A pilot flies in a straight path for 4 hours. He then makes a course correction, heading 20° to the right of his original course, and flies 4 hours in the new direction. If he maintains a constant speed of 115 miles per hour, how far is he from his starting position? Round your final answer to the nearest mile.

For the function f(x)=3x−6

(a) f(a) = 3a - 6

(b) f(a+h) = 3a + 3h - 6

(c) f(a+h) - f(a) = 3h

(d) (f(a+h) - f(a)) / h = 3

Given the function f(x) = 3x - 6, let's compute the requested expressions:

(a) Compute f(a):

To compute f(a), substitute the value of a into the function f(x):

f(a) = 3a - 6

(b) Compute and simplify f(a+h):

To compute f(a+h), substitute the value of a+h into the function f(x):

f(a+h) = 3(a+h) - 6

= 3a + 3h - 6

(c) Compute and simplify f(a+h) - f(a):

To compute f(a+h) - f(a), substitute the expressions for f(a+h) and f(a) into the equation:

f(a+h) - f(a) = (3a + 3h - 6) - (3a - 6)

= 3a + 3h - 6 - 3a + 6

= 3h

(d) Compute and simplify (f(a+h) - f(a)) / h:

To compute (f(a+h) - f(a)) / h, substitute the expression for f(a+h) - f(a) into the equation:

(f(a+h) - f(a)) / h = (3h) / h

= 3

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As an integer the evaluation comes out to be 5 for the expression +3/24-2%1

Expression is: 5 + 3/24 - 2 % 1.

We will solve this expression step by step:

1) we will solve the modulo operation: 2 % 1 = 0.

2) we will solve the division operation: 3/24 = 0.125.

Now, we will substitute the values in the given expression: 5 + 0.125 - 0 = 5.125.

Since we need to evaluate the expression as an integer, we will round it off to the nearest integer.5.125 is closer to 5 than to 6.

Therefore, we will round it down to 5.Hence, the integer value of the given expression is 5.

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Tangent value of zero.

The tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle containing that angle. In this case, we have a point P(-8, 0) on the terminal arm of angle θ in standard position. Since the x-coordinate is negative and the y-coordinate is zero, we can determine that the point is located on the x-axis, specifically to the left of the origin.

When the y-coordinate is zero, it means that the length of the opposite side of the angle is zero. This implies that there is no vertical displacement from the x-axis. Since the tangent of an angle is defined as the ratio of the opposite side to the adjacent side, and the adjacent side is represented by the x-coordinate,

we have y/x = 0/x = 0.

Therefore, "the tangent of the angle θ at the point P(-8, 0) is zero". This indicates that the angle has no vertical displacement relative to the x-axis. The terminal arm lies entirely on the x-axis, resulting in a tangent value of zero.

In summary, tan θ = 0 because the point P(-8, 0) is situated on the x-axis, to the left of the origin, with no vertical displacement. Division by zero is undefined in mathematics, so the tangent value is zero rather than being undefined.

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The equation of the circle is:(x - 1)² + (y - 2)² = 17

To find the equation of a circle with center at (1, 2) and passing through (-3, 3), we need to use the formula for the standard form of the equation of a circle.

A circle with center (h, k) and radius r is given by the equation:(x - h)² + (y - k)² = r²

Substituting the given values, we have:(x - 1)² + (y - 2)² = r²

We can now find the value of r using the fact that the circle passes through the point (-3, 3):

(x - 1)² + (y - 2)² = r²(-3 - 1)² + (3 - 2)² = r²16 + 1 = r²17 = r²

So the equation of the circle is:(x - 1)² + (y - 2)² = 17

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The speed of the car is 13/16 mile per minute.

To find the speed of the car per minute, we need to divide the distance traveled by the time taken.

Given that the car traveled 1/8 mile in 2/13 minutes, we can calculate the speed as follows:

Speed = Distance / Time

Speed = (1/8 mile) / (2/13 min)

To divide by a fraction, we can multiply by the reciprocal:

Speed = (1/8 mile) * (13/2 min)

Simplifying the expression:

Speed = (1 * 13) / (8 * 2) mile/min

Speed = 13/16 mile/min

Therefore, the speed of the car is 13/16 mile per minute.

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The velocity function v(t) is 6t + 3. The acceleration function a(t) is constant and equal to 6. When t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.

To find the velocity and acceleration, we need to differentiate the position function s(t) with respect to time t.a) Velocity (v(t)): The velocity is the derivative of the position function s(t) with respect to time t.v(t) = d/dt [s(t)]

Given s(t) = 3t^2 + 3t, we can differentiate it to find the velocity:

v(t) = d/dt [3t^2 + 3t]

To differentiate, we apply the power rule of differentiation: v(t) = 6t + 3

Therefore, the velocity function v(t) is 6t + 3.

b) Acceleration (a(t)): The acceleration is the derivative of the velocity function v(t) with respect to time t. a(t) = d/dt [v(t)]

Given v(t) = 6t + 3, we can differentiate it to find the acceleration:

a(t) = d/dt [6t + 3]

The derivative of a constant term is zero, so the derivative of 3 is 0:

a(t) = 6

Therefore, the acceleration function a(t) is constant and equal to 6.

c) Velocity and acceleration when t = 4 sec:

To find the velocity and acceleration at t = 4 seconds, we substitute t = 4 into the respective functions: At t = 4 sec: v(4) = 6(4) + 3

v(4) = 24 + 3

v(4) = 27 ft/s ,a(4) = 6

Therefore, when t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.

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The 2-, 10-, and 100-year peak floods for a normal distribution are 5500 cfs.

Given a mean and standard deviation of 3,500 and 2,000 cfs, respectively, the 2-, 10-, and 100-year peak floods for a normal distribution can be found as follows:

Formula used in finding the peak flood is as follows:

Q_T= Q_m + K_T

σ

Where Q_T is the flow for a given period,

Q_m is the mean flow, K_T is the coefficient of skewness, and σ is the standard deviation of the flows.

For a normal distribution, K_T= frac{\text{duration of period in years}-1}{2}\times\frac{\text{duration of period in years}+1}{2}

Substitute the mean and standard deviation to the formula above:

When the period of interest is 2 years, the coefficient of skewness is calculated below:

[{{K}_{T}}=\frac{\text{(2-1)(2+1)}}{2}=1\]

Also, K_{T} is 1 for the 10-year and 100-year flood.

When these values are computed, we get the following values:

Q_{2}=3500+1(2000)=5500 \text{ cfs}

Q_{10}=3500+1(2000)=5500 \text{ cfs}

Q_{100}=3500+1(2000)=5500 \text{ cfs}

Therefore, the 2-, 10-, and 100-year peak floods for a normal distribution are 5500 cfs.

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Given that ℓ₁ and ℓ₂ are parallel lines, and ℓ₃ is a third line, distinct from ℓ₁.

Let's assume that ℓ₃ intersects ℓ₁. Let the intersection point be A,

Then, we will have the following: AB||CD, and ℓ₁⊥AB, ℓ₂⊥CD (As they are parallel lines).

Therefore, ℓ₂ is perpendicular to CD, and AB is perpendicular to ℓ₁. Also, AB is perpendicular to CD (because they intersect perpendicularly at A).

Therefore, CD is perpendicular to both ℓ₁ and ℓ₂. Hence, ℓ₂ intersects ℓ₃.

Thus, it is proved that if ℓ₃ intersects ℓ₁ then it also must intersect ℓ₂.

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Given the equation 18 sin²(x) = 9, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.

To solve the equation, we start by dividing both sides by 18 to isolate the sin²(x):

sin²(x) = 9/18

sin²(x) = 1/2

Next, we use the trigonometric identity sin²θ + cos²θ = 1. By substituting sin²(x) with 1 - cos²(x), we have:

1 - cos²(x) = 1/2

Rearranging the equation, we get:

cos²(x) = 1 - 1/2

cos²(x) = 1/2

Taking the square root of both sides, we have:

cos(x) = ±√(1/2)

cos(x) = ±1/√2

cos(x) = ±√2/2

Since cosine is positive in the first and fourth quadrants, and negative in the second and third quadrants, we have two solutions:

x = π/4 and x = 3π/4

We can confirm that these solutions lie in the interval [0, 2].

Therefore, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.

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The given expression in terms of a, b, and c is a - b - c.

Using the trigonometric identities, we can express the given expression in terms of a, b, and c:

sin(t+2π) − cos(t+10π) + tan(t+5π)

Using the periodicity of sine and cosine functions, sin(t+2π) is equal to sin(t) and cos(t+10π) is equal to cos(t). We can substitute these values:

sin(t) − cos(t) + tan(t+5π)

Using the trigonometric identity tan(t+π) = -tan(t), we can rewrite tan(t+5π) as -tan(t):

sin(t) − cos(t) - tan(t)

Now, substituting a for sin(t), b for cos(t), and c for tan(t), we have:

a - b - c

So, A - B - C is the provided phrase in terms of a, b, and c.

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The exact values of the trigonometric functions are:

Sec(θ) = 2

Csc(θ) = (-2√3)/3

Tan(θ) = -√3

Cot(θ) = (-√3)/3

If θ = -7π/3, then the values of the trigonometric functions are as follows:

Sec(θ) = 1/cos(θ) = 1/cos(-7π/3) = 1/(cos(π/3)) = 1/(1/2) = 2

Csc(θ) = 1/sin(θ) = 1/sin(-7π/3) = 1/(sin(-π/3)) = 1/(-√3/2) = -2/√3 = (-2√3)/3

Tan(θ) = sin(θ)/cos(θ) = sin(-7π/3)/cos(-7π/3) = (sin(-π/3))/(cos(-π/3)) = (-√3/2)/(1/2) = -√3

Cot(θ) = 1/tan(θ) = 1/(-√3) = -1/√3 = (-√3)/3

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The solution to the given Bernoulli's differential equation xy - dy/dx = y³ e⁻ˣ^² is  y = -1/(3Ei(-x^2) + C).

To solve the given Bernoulli's differential equation, we can use a substitution. Let's substitute y = u^(1-n) where n is not equal to 0 and 1.

Here's how we solve it step-by-step:

1. Start by differentiating both sides of the equation with respect to x:
  d/dx (xy - dy/dx) = d/dx (y^3 e^(-x^2))

2. Simplify the left side using the product rule:
  y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * d/dx (e^(-x^2))

3. Differentiate the right side using the chain rule:
  y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x)

4. Rearrange the equation to isolate dy/dx terms on one side:
  x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x) - y

5. Multiply both sides of the equation by dx:
  xdy - ydx = -2x * 3y^2 e^(-x^2) dx - ydx

6. Simplify the equation by canceling out the common terms:
  xdy = -6xy^2 e^(-x^2) dx

7. Divide both sides of the equation by x * y^2 to separate variables:
  (1/y^2) dy = -6e^(-x^2) dx/x

8. Integrate both sides of the equation:
  ∫(1/y^2) dy = ∫-6e^(-x^2) dx/x

9. The left side of the equation can be integrated as follows:
  ∫(1/y^2) dy = -1/y

10. The right side of the equation requires a substitution. Let's substitute u = -x^2, then du/dx = -2x:
  ∫-6e^(-x^2) dx/x = -6 ∫e^u du/-2u
                     = 3 ∫e^u du/u

11. The integral on the right side is a special function called the exponential integral, Ei(u). So we have:
  -1/y = 3Ei(u) + C

12. Substitute back u = -x^2:
  -1/y = 3Ei(-x^2) + C

13. Rearrange the equation to solve for y:
  y = -1/(3Ei(-x^2) + C)

That's the solution to the given Bernoulli's differential equation. Remember to consider any initial conditions or constraints to determine the value of the constant C.

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The amount of money that will be in the account immediately after your grandmother makes the deposit on your 18th birthday is $29,960.28.

We need to find the amount of money that will be in the savings account after your grandmother makes the deposit on your 18th birthday.

We have the following data:

Present Value (PV) = $1000

Number of payments (N) = 18

Interest rate (I/Y) = 3% per annum.

Using the BAII financial calculator, we can find the future value of this savings account as follows:

1: Clear the calculator. Press the [2nd] [FV] key and then the [C] key.

2: Enter the present value of the savings account. Press the [1000] key and then the [+/-] key.Step 3: Enter the number of payments. Press the [18] key and then the [N] key.

4: Enter the interest rate per period. The interest rate is given in annual terms, so we need to divide it by the number of compounding periods per year. The account pays an interest rate of 3%, so we divide it by 1, since the interest is compounded annually.

Press the [3] key, then the [÷] key, then the [1] key, and finally the [%] key.

5: Calculate the future value. Press the [FV] key. The calculator should display $29,960.28 (rounded to the nearest cent).

Therefore, the amount of money that will be in the account immediately after your grandmother makes the deposit on your 18th birthday is $29,960.28.

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The dimensions of the original rectangle are _______ meters (width) and _______ meters (length). [Please provide the calculated values here.]

1. Given:
- The length of the original rectangle is 3 times its width.
- The area of the new rectangle is 68 square meters after increasing the length by 2 meters and decreasing the width by 1 meter. 2. Required:
- The dimensions of the original rectangle. 3. Equation:
Let's represent the width of the original rectangle as 'w' meters. Then, the length of the original rectangle would be '3w' meters.
The area of a rectangle is calculated by multiplying its length by its width:
Area = Length * Width

4. Solution:
Given that the area of the new rectangle is 68 square meters, we can set up the equation:
(3w + 2) * (w - 1) = 68. Expanding the equation, we get: 3w^2 - w - 70 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. After finding the value of 'w', we can substitute it back into the equation to find the length.

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The distance between City A and City C, when an automobile driven at speed of 65 mph takes 1.8 hours to drive from City A to City B and takes 1.2hours to drive from City B to City C, is approximately 287.87 miles.

Given that the bearing from City A to City B is N 56° E and the bearing from City B to City C is S 34° E. Also, the time taken by the automobile to drive from A to B and from B to C is 1.8 hours and 1.2 hours, respectively.

Let's calculate the distance between City A and City B. Let CB = x, then AB = x cosec 56° = x / sin 56°.

Now, let's calculate the distance between City B and City C.BC = x cosec 34° = x / sin 34°

Thus, the distance between City A and City C is AB + BC

Now, AB = x / sin 56° and BC = x / sin 34°. Therefore, the distance between City A and City C = x / sin 56° + x / sin 34° = 110.53 + 177.34 ≈ 287.87 miles (approximately). Therefore, the required distance between City A and City C is approximately 287.87 miles.

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After 31.6 minutes have passed, approximately 40.3 mg of lead-214 remains in the sample. This can be determined using the decay equation lnA = lnA₀ - kt, where A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.

The given equation, lnA = lnA₀ - kt, represents the decay of the radioactive isotope lead-214. In this equation, A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.

To find the amount of lead-214 remaining after 31.6 minutes, we can plug in the given values into the equation. We are given that A₀, the original amount of lead-214 in the sample, is 51.3 mg. The value of k for lead-214 is 2.59×[tex]10^(^-^2^)[/tex][tex]minutes^(^-^1^)[/tex], as mentioned in the question. Finally, t is 31.6 minutes.

Substituting these values into the equation, we have:

lnA = ln(51.3) - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)

Evaluating the right side of the equation, we get:

lnA ≈ 3.937 - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)

    ≈ 3.937 - 0.8164

    ≈ 3.1206

To find A, we need to exponentiate both sides of the equation using the natural logarithm base, e:

[tex]e^(^l^n^A^)[/tex] = [tex]e^(^3^.^1^2^0^6^)[/tex]

A ≈ 22.63 mg

Therefore, after 31.6 minutes have passed, approximately 22.63 mg of lead-214 remains in the sample.

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The a = 0.Substituting the values of a, b, and c in the general equation, we get:y = 0x² + 2x + 3The quadratic function is:y = 2x + 3Answer: The quadratic function is y = 2x + 3.

The given table illustrates the values of a quadratic function. Here is how you can find the quadratic function:Step 1: Write the general form of a quadratic function y = ax² + bx + c, where y is the dependent variable and x is the independent variable. a, b, and c are constants that affect the shape and position of the parabola.Step 2: Substitute the values from the table for x and y to form a system of equations.Step 3: Solve the system of equations to find the values of a, b, and c. Once you have found these values, substitute them into the quadratic equation to get the quadratic function.

The given table is as follows:x   |  0   |  2   |  4  |   6y   |  3   |  1   |  -1  |   -3Step 2:Form a system of equations using the values in the table. Here are the equations:y = a(0)² + b(0) + cy = a(2)² + b(2) + cy = a(4)² + b(4) + cy = a(6)² + b(6) + cStep 3:Solve the system of equations.Using the first equation, y = c. Hence, we have:y = 0²a + 0b + c3 = cThe value of c is 3.Using the second equation, we have:y = 2²a + 2b + 3y = 4a + 2b + 3Subtracting the two equations, we get:- 2a - b = - 2a + b = 2b = 4Therefore, b = 2.Substituting the values of b and c into the first equation, we get:3 = a(0)² + 2(0) + 3

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The formula for the function g is g(x) = x^2 - 9x + 9.

To find a formula for the function g, we can analyze the given equation. The equation g(x) = x^2 - 9x + 9 represents a quadratic function.

The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

Comparing the given equation to the general form, we can determine the values of a, b, and c.

From the given equation, we have a = 1, b = -9, and c = 9. Therefore, the formula for the function g is:

g(x) = x^2 - 9x + 9

So, the formula for the function g is g(x) = x^2 - 9x + 9.

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Question-

The graphs of ( f ) and ( g ) are given. Find a formula for the function ( g )

1) The slope of y = 6x − 2 and y = 6x + 7 are both 6, thus, making them parallel.

2) The slope of y = 2x + 4 and x + 2y + 10 = 0 is 2 and -1/2, respectively. They are perpendicular.

3) The slope of y = 8x − 1 is 8 and of 7x − y − 1 = 0 is 7 and they are neither parallel nor perpendicular with each other.

1. To find the slope of the given lines, we must write their equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

y = 6x - 2 has a slope of 6

y = 6x + 7 has a slope of 6.

The two equations have the same slopes but different y-intercept. Therefore, the two lines are parallel.

2. y = 2x + 4 has a slope of 2.

Rearranging x + 2y + 10 = 0 into slope-intercept form gives 2y = -x - 10, so y = (-1/2)x - 5, which has a slope of -1/2.

-1/2 is the negative reciprocal of 2. Therefore, the two lines are perpendicular.

3. y = 8x - 1 has a slope of 8.

Rearranging 7x - y - 1 = 0 into slope-intercept form gives y = 7x - 1, which has a slope of 7.

The slopes of the two equation are neither the same nor the negative reciprocal of one another. Therefore, the two lines are neither parallel nor perpendicular.

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Given information:Slope = - 1/3Passing through (1, -5)To write the equation of a line in point-slope form, we use the formula as follows:y - y1 = m(x - x1)Where m is the slope and (x1, y1) are the coordinates of the given point.Substituting the given values in the above formula, we have;y - (-5) = -1/3(x - 1)y + 5 = -1/3x + 1/3y = -1/3x + 1/3 - 5y = -1/3x - 14/3Thus, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3. Therefore, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3.

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Answer:any number less than or equal to 0.

Step-by-step explanation:

To determine the value of y for the given inequality, let's solve it step by step:

3(y + 1/4) ≤ 3/4

First, let's simplify the left side of the inequality:

3y + 3/4 ≤ 3/4

Next, let's isolate the term with y by subtracting 3/4 from both sides of the inequality:

3y ≤ 3/4 - 3/4

This simplifies to:

3y ≤ 0

To solve for y, divide both sides of the inequality by 3:

y ≤ 0/3

y ≤ 0

Therefore, the value of y that satisfies the inequality is any number less than or equal to 0.

Answer:

number less than or equal to 0.

Step-by-step explanation:

The answers are:  (I) F, (II) F, (III) T

(I) False. Saltwater is an example of a homogeneous mixture, also known as a solution, where the salt particles are evenly distributed throughout the water. It is not possible to separate salt from water by simple physical means like filtration. To separate the salt from saltwater, a process like evaporation or distillation is required.

(II) False. Coulomb's Law states that the interaction between charges decreases as the distance between them increases. The law describes the electrostatic force between two charged objects, which follows an inverse square relationship. As the distance between charges increases, the force of interaction decreases.

(III) True. The measurement of 101.0 g has three significant figures. In scientific notation, significant figures are the digits that carry meaningful information about the measurement. Non-zero digits and zeros between non-zero digits are considered significant. In this case, all three digits (1, 0, and 1) are non-zero and, therefore, significant.

In summary, saltwater cannot be separated by simple physical means, Coulomb's Law states that the interaction between charges decreases as the distance increases, and the measurement of 101.0 g has three significant figures.

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The student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.

The distance flown due west from Dallas is 100 miles, and the distance flown due north from Wichita Falls is 75 miles. These distances form the two sides of a right-angled triangle.

Using the Pythagorean theorem, we can calculate the hypotenuse (the straight-line distance) as follows:

Hypotenuse² = (Distance due west)² + (Distance due north)²

Hypotenuse² = 100² + 75²

Hypotenuse² = 10000 + 5625

Hypotenuse² = 15625

Taking the square root of both sides gives us:

Hypotenuse = √15625

Hypotenuse ≈ 125

Therefore, the student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.

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The rational numbers in the expressions are: √3 - 2√5,  18/24,19/24,20/24,21/24,and 24/30,  1363636/1000000,  432323/1000000,  59, 60 + 10√2 + 6√3 + √6,  = 28 + 12√11

Surd is a mathematical concept that refers to a number that cannot be simplified to remove a square root or cube root. For example, 2 is a surd because the square root of 2 cannot be simplified further. Surds are used to simplify calculations and can be used in various fields such as mathematics and physics.

The given parameters are

1)  14/[√108 - √96  +√192 -√54]

14/[6√3 - 4√6 + 8√3  - 3√6]

This implies that

14/ 14√3 -7√6

This is = √3 - 2√5

2) Five rational numbers between 3/4 and 4/5 are

A rational number is a type of real number that can be written as a fraction, where both the numerator and denominator are integers, and the denominator is not equal to zero

These include

Since we want five numbers, we write 3/5 and 4/5  So multiply in numerator and denominator by 5+1 =6  we get

3/4 * 6/6 = 18/24,  and 4/5 * 6/6 = 24/30

the rational numbers are

18/24,19/24,20/24,21/24,and 24/30

3_ The value (i) 1.363636 in the form p/q where p and q are integers is

1.363636/1000000

1363636/1000000

(ii)  0.4323232

= 0.432323/1000000

= 432323/1000000

4)  (i)  (8 + √5)(8+√5)

= 64 - 8√5 + 8√5 -√25

⇒64 -5

= 59

ii) (10 + √3)(6 + √2)

60 + 10√2 + 6√3 + √6

iii)  (√3 + √11)²  +  (√3 + √11)²

Simplifying this to have

(√3 + √11)(√3 + √11)  +  (√3 + √11)(√3 + √11)

[3 +3√11 + 3√11 + 11]  +  [3 +3√11 + 3√11 + 11]

= (14 + 6√11)  +  (14 + 6√11)

= 28 + 12√11

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The equation of the axis of symmetry for the parabola whose equation is given by f(x) = 2(x + 3)² - 4 is x = -3 and the second point on the parabola whose y-coordinate is the same as the given point (-2, -2) is (-3, -2).

The standard form of a quadratic equation is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Therefore, comparing f(x) with the standard form, we get the vertex form of the equation as: f(x) = 2(x + 3)² - 4 = 2(x - (-3))² - 4. So, the vertex of the parabola is (-3, -4).

The axis of symmetry is the line that passes through the vertex of the parabola and is parallel to the y-axis. Hence, the axis of symmetry is x = -3.

To find a second point on the parabola whose y-coordinate is the same as the given point, we can use the reflection property of the vertex. Since the parabola is symmetric about the axis of symmetry, the y-coordinate of the point that is the same distance from the axis of symmetry as (-2, -2) will be the same as the y-coordinate of (-2, -2).

The distance between x-coordinate of (-2, -2) and the axis of symmetry x = -3 is 1 unit. Therefore, the x-coordinate of the required point will be 1 unit to the right of the axis of symmetry. Since the parabola is symmetric about the axis of symmetry, the x-coordinate of the point that is 1 unit to the right of the axis of symmetry will be the same as the x-coordinate of the point that is 1 unit to the left of the axis of symmetry.

The x-coordinate of the point that is 1 unit to the left of the axis of symmetry is -3 - 1 = -4. Hence, the x-coordinate of the required point is -4 + 1 = -3. The y-coordinate of the required point is the same as the y-coordinate of (-2, -2). Hence, the required point is (-3, -2).

Therefore, the equation of the axis of symmetry is x = -3 and the second point on the parabola whose y-coordinate is the same as the given point (-2, -2) is (-3, -2).

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Complete Question:  

Find the axis of symmetry for the parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point f(x) 20x +3)2 -4; (-2,-2).

a. The value of \( f(-9) \) is 63.

b. The simplified form of Function \( f(x+3) \) is \( x²+ 11x + 17 \).

c. The simplified form of \( f(-x) \) is \( x² - 5x - 5 \).

a. To evaluate \( f(-9) \), we substitute -9 for x in the given function:

  \( f(-9) = (-9)² + 5(-9) - 5 = 81 - 45 - 5 = 63 \)

b. To evaluate \( f(x+3) \), we substitute \( (x+3) \) for x in the given function:

  \( f(x+3) = (x+3)² + 5(x+3) - 5 \)

  Expanding and simplifying the expression, we have:

  \( f(x+3) = x² + 6x + 9 + 5x + 15 - 5 = x² + 11x + 17 \)

c. To evaluate \( f(-x) \), we substitute -x for x in the given function:

  \( f(-x) = (-x)² + 5(-x) - 5 = x² - 5x - 5 \)

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The solutions to the equation 5x^2 - 35 = 0 are x = ±√7.

To solve this quadratic equation, we can first isolate the variable by moving the constant term to the other side:

5x^2 = 35

Next, we divide both sides of the equation by 5 to solve for x^2:

x^2 = 7

To find the value of x, we take the square root of both sides:

√(x^2) = ±√7

Since we took the square root, we need to consider both the positive and negative square roots, giving us two solutions:

x = ±√7

Therefore, the solutions to the equation 5x^2 - 35 = 0 are x = ±√7.

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a) The next 3 terms are aₙ + 3, aₙ + 6, and aₙ + 9, where aₙ is the last term in the sequence.
b) a₅₂ = 154 and a₇₅₈ = 2,272, using the formula aₙ = a₁ + (n - 1)d, with a₁ = 1 and d = 3.

a) To find the next 3 terms in the given sequence, we observe that each term is obtained by adding 3 to the previous term. Therefore, we can continue the pattern by adding 3 to the last term. In general, if the last term is denoted as aₙ, then the next three terms would be aₙ + 3, aₙ + 6, and aₙ + 9.

b) To find a specific term in the sequence, we can use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, n is the term number, and d is the common difference. By substituting the given values (a₁ = 1 and d = 3) into the formula, we can find a₅₂ and a₇₅₈. Applying the formula, we find that a₅₂ = 154 and a₇₅₈ = 2,272.

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The width of the rectangle is 5/2 cm, which can be written as 2 1/2 cm or 2.5 cm.

Let's denote the width of the rectangle as "w" (in cm). We are given that the length of the rectangle is 4 1/4 cm.

The perimeter of a rectangle is calculated by adding up the lengths of all four sides. In this case, we are given that the perimeter is 13 1/2 cm.

The formula for the perimeter of a rectangle is:

Perimeter = 2(length + width)

Substituting the given values:

13 1/2 = 2(4 1/4 + w)

To simplify the equation, we convert all mixed numbers to improper fractions:

Perimeter = 2(17/4 + w)

Next, we distribute the 2:

13 1/2 = 34/4 + 2w

Now, let's simplify the equation by finding a common denominator:

13 1/2 = 8 1/2 + 2w

We subtract 8 1/2 from both sides to isolate 2w:

5 = 2w

Finally, we divide both sides by 2 to solve for w:

w = 5/2

Therefore, the width of the rectangle is 5/2 cm, which can be written as 2 1/2 cm or 2.5 cm.

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(a) d(5) = 75. Joe travels 75 miles in 5 hours.

(b) Joe travels 7.5 miles in 30 minutes.

(a) Evaluating d(5) means plugging in the value of 5 for t in the equation d(t) = 15t and calculating the resulting distance.

Substituting t = 5 into the equation, we get d(5) = 15 * 5 = 75.

Therefore, d(5) = 75. This means that Joe travels a distance of 75 miles in 5 hours.

(b) We need to determine the distance Joe travels in 30 minutes.

Since the time is given in hours, we convert 30 minutes to hours by dividing by 60 (since there are 60 minutes in an hour): 30 minutes / 60 = 0.5 hours.

Now, we can substitute t = 0.5 into the distance equation: d(0.5) = 15 * 0.5 = 7.5.

Therefore, Joe travels a distance of 7.5 miles in 30 minutes.

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