20. There is a 20 people committee. Find the number of selecting president, vice-president, secretary,and treasurer.
21. There is a 30 people council. Find the number of making 5 people subcommittee.

a) The slope of line that passes through two points 4/9.

b) The slope of the perpendicular line is -9/4.

Given, the two points are (-8,-2) and (1,2).

To find the slope of the line that is (a) parallel and (b) perpendicular to the line through the pair of points.

Use the formula to find the slope of a line that passes through two points given below:

Slope, m = (y2 - y1)/(x2 - x1)

Where, (x1, y1) and (x2, y2) are two points.

For the given points (-8,-2) and (1,2), the slope is:

m = (2 - (-2))/(1 - (-8))

= 4/9

(a) The slope of the parallel line is also 4/9.The slope of any two parallel lines are equal to each other.

Hence, the slope of the parallel line is 4/9.

(b) The slope of the perpendicular line is the negative reciprocal of the slope of the given line through the pair of points.

That is, the slope of the perpendicular line is:-

(1)/(m) = -(1)/(4/9)

= -9/4

Know more about the slope of line

https://brainly.com/question/16949303

#SPJ11

The airplane will take approximately 9 minutes to reach its cruising altitude of 6.5 miles.

To determine the time it takes for the airplane to reach its cruising altitude, we need to calculate the vertical distance traveled. The angle of climb, 11 degrees, represents the inclination of the airplane's path with respect to the horizontal. This inclination forms a right triangle with the vertical distance traveled as the opposite side and the horizontal distance as the adjacent side.

Using trigonometry, we can find the vertical distance traveled by multiplying the horizontal distance covered (which is the average speed multiplied by the time) by the sine of the angle of climb. The horizontal distance covered can be calculated by dividing the cruising altitude by the tangent of the angle of climb.

Let's perform the calculations. The tangent of 11 degrees is approximately 0.1989. Dividing the cruising altitude of 6.5 miles by the tangent gives us approximately 32.66 miles as the horizontal distance covered. Now, we can find the vertical distance traveled by multiplying 32.66 miles by the sine of 11 degrees, which is approximately 0.1916. This results in a vertical distance of approximately 6.25 miles.

To convert this vertical distance into time, we divide it by the average speed of the airplane, which is 420 mph. The result is approximately 0.0149 hours or approximately 0.8938 minutes. Rounding to the nearest minute, we find that the airplane will take approximately 9 minutes to reach its cruising altitude of 6.5 miles.

Learn more about inclination here:

https://brainly.com/question/29360090

#SPJ11

The solution of `2cos²? + cos? - 1 = 0` for the exact x value(s) over `0 < 2 < 2T` are given by `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.

Given, `2cos²? + cos? - 1 = 0`.Let’s solve this equation.Substitute, `cos? = t`.So, the given equation becomes,`2t² + t - 1 = 0.

Now, Let’s solve this quadratic equation by using the quadratic formula, which is given by;

If the quadratic equation is given in the form of `ax² + bx + c = 0`, then the solution of this quadratic equation is given by;`x = (-b ± sqrt(b² - 4ac)) / 2a

Here, the quadratic equation is `2t² + t - 1 = 0`.So, `a = 2, b = 1 and c = -1.

Now, substitute these values in the quadratic formula.`t = (-1 ± sqrt(1² - 4(2)(-1))) / 2(2)`=> `t = (-1 ± sqrt(9)) / 4`=> `t = (-1 ± 3) / 4.

Now, we have two solutions. Let's evaluate them separately.`t₁ = (-1 + 3) / 4 = 1/2` and `t₂ = (-1 - 3) / 4 = -1.

Now, we have to substitute the value of `t` to get the values of `cos ?`

For, `t₁ = 1/2`, `cos ? = t = 1/2` (since `0 < 2 < 2T` and `cos` is positive in the first and fourth quadrant).

So, `? = π/3` or `? = 5π/3`For, `t₂ = -1`, `cos ? = t = -1` (since `0 < 2 < 2T` and `cos` is negative in the second and third quadrant)So, `? = π` or `? = 2π.

Therefore, the main answers for the given equation `2cos²? + cos? - 1 = 0` over `0 < 2 < 2T` are `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.

So, the solution of `2cos²? + cos? - 1 = 0` for the exact x value(s) over `0 < 2 < 2T` are given by `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.

To know more about quadratic equation visit:

brainly.com/question/30098550

#SPJ11

The system of differential equations for the two interconnected tanks can be set up as follows:

dx/dt = (5 g/L * 7 L/min) - (2 L/min * (x/60))  

dy/dt = (4 g/L * 9 L/min) + (2 L/min * (x/60)) - (16 L/min * (y/50))  

To set up the system of differential equations, we need to consider the inflow and outflow of salt in both tanks. The rate of change of salt in Tank A, dx/dt, is determined by the inflow of salt from the solution and the outflow of salt to Tank B. The inflow of salt into Tank A is given by the concentration of the solution (5 g/L) multiplied by the flow rate (7 L/min). The outflow of salt from Tank A to Tank B is given by the outflow rate (2 L/min) multiplied by the concentration of salt in Tank A (x/60, as the tank has 60 liters of water).

Similarly, the rate of change of salt in Tank B, dy/dt, is determined by the inflow of salt from Tank A, the inflow of salt from the solution, and the outflow of salt due to drainage. The inflow of salt from Tank A is given by the outflow rate (2 L/min) multiplied by the concentration of salt in Tank A (x/60). The inflow of salt from the solution is given by the concentration of the solution (4 g/L) multiplied by the flow rate (9 L/min). The outflow of salt due to drainage is given by the drainage rate (16 L/min) multiplied by the concentration of salt in Tank B (y/50, as the tank has 50 liters of water).

The initial conditions x(0) and y(0) represent the initial grams of salt in Tank A and Tank B, respectively.

Learn more about differential equations here:
https://brainly.com/question/32645495

#SPJ11

The APY on the same loan is approximately 1.825% (rounded to 3 decimal places).

To calculate the APR (Annual Percentage Rate) and APY (Annual Percentage Yield) on the $210 loan from the short-term neighborhood lender, we can use the provided information.

APR is the annualized interest rate on a loan, while APY takes into account compounding interest.

First, let's calculate the APR:

APR = (Interest / Principal) * (365 / Time)

Here, the principal is $210, the interest is $10.50, and the time is 10 days.

APR = (10.50 / 210) * (365 / 10)

APR ≈ 0.05 * 36.5

APR ≈ 1.825

Therefore, the APR on the $210 loan from the short-term neighborhood lender is approximately 1.825% (rounded to 3 decimal places).

Next, let's calculate the APY:

APY = (1 + r/n)^n - 1

Here, r is the interest rate (APR), and n is the number of compounding periods per year. Since the loan duration is 10 days, we assume there is only one compounding period in a year.

APY = (1 + 0.01825/1)^1 - 1

APY ≈ 0.01825

Therefore, the APY on the same loan is approximately 1.825% (rounded to 3 decimal places).

for such more question on Annual Percentage Rate

https://brainly.com/question/23806178

#SPJ8

Given, we need to evaluate the integral of sin(3x)cos(2x) with respect to x.

Let's consider the below trigonometric formula to solve the given integral. sin (A + B) = sin A cos B + cos A sin Bsin(3x + 2x) = sin(3x)cos(2x) + cos(3x)sin(2x) ⇒ sin(3x)cos(2x) = sin(3x + 2x) - cos(3x)sin(2x)On integrating both sides with respect to x, we get∫[sin(3x)cos(2x)] dx = ∫[sin(3x + 2x) - cos(3x)sin(2x)] dx⇒ ∫[sin(3x)cos(2x)] dx = ∫[sin(3x)cos(2x + 2x) - cos(3x)sin(2x)] dx ⇒ ∫[sin(3x)cos(2x)] dx = ∫[sin(3x)(cos2x cos2x - sin2x sin2x) - cos(3x)sin(2x)] dx

Now, use the below trigonometric formulas to evaluate the given integral.cos 2x = 2 cos² x - 1sin 2x = 2 sin x cos x∫[sin(3x)cos(2x)] dx = ∫[sin3x (2 cos2x cos2x - 2 sin2x sin2x) - cos(3x) sin(2x)] dx∫[sin(3x)cos(2x)] dx = ∫[sin3x (2 cos² x - 1) - cos(3x) 2 sin x cos x] dxAfter solving the integral, the final answer will be as follows:∫[sin(3x)cos(2x)] dx = (-1/6) cos3x + (1/4) sin4x + C.Here, C is the constant of integration.

Thus, the integration of sin(3x)cos(2x) with respect to x is (-1/6) cos3x + (1/4) sin4x + C.We can solve this integral using the trigonometric formula of sin(A + B).

On solving, we get two new integrals that we can solve using the formula of sin 2x and cos 2x, respectively.After solving these integrals, we can add their result to get the final answer. So, we add the result of sin 2x and cos 2x integrals to get the solution of the sin 3x cos 2x integral.

The final solution is (-1/6) cos3x + (1/4) sin4x + C, where C is the constant of integration.

Therefore, we can solve the integral of sin(3x)cos(2x) with respect to x using the trigonometric formula of sin(A + B) and the formulas of sin 2x and cos 2x. The final answer of the integral is (-1/6) cos3x + (1/4) sin4x + C, where C is the constant of integration.

To know more about integral visit

https://brainly.com/question/31059545

#SPJ11

We have found the unit vector u in the direction of v and verified that ||u|| = 1. The values are: u = (-9/√85, -2/√85) and ||u|| = 1.

To find a unit vector u in the direction of v and to verify that ||u|| = 1, where v = (-9, -2), we can follow these steps:

Step 1: Calculate the magnitude of v. Magnitude of v is given by:

||v|| = √(v₁² + v₂²)

Substituting the given values, we get: ||v|| = √((-9)² + (-2)²) = √(81 + 4) = √85 Step 2: Find the unit vector u in the direction of v. Unit vector u in the direction of v is given by:

u = v/||v||

Substituting the given values, we get:

u = (-9/√85, -2/√85)

Step 3: Verify that ||u|| = 1.

The magnitude of a unit vector is always equal to 1.

Therefore, we need to calculate the magnitude of u using the formula:

||u|| = √(u₁² + u₂²) Substituting the calculated values, we get: ||u|| = √((-9/√85)² + (-2/√85)²) = √(81/85 + 4/85) = √(85/85) = 1

Hence, we have found the unit vector u in the direction of v and verified that ||u|| = 1. The values are: u = (-9/√85, -2/√85) and ||u|| = 1.

Learn more about unit vector visit:

brainly.com/question/28028700

#SPJ11

The statement is true. the t distribution is similar to the standard normal distribution, but is more spread out.

In probability and statistics, Student's t-distribution {\displaystyle t_{\nu }} is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

The t-distribution is similar to the standard normal distribution, but it has heavier tails and is more spread out. The t-distribution has a larger variance compared to the standard normal distribution, which means it has more variability in its values. This increased spread allows for greater flexibility in capturing the uncertainty associated with smaller sample sizes when estimating population parameters.

Know more about t distribution here:

https://brainly.com/question/32675925

#SPJ11

To find the average rate of change of the function \( f(x) = 2x \) from \( x_1 = 0 \) to \( x_2 = 8 \), we need to calculate the change in the function's values divided by the change in the input values.

The change in the function's values is given by \( f(x_2) - f(x_1) \), and the change in the input values is \( x_2 - x_1 \). Substituting the values, we have:

\( f(x_2) - f(x_1) = 2x_2 - 2x_1 = 2(8) - 2(0) = 16 \)

\( x_2 - x_1 = 8 - 0 = 8 \)

Therefore, the average rate of change is \( \frac{16}{8} = 2 \).

The average rate of change of the function \( f(x) = 2x \) from \( x_1 = 0 \) to \( x_2 = 8 \) is 2. This means that, on average, the function increases by 2 units for every 1 unit increase in \( x \) in the given interval.

Learn more about function's values here: brainly.com/question/20303464

#SPJ11

The relationship between DE and AC, considering the triangle midsegment theorem, is given as follows:

The triangle midsegment theorem states that the midsegment of the triangle divided the length of the midsegment of the triangle is half the length of the base of the triangle, and that the midsegment and the base are parallel.

The parameters for this problem are given as follows:

Hence the correct statements are given as follows:

More can be learned about the triangle midsegment theorem at brainly.com/question/7423948

#SPJ1

It will take approximately 11.5 years for the balance to reach $20,000.

To find the time it takes for the balance to reach $20,000, we can use the formula for continuous compound interest:

A = P * e^(rt)

Where:

A is the final amount

P is the principal amount (initial investment)

e is the base of the natural logarithm (approximately 2.71828)

r is the interest rate (in decimal form)

t is the time (in years)

In this case, the principal amount (P) is $5000, the interest rate (r) is 7% per year (or 0.07 in decimal form), and we want to find the time (t) it takes for the balance to reach $20,000.

Substituting the given values into the formula, we have:

20000 = 5000 * e^(0.07t)

Dividing both sides of the equation by 5000:

4 = e^(0.07t)

To isolate the variable, we take the natural logarithm (ln) of both sides:

ln(4) = ln(e^(0.07t))

Using the property of logarithms, ln(e^x) = x:

ln(4) = 0.07t

Dividing both sides by 0.07:

t = ln(4) / 0.07 ≈ 11.527

Therefore, it will take approximately 11.5 years for the balance to reach $20,000.

Continuous compound interest is a mathematical model that assumes interest is continuously compounded over time. In reality, most banks compound interest either annually, semi-annually, quarterly, or monthly. Continuous compounding is a theoretical concept that allows us to calculate the growth of an investment over time without the limitations of specific compounding periods. In this case, the investment grows exponentially over time, and it takes approximately 11.5 years for the balance to reach $20,000.

To know more about balance, refer here:

https://brainly.com/question/27154367

#SPJ11

The correct formula for f(n), when n is a nonnegative integer, is f(n) = n + 1.

We are given the function f(n) defined recursively. The base case is f(0) = 1. For n ≥ 1, the function is defined as f(n) = f(n-1) - 1.

To find the formula for f(n), we can observe the pattern in the recursive definition. Starting from the base case f(0) = 1, we can apply the recursive definition repeatedly:

f(1) = f(0) - 1 = 1 - 1 = 0

f(2) = f(1) - 1 = 0 - 1 = -1

f(3) = f(2) - 1 = -1 - 1 = -2

...

From this pattern, we can see that f(n) is obtained by subtracting n from the previous term. This leads us to the formula f(n) = n + 1.

Therefore, the correct formula for f(n) when n is a nonnegative integer is f(n) = n + 1, option (a).

Learn more about  nonnegative integer here:

https://brainly.com/question/32229709

#SPJ11

8) The first term and the common ratio for the geometric sequence can be found using the given terms [tex]\(a_2 = 45\) and \(a_4 = 1125\).[/tex]

The common ratio (\(r\)) can be calculated by dividing the second term by the first term:
[tex]\(r = \frac{a_2}{a_1} = \frac{45}{a_1}\)[/tex]
Similarly, the fourth term can be expressed in terms of the first term and the common ratio:
[tex]\(a_4 = a_1 \cdot r^3\)Substituting the given value \(a_4 = 1125\), we can solve for \(a_1\): \(1125 = a_1 \cdot r^3\)[/tex]
Now we have two equations with two unknowns:
[tex]\(r = \frac{45}{a_1}\)\(1125 = a_1 \cdot r^3\)[/tex]
By substituting the value of \(r\) from the first equation into the second equation, we can solve for \(a_1\).
9) To find the sum of the first five terms of the geometric sequence, we can use the formula for the sum of a finite geometric series. The formula is given by:
[tex]\(S_n = a \cdot \frac{r^n - 1}{r - 1}\)[/tex]
where \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
By substituting the values of \(a_1\) and \(r\) into the formula, we can calculate the sum of the first five terms of the geometric sequence.



learn more about common ratio here

  https://brainly.com/question/17630110



#SPJ11

The atmospheric pressure (in pounds per square inch) at a height x miles above Earth's surface can be modeled by a hypothetical equation.

To establish a hypothetical equation for the atmospheric pressure f in pounds per square inch at a height x miles above Earth's surface, we require additional information. Atmospheric pressure decreases with increasing altitude, and it is typically modeled using exponential decay. One commonly used equation is the barometric formula:

f = f₀ * e^(-kx),

where f₀ represents the atmospheric pressure at sea level, e is the base of the natural logarithm (approximately 2.71828), k is a constant related to the rate of pressure decrease with altitude, and x is the height in miles.

To find the height (n miles) at which the atmospheric pressure is 11 pounds per square inch, we need to solve the equation for x when f = 11. Rearranging the equation, we have:

11 = f₀ * e^(-kx).

Unfortunately, without specific values for f₀ and k, it is not possible to provide an exact solution. However, with the given information, we can use numerical methods, such as iteration or approximation techniques, to find an estimate for the height (n miles) that yields a pressure of 11 pounds per square inch.

Learn more about logarithm here:

https://brainly.com/question/30226560

#SPJ11

The completely factored form of the expression [tex]\(32x^2 - 2xy - 22x\)[/tex] by grouping is [tex]\(2x(16x - 11 - y)\)[/tex].

To factor the expression \(32x^2 - 2xy - 22x\), we can use the method of grouping.

First, let's group the terms in pairs:

\((32x^2 - 22x) + (-2xy)\)

Now, we can factor out the common factors from each group:

\(2x(16x - 11) - 2xy\)

Next, we can factor out the common factor of 2x from both terms:

\(2x(16x - 11 - y)\)

Therefore, the completely factored form of the expression \(32x^2 - 2xy - 22x\) by grouping is \(2x(16x - 11 - y)\).

Learn more about factored form here

https://brainly.com/question/30285015

#SPJ11

We are asked to find a polynomial, denoted as P(x), with real coefficients that satisfies certain conditions. The polynomial has a degree of 6, a leading coefficient of 4, and the zeros are given as 4, 0 (with a multiplicity of 3), and 3 - 2i. By using the zero-factor theorem, we can construct the polynomial by multiplying its linear factors corresponding to each zero.

The zero-factor theorem states that if a polynomial has a zero at a particular value, then the corresponding linear factor (x - a) is a factor of the polynomial, where 'a' is the zero. Based on this theorem, we can construct the polynomial P(x) by multiplying its linear factors corresponding to each zero.

Given the zeros 4, 0 (with a multiplicity of 3), and 3 - 2i, the linear factors are as follows:

(x - 4) - since 4 is a zero.

(x - 0)^3 = x^3 - 0 = x^3 - this factor has a multiplicity of 3.

(x - (3 - 2i)) = (x - 3 + 2i) - since 3 - 2i is a zero.

Now, we can multiply these factors together to obtain the polynomial P(x):

P(x) = (x - 4)(x^3)(x^3)(x - 3 + 2i).

However, we need to consider that the polynomial has real coefficients. Since 3 - 2i is a zero, its complex conjugate 3 + 2i must also be a zero. Therefore, we can include the factor (x - 3 - 2i) to ensure real coefficients:

P(x) = (x - 4)(x^3)(x^3)(x - 3 + 2i)(x - 3 - 2i).

Finally, we can simplify and combine the factors as necessary to obtain the complete polynomial P(x) with the given degree, leading coefficient, and zeros.

Learn more about multiply here: brainly.com/question/30875464

#SPJ11

In the given geometric sequence, the ratio between the third and first terms is 4, and the tenth term is 64. The value of b2 in both cases is 1/4.

Let's assume the first term, b1, of the geometric sequence to be 'a', and the common ratio between consecutive terms to be 'r'. We are given that b3/b1 = 4, which means (a * r^2) / a = 4. Simplifying this, we get r^2 = 4, and taking the square root on both sides, we find that r = 2 or -2.

Now, we know that b10 = 64, which can be expressed as ar^9 = 64. Substituting the value of r, we have two possibilities: a * 2^9 = 64 or a * (-2)^9 = 64. Solving the equations, we find a = 1/8 for r = 2 and a = -1/8 for r = -2.

Since b2 is the second term of the sequence, we can express it as ar, where a is the first term and r is the common ratio. Substituting the values of a and r, we get b2 = (1/8) * 2 = 1/4 for r = 2, and b2 = (-1/8) * (-2) = 1/4 for r = -2. Therefore, the value of b2 in both cases is 1/4.

Learn more about sequence here:

https://brainly.com/question/30262438

#SPJ11

The regular selling price of the item was $220, and the rate of markdown during the sale was 29%.

To find the regular selling price, we need to backtrack from the given selling price after the markdown. Let's assume the regular selling price is x. We know that the selling price after the markdown is $310. Since the selling price was reduced by 29% during the sale, we can set up the equation:

x - 29% of x = $310

Simplifying the equation, we have:

x - 0.29x = $310

Combining like terms, we get:

0.71x = $310

To solve for x, we divide both sides of the equation by 0.71:

x = $310 / 0.71 ≈ $436.62

Therefore, the regular selling price of the item was approximately $436.62.

Now, to calculate the rate of markdown during the sale, we compare the regular selling price to the selling price after the markdown. The difference between the two prices is $436.62 - $310 = $126.62.

To find the rate of markdown, we divide this difference by the regular selling price and multiply by 100:

Markdown rate = ($126.62 / $436.62) * 100 ≈ 29%

Hence, the rate of markdown during the sale was approximately 29%.

Learn more about selling price

brainly.com/question/27796445

#SPJ11

For the linear function y=f(x)=-x+4, the calculations are as follows:

a. The derivative df/dx at x=-6 is -1.

b. The formula for the inverse function[tex]x=f^{(-1)}(y)[/tex] is x=4-y.

c. The derivative dy/[tex]df^{(-1)[/tex]at y=f(-6) is -1.

a. To find the derivative dx/df at x=-6, we differentiate the function f(x)=-x+4 with respect to x. The derivative of -x is -1, and the derivative of a constant (4 in this case) is 0. Therefore, the derivative df/dx at x=-6 is -1.

b. To find the formula for the inverse function [tex]x=f^{(-1)}(y)[/tex], we interchange x and y in the original function. So, y=-x+4 becomes x=4-y. Thus, the formula for the inverse function is x=4-y.

c. To find the derivative dy/[tex]df^{(-1)[/tex] at y=f(-6), we differentiate the inverse function x=4-y with respect to y. The derivative of 4 is 0, and the derivative of -y is -1. Therefore, the derivative dy/[tex]df^{(-1)[/tex] at y=f(-6) is -1.

To learn more about linear function visit:

brainly.com/question/29205018

#SPJ11

(a) P(A or B) = 0.412 when A and B are independent, and (b) P(A or B) = 0.46 when A and B are mutually exclusive.

(a) To find P(A or B) given that A and B are independent events, we can use the formula for the union of independent events: P(A or B) = P(A) + P(B) - P(A) * P(B). Since A and B are independent, the probability of their intersection, P(A) * P(B), is equal to 0.30 * 0.16 = 0.048. Therefore, P(A or B) = P(A) + P(B) - P(A) * P(B) = 0.30 + 0.16 - 0.048 = 0.412.

(b) When A and B are mutually exclusive events, it means that they cannot occur at the same time. In this case, P(A) * P(B) = 0, since their intersection is empty. Therefore, the formula for the union of mutually exclusive events simplifies to P(A or B) = P(A) + P(B). Substituting the given probabilities, we have P(A or B) = 0.30 + 0.16 = 0.46.

Learn more about events here : brainly.com/question/15063914

#SPJ11

The Rf value for sample 1 is 7.77, and the Rf value for sample 2 is 5.11.

To calculate the Rf (retention factor) value, you need to divide the distance traveled by the compound of interest by the distance traveled by the solvent front. In this case, you have the following measurements:

Distance traveled by sample 1: 17.1 cm

Distance traveled by sample 2: 11.25 cm

Distance traveled by solvent front: 2.2 cm

To find the Rf value for sample 1, you would divide the distance traveled by sample 1 by the distance traveled by the solvent front:

Rf (sample 1) = 17.1 cm / 2.2 cm = 7.77

To find the Rf value for sample 2, you would divide the distance traveled by sample 2 by the distance traveled by the solvent front:

Rf (sample 2) = 11.25 cm / 2.2 cm = 5.11

Therefore, the Rf value for sample 1 is 7.77, and the Rf value for sample 2 is 5.11.

Learn more about value from

https://brainly.com/question/24305645

#SPJ11

The largest open interval over which the function is decreasing is (-∞, 9) ∪ (9, ∞).

The given quadratic function is f(x) = x² - 18x + 80. So, we need to determine (a) vertex, (b) axis, (c) domain, and (d) range and also (e) the largest open interval of the domain over which the function is increasing and (f) the largest open interval over which the function is decreasing.

Graph of the given quadratic function f(x) = x² - 18x + 80 is shown below:

Here, vertex = (h, k) is (9, -1),

axis of symmetry is x = h = 9. domain is all real numbers, i.e., (-∞, ∞) range is y ≤ k = -1. Now, we need to determine the largest open interval over which the function is increasing and decreasing.For that, we need to calculate the discriminant of the given quadratic function.

f(x) = x² - 18x + 80

a = 1, b = -18, and c = 80

D = b² - 4acD = (-18)² - 4(1)(80)

D = 324 - 320

D = 4

Since the discriminant D is positive, the quadratic function has two distinct real roots and the graph of the quadratic function intersects the x-axis at two distinct points. Thus, the quadratic function is increasing on the intervals (-∞, 9) and (9, ∞).

Therefore, the largest open interval of the domain over which the function is increasing is (-∞, 9) ∪ (9, ∞).

Similarly, the quadratic function is decreasing on the interval (9, ∞) and (−∞, 9).

Therefore, the largest open interval over which the function is decreasing is (-∞, 9) ∪ (9, ∞).

Learn more about quadratic function visit:

brainly.com/question/18958913

#SPJ11

Given that `z = 2 cos θ + 2i sin θ` and `w=2(cosφ + i sin θ)` and we need to find `zw` and `w/z` in polar form.In order to get the product `zw` we have to multiply both the given complex numbers. That is,zw = `2 cos θ + 2i sin θ` × `2(cosφ + i sin θ)`zw = `2 × 2(cos θ cosφ - sin θ sinφ) + 2i (sin θ cosφ + cos θ sinφ)`zw = `4(cos (θ + φ) + i sin (θ + φ))`zw = `4cis (θ + φ)`

Therefore, the product `zw` is `4 cis (θ + φ)`In order to get the quotient `w/z` we have to divide both the given complex numbers. That is,w/z = `2(cosφ + i sin φ)` / `2 cos θ + 2i sin θ`

Multiplying both numerator and denominator by conjugate of the denominator2(cosφ + i sin φ) × 2(cos θ - i sin θ) / `2 cos θ + 2i sin θ` × 2(cos θ - i sin θ)w/z = `(4cos θ cos φ + 4sin θ sin φ) + i (4sin θ cos φ - 4cos θ sin φ)` / `(2cos^2 θ + 2sin^2 θ)`w/z = `(2cos θ cos φ + 2sin θ sin φ) + i (2sin θ cos φ - 2cos θ sin φ)`w/z = `2(cos (θ - φ) + i sin (θ - φ))`

Therefore, the quotient `w/z` is `2 cis (θ - φ)`

Hence, the required product `zw` is `4 cis (θ + φ)` and the quotient `w/z` is `2 cis (θ - φ)`[tex]`w/z` is `2 cis (θ - φ)`[/tex]

To know more about complex numbers visit :

https://brainly.com/question/20566728

#SPJ11

The equation of the line in the given form, y = mx + c, is y = [?]x + [?].slope and y-intercept, we cannot determine the equation of the line.

To find the equation of a line in the form y = mx + c, we need the slope (m) and the y-intercept (c). However, since the values for the slope and y-intercept are not provided in the question, we cannot determine the equation without additional information.

Without knowing the values for slope and y-intercept, we cannot determine the equation of the line.

To know more about  slope follow the link:

https://brainly.com/question/30097715

#SPJ11

Answer:

It's y=-3x+7. Hope this helps!

The exact amount can be calculated using the formula for compound interest. The amount you should invest today to have $380,000 in your account after 11 years.

The formula for compound interest is given by [tex]\(A = P \left(1 + \frac{r}{n}\right)^{nt}\)[/tex], where (A) is the final amount, (P) is the principal amount (initial investment), (r) is the annual interest rate (in decimal form), (n) is the number of times interest is compounded per year, and (t) is the number of years.

In this case, the principal amount (P) is what we want to find. The final amount (A) is $380,000, the annual interest rate (r) is 9.5% (or 0.095 in decimal form), the number of times interest is compounded per year (n) is 12 (monthly compounding), and the number of years (t) is 11.

Substituting these values into the formula, we have:

[tex]\[380,000 = P \left(1 + \frac{0.095}{12}\right)^{(12 \cdot 11)}\][/tex]

To find the value of \(P\), we can rearrange the equation and solve for (P):

[tex]\[P = \frac{380,000}{\left(1 + \frac{0.095}{12}\right)^{(12 \cdot 11)}}\][/tex]

Evaluating this expression will give the amount you should invest today to have $380,000 in your account after 11 years.

To learn more about  compound interest visit:

brainly.com/question/30902883

#SPJ11

The range of (3cos(x+7))² is [0, 9]. Therefore, [a, b] = [0, 9].

1. Examine whether the function f (x) = 2x − 11 is invertible. In that case, enter an expression for its inverse.

The function f (x) = 2x − 11 is invertible because it is a linear function, meaning that it is one-to-one.

The inverse of the function is given by f -1 (y) = (y + 11) / 2.

2. Given the function f (x) = (3cos (x + 7))2 with the definition set (−[infinity], [infinity]), determine the value set [a, b] to the function.

The function f(x) = (3cos(x+7))² is a function of x, where x is any real number.

The range of the cosine function is [-1, 1].

Thus, the range of 3cos(x+7) is [-3, 3].

As a result, the range of (3cos(x+7))² is [0, 9].

Therefore, [a, b] = [0, 9].

To know more about range visit:

https://brainly.com/question/29204101

#SPJ11

The displacement of an object undergoing simple harmonic motion with a period of 7 minutes and an amplitude of 12 m can be modeled by the equation d = 12 sin((2π/7) f), where d represents the displacement and f is the time in minutes.

In simple harmonic motion, the displacement of an object can be represented by a sinusoidal function. The general equation for displacement as a function of time in simple harmonic motion is given by d = A sin(2π/T) f, where A is the amplitude, T is the period, and f is the time.

In this case, the period of the motion is 7 minutes, and the amplitude is 12 m. Thus, we can substitute these values into the general equation to obtain d = 12 sin((2π/7) f).

The negative displacement of -12 m at time f = 0 minutes indicates that the object is at its maximum displacement in the negative direction at the start. Since the equation represents a sinusoidal function, the initial positive direction of motion can be determined by examining the phase of the sine function at f = 0. In this case, the displacement is negative, which implies a phase shift of half a period (180 degrees or π radians) to account for the initial position. Therefore, the object is moving in the positive direction at the start.

In conclusion, the equation modeling the displacement d as a function of time f for the given simple harmonic motion is d = 12 sin((2π/7) f), where d represents the displacement and f is the time in minutes.

Learn more about  displacement here:

https://brainly.com/question/31682467

#SPJ11

(a) (f ◦ g)(x) = 2x² + 1, domain: all real numbers.

(b) (g ◦ f)(x) = 4x² + 4x + 1, domain: all real numbers.

(c) (f ◦ f)(x) = 4x + 3, domain: all real numbers.

(d) (g ◦ g)(x) = x⁴, domain: all real numbers.

To find the composite functions and their domains for the given functions f(x) = 2x + 1 and g(x) = x², we need to substitute one function into another and evaluate the resulting expression. Let's calculate each composite function and determine their domains:

(a) (f ◦ g)(x) = f(g(x))

Substituting g(x) into f(x), we get:

(f ◦ g)(x) = f(g(x)) = f(x²) = 2(x²) + 1 = 2x² + 1

The domain of (f ◦ g)(x) is the same as the domain of g(x), which is all real numbers.

(b) (g ◦ f)(x) = g(f(x))

Substituting f(x) into g(x), we have:

(g ◦ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1

The domain of (g ◦ f)(x) is the same as the domain of f(x), which is all real numbers.

(c) (f ◦ f)(x) = f(f(x))

Substituting f(x) into itself, we get:

(f ◦ f)(x) = f(f(x)) = f(2x + 1) = 2(2x + 1) + 1 = 4x + 3

The domain of (f ◦ f)(x) is the same as the domain of f(x), which is all real numbers.

(d) (g ◦ g)(x) = g(g(x))

Substituting g(x) into itself, we have:

(g ◦ g)(x) = g(g(x)) = g(x²) = (x²)² = x⁴

The domain of (g ◦ g)(x) is the same as the domain of g(x), which is all real numbers.

For more such information on: domain

https://brainly.com/question/30096754

#SPJ8

a. The equation in slope-intercept form is y = -2x + 2.

b. A table for the equation is shown below.

c. A graph of the points with a line for the inequality is shown below.

d. The solution area for the inequality has been shaded.

e. Yes, the test point (0, 0) satisfy the conditions of the original inequality.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + b

Where:

Part a.

In this exercise, we would change each of the inequality to an equation in slope-intercept form by replacing the inequality symbols with an equal sign as follows;

2x + y ≤ 2

y = -2x + 2

Part b.

Next, we would complete the table for each equation based on the given x-values as follows;

x       -1        0        1

y        4        2       0

Part c.

In this scenario, we would use an online graphing tool to plot the inequality as shown in the graph attached below.

Part d.

The solution area for this inequality y ≤ -2x + 2 has been shaded and a possible solution is (-1, 1).

Part e.

In conclusion, we would use the test point (0, 0) to evaluate the original inequality.

2x + y ≤ 2

2(0) + 0 ≤ 2

0 ≤ 2 (True).

Read more on inequality here: brainly.com/question/10413737

#SPJ1

The general solution of the given differential equation is [tex]y = Ce^(-3x) + Cze^(2x) + 9/(1+10x) + (35/32)sin(x) + (45/32)cos(x).[/tex]

To find the general solution of the given differential equation, we will follow the procedures developed in this chapter. The differential equation is presented in the form y'' - 7y' + 10y = 9 + 5sin(x). In order to solve this equation, we will first find the complementary function and then determine the particular integral.

Complementary Function

The complementary function represents the homogeneous solution of the differential equation, which satisfies the equation when the right-hand side is equal to zero. To find the complementary function, we assume y = e^(rx) and substitute it into the differential equation. Solving the resulting characteristic equation [tex]r^2[/tex] - 7r + 10 = 0, we obtain the roots r = 3 and r = 4. Therefore, the complementary function is given by[tex]y_c = Ce^(3x) + C'e^(4x)[/tex], where C and C' are arbitrary constants.

Particular Integral

The particular integral represents a specific solution that satisfies the non-homogeneous part of the differential equation. In this case, the non-homogeneous part is 9 + 5sin(x). To find the particular integral, we use the method of undetermined coefficients. Since 9 is a constant term, we assume a constant solution, y_p1 = A. For the term 5sin(x), we assume a solution of the form y_p2 = Bsin(x) + Ccos(x). Substituting these solutions into the differential equation and solving for the coefficients, we find that A = 9/10, B = 35/32, and C = 45/32.

General Solution

The general solution of the differential equation is the sum of the complementary function and the particular integral. Therefore, the general solution is y = [tex]Ce^(3x) + C'e^(4x) + 9/(1+10x) + (35/32)sin(x) + (45/32)cos(x[/tex]), where C, C', and the coefficients A, B, and C are arbitrary constants.

Learn more about differential equation

brainly.com/question/32645495

#SPJ11