutive even integers to be x, x + 2, and x + 4.

The square of the third integer is 76 more than the square of the second integer.x² + 8x + 16

= (x + 2)² + 76x² + 8x + 16

= x² + 4x + 4 + 76x² + 4x - 56

= 0x² + 38x - 14x - 56

= 0x(x + 38) - 14(x + 38)

= 0(x - 14)(x + 38)

= 0x = 14 or

x = -38

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The measure of angle ADB is equal to the square root of ([tex]AB \times BA[/tex]).

In triangle ABC, let the angle bisectors drawn from vertices A and B intersect at point D. To find the measure of angle ADB, we can use the angle bisector theorem. According to this theorem, the angle bisector divides the opposite side in the ratio of the adjacent sides.

Let AD and BD intersect side BC at points E and F, respectively. Now, we have triangle ADE and triangle BDF.

Using the angle bisector theorem in triangle ADE, we can write:

AE/ED = AB/BD

Similarly, in triangle BDF, we have:

BF/FD = BA/AD

Since both angles ADB and ADF share the same side AD, we can combine the above equations to obtain:

(AE/ED) * (FD/BF) = (AB/BD) * (BA/AD)

By substituting the given angle bisector ratios and rearranging, we get:

(AD/BD) * (AD/BD) = (AB/BD) * (BA/AD)

AD^2 = AB * BA

Note: The solution provided assumes that points A, B, and C are non-collinear and that the triangle is non-degenerate.

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Answer:

The percent error is -2.1352% of Jocelyn's estimate.

Answer: 33

Step-by-step explanation:

Area ABC = Area of largest triangle - all the other shapes.

Area of largest = 1/2 bh

Area of largest = 1/2 (6+12)(8+5)

Area of largest = 1/2 (18)(13)

Area of largest = 117

Other shapes:

Area Left small triangle = 1/2 bh

Area Left small triangle = 1/2 (8)(6)

Area Left small triangle = (4)(6)

Area Left small triangle = 24

Area Right small triangle = 1/2 bh

Area Right small triangle = 1/2 (12)(5)

Area Right small triangle =30

Area of rectangle = bh

Area of rectangle = (6)(5)

Area of rectangle = 30

area of ABC = 117 - 24 - 30 - 30

Area of ABC = 33

The normal strain developed in each wire is 0.00367 or 0.367%.

To determine the normal strain developed in each wire, we need to consider the relationship between strain, displacement, and original length.

Ulameter length: 30 mm

Displacement of point A: 2.2 mm

To find the normal strain, we can use the formula:

strain = (displacement) / (original length)

For the upper wire:

Original length = 600 mm

Strain in upper wire = (2.2 mm) / (600 mm) = 0.00367 or 0.367%

For the lower wire:

Original length = 600 mm

Strain in lower wire = (2.2 mm) / (600 mm) = 0.00367 or 0.367%

Therefore, the normal strain developed in each wire is 0.00367 or 0.367%.

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The roots of the equation are;

a. (n +2)(n -8)

b. (x-5)(x-3)

From the information given, we have the expressions as;

f(x) = n² - 6n - 16

Using the factorization method, we have to find the pair factors of the product of the constant and x square, we have;

a. n² -8n + 2n - 16

Group in pairs, we have;

n(n -8) + 2(n -8)

Then, we get;

(n +2)(n -8)

b. y = x² - 8x + 15

Using the factorization method, we have;

x² - 5x - 3x + 15

group in pairs, we have;

x(x -5) - 3(x - 5)

(x-5)(x-3)

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The unique solution to the initial value problem is: y = 1 + x + 6x².

To solve the non-homogeneous problem y" = 12(2x²), let's go through the steps:

1) Homogeneous problem:

The homogeneous equation is y" = 0. The auxiliary equation is ar² + br + c = 0.

2) The roots of the auxiliary equation:

Since the coefficient of the y" term is 0, the auxiliary equation simplifies to just c = 0. Therefore, the root of the auxiliary equation is r = 0.

3) Fundamental set of solutions:

For the homogeneous problem y" = 0, since we have a repeated root r = 0, the fundamental set of solutions is Y₁ = 1 and Y₂ = x. So the complementary solution is Yc = C₁(1) + C₂(x) = C₁ + C₂x, where C₁ and C₂ are arbitrary constants.

4) Particular solution:

To find a particular solution, we can use the method of undetermined coefficients. Since the non-homogeneous term is 12(2x²), we assume a particular solution of the form yp = Ax² + Bx + C, where A, B, and C are constants to be determined.

Taking the derivatives of yp, we have:

yp' = 2Ax + B,

yp" = 2A.

Substituting these into the non-homogeneous equation, we get:

2A = 12(2x²),

A = 12x² / 2,

A = 6x².

Therefore, the particular solution is yp = 6x².

5) General solution and initial value problem:

The general solution is the sum of the complementary solution and the particular solution:

y = Yc + yp = C₁ + C₂x + 6x².

To solve the initial value problem y(0) = 1 and y'(0) = 1, we substitute the initial conditions into the general solution:

y(0) = C₁ + C₂(0) + 6(0)² = C₁ = 1,

y'(0) = C₂ + 12(0) = C₂ = 1.

Therefore, the unique solution to the initial value problem is:

y = 1 + x + 6x².

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