enrollment decreased by 4% last semesterthen increased by 6% this semester enrollment higher or lower than it was at the beginning of last semester

Question:

Write a linear function f with f (- 1/2) = 1 and f (0) = -4

The linear function f with f (- 1/2) = 1 and f (0) = -4 would be ; y = -5x -4.

What is a linear equation?

A linear equation is an equation in which each term has at max one degree. Linear equations in variables x and y can be written in the form

y = mx + c

Linear equation with two variables, when graphed on the cartesian plane with axes of those variables, give a straight line.

We are asked to write the linear function f with f (- 1/2) = 1 and f (0) = -4

Let the equation in variables x and y can be written in the form y = mx + c

So f (- 1/2) = 1

this gives, 1 = -1/2m+c      -----------eq 1

Also f (0) = -4

This gives -4 = c.            --------------eq2

Now Putting the value of c in the equation in eq1 we get m=0.

So 1 = -1/2m+c  

1 = -1/2m - 4

m = -5

Then we get;

y = -5x -4.

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The sum of the series \(\sum_{k=0}^\infty \frac{3}{7^{k} }\)  is 7/2. Therefore, the correct answer is option C. The sum of a geometric series can be found only if the ratio is between -1 and 1.

To find the sum of the series \(\sum_{k=0}^\infty \frac{3}{7^{k} }\), we can use the formula for the sum of an infinite geometric series, which is \(\frac{a}{1-r}\), where a is the first term and r is the common ratio.

In this case, the first term is \(\frac{3}{7^0}=3\) and the common ratio is \(\frac{1}{7}\). Substituting these values into the formula, we get:

\(\frac{3}{1-\frac{1}{7}}=\frac{3}{\frac{6}{7}}=\frac{7}{2}\)

Therefore, the sum of the series is c. 7/2. Alternatively, we can also find the sum of the series by adding up the terms:

\(\frac{3}{1}+\frac{3}{7}+\frac{3}{49}+\frac{3}{343}+...\approx 4.5\)

This method involves adding up an infinite number of terms, so it may not always be practical or accurate. Using the formula for the sum of an infinite geometric series is a more efficient and reliable method. Therefore, the correct answer is option C.

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Complete Question:

Find the sum of the series:

\(\sum_{k=0}^\infty \frac{3}{7^{k} }\)

a. 7/3

b. 21/2

c. 7/2

d. 21/4

e. 7

The factored form of the given algebraic expression by factoring out the Greatest Common Factor is 6y^2 (3y^2 + 4y + 1). Option A is correct.

We are given an algebraic expression:

18y^4 + 24y^3 + 6y^2

We need to factorize by factoring out the Greatest Common Factor.

In the given algebraic expression, we can see that;

6 is the Greatest Common Factor of 18, 24, and 6.

y^2 is the Greatest Common Factor of y^4, y^3 and y^2.

So, we will take the common 6y^2 from the given expression;

18y^4 + 24y^3 + 6y^2

= 6y^2 (3y^2 + 4y + 1)

So, option A is correct.

Thus, the factored form of the given algebraic expression by factoring out the Greatest Common Factor is 6y^2 (3y^2 + 4y + 1). Option A is correct.

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Answer:

\(r = \$14.71\)

Step-by-step explanation:

Given

Normal Hour = 32 hours

Overtime = Hours above 32

Rate for Overtime = 1.4 times normal rate

Earnings = $535.62

Required

Determine the normal hour pay

First, we need to determine the hours worked overtime.

This is:

\(Overtime = 43 - 32\)

\(Overtime = 11\)

The equation that binds all the parameters is:

\(Earnings = Normal\ Hour * Normal\ Rate + Overtime * Overtime\ Rate\)

This gives:

\(535.62 = 21 * r + 11 * 1.4r\)

\(535.62 = 21r + 15.4r\)

\(535.62 = 36.4r\)

Solve for r

\(r = 535.62/36.4\)

\(r = \$14.71\)

Answer:

C. 3

Step-by-step explanation:

One of the (many) properties of definite integrals is
\(\mbox{\large \int\limits _{a}^{b}f(x)\,dx + \int\limits _{b}^{c}f(x)\,dx = \int\limits }_{a}^{c}f(x)\,dx\)

Therefore
\(\mbox{\large \int\limits _{-2}^{5}f(x)\,dx + \int\limits _{5}^{2}f(x)\,dx = \int\limits }_{-2}^{2}f(x)\,dx\)

Given

\(\mbox{\large \int\limits _{-2}^{5}f(x)\,dx = 11}}}\para\)

and

\(\mbox{\large \int\limits _{-2}^{2}f(x)\,dx = 14}}\)

We get

\(\mbox{\large 11+ \int\limits _{5}^{2}f(x)\,dx} = 14\)

Subtracting 11 from both sides we get

\(\mbox{\large \int\limits _{5}^{2}f(x)\,dx} = 14 - 11 = 3\\\)

Answer: Choice C which is 3

Answer:

360 ft

Step-by-step explanation:

1 min = 60 sec

60/6 = 10

12 x 10 = 120

120 yd = 120 x 3 ft

120 yd = 360 ft

Answer:

$870.00 is the motorcycle price after a 40% discount

Step-by-step explanation:

discount = original price x discount % / 100

discount = 1450 x 40 / 100

discount = 1450 x 0.4

discount = $580.00

Final Price = Original Price - Discount

Final Price = 1450 - 580

Final Price = $870

Answer: 8ft

Step-by-step explanation:

Answer:

The width is 8 feet.

Step-by-step explanation:

P=2L x 2W

2L=24

40-24=16

16=2W

1W=8

Answer:

Step-by-step explanation:

Let width be w.

Let the length equal 3+2w.

Since w=5 cm,

3+2w=13 cm

Area of rectangle= length× width

Area=13×5

Area=65 cm2

So the answer is the 2nd option. Hope it helps!

Answer and Step-by-step explanation:

1. Given

This part is given at the beginning of the problem.

2. Corresponding Angles Postulate

Because the lines are parallel, these two angles will be congruent by the corresponding angles postulate.

3. Same-Side Interior Angles Theorem

The same-side interior angles theorem has it so that the angles within parallel lines will add up to 180.

4. Substitution

We can use substitution here because we already determined that the measure of angle UTY is congruent to the measure of angle HGY.

I hope this helps!

#teamtrees #PAW (Plant And Water)

Answer:

Below.

Step-by-step explanation:

m <UTY + m <UTR = 180 degrees ( Angles on a straight line YR)

But m < UTR = m<RWX ( Corresponding angles  in the parallel lines SU and VX)

So substituting for m< UTR in the first equation:

m <UTY + m<RWX  = 180.

Answer:

3 looks right but I am not so sure

Step-by-step explanation:

sorry if i am wring

The mean weight for each group and the difference between them, calculate the mean weight for group a and group b separately, and then subtract the mean of group b from the mean of group a.

To calculate the mean weight for each group (xa and xb), follow these steps:

1. Add up all the weights of group a and divide by the total number of data points in group a to find the mean weight (xa).
2. Repeat the same process for group b, adding up all the weights and dividing by the total number of data points to find the mean weight (xb).
3. Finally, subtract the mean of group b (xb) from the mean of group a (xa) to find the difference between the two means (xa - xb).

In summary, to find the mean weight for each group and the difference between them, calculate the mean weight for group a and group b separately, and then subtract the mean of group b from the mean of group a.

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The statement which is true by the corresponding angles theorem is option (B) ∠2 ≅ ∠6

The corresponding angles theorem states that when a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.

In this case, lines c and d are parallel and transversal p intersects them, creating eight angles: ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, and ∠8.

The corresponding angles are

∠1 and ∠5

∠2 and ∠6

∠3 and ∠7

∠4 and ∠8

Since lines c and d are parallel, we know that ∠2 and ∠6 are corresponding angles. Therefore, by the corresponding angles theorem, we can conclude that ∠2 ≅ ∠6

Therefore, the correct option is (B) ∠2 ≅ ∠6

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The given question is incomplete, the complete question is:

Lines c and d are parallel lines cut by transversal p.

Which must be true by the corresponding angles theorem?

A) ∠1 ≅ ∠7

B) ∠2 ≅ ∠6

C) ∠3 ≅ ∠5

D) ∠5 ≅ ∠7

Answer:

Austin is the capital of Texas

Step-by-step explanation:

Answer:

(4,6,4,-8,-3)

Step-by-step explanation:

Domain means of function is the set of all possible inputs for function

Answer:

Distributive

Step-by-step explanation:

Answer:3(4+5)= 3(4)+3(5)

Step-by-step explanation:

The Fourier series representation of the 2π periodic signal f(x) = x for 0 < x < π is (π/4) + Σ[(-1/n) \((-1)^n\) sin(nω₀x)].

To determine the Fourier series representation of the periodic signal f(x) = x for 0 < x < π with a period of 2π, we can use the following steps:

Determine the coefficients a₀, aₙ, and bₙ:

a₀ = (1/π) ∫[0,π] f(x) dx

= (1/π) ∫[0,π] x dx

= (1/π) [x²/2] ∣ [0,π]

= (1/π) [(π²/2) - (0²/2)]

= π/2

aₙ = (1/π) ∫[0,π] f(x) cos(nω₀x) dx

= (1/π) ∫[0,π] x cos(nω₀x) dx

bₙ = (1/π) ∫[0,π] f(x) sin(nω₀x) dx

= (1/π) ∫[0,π] x sin(nω₀x) dx

Simplify and evaluate the integrals:

For aₙ:

aₙ = (1/π) ∫[0,π] x cos(nω₀x) dx

For bₙ:

bₙ = (1/π) ∫[0,π] x sin(nω₀x) dx

Write the Fourier series representation:

f(x) = a₀/2 + Σ[aₙcos(nω₀x) + bₙsin(nω₀x)]

where Σ represents the summation from n = 1 to ∞.

To evaluate the integrals for aₙ and bₙ and determine the specific values of the coefficients, let's calculate them step by step:

For aₙ:

aₙ = (1/π) ∫[0,π] x cos(nω₀x) dx

Using integration by parts, we have:

u = x (derivative = 1)

dv = cos(nω₀x) dx (integral = (1/nω₀) sin(nω₀x))

Applying the integration by parts formula, we get:

∫ u dv = uv - ∫ v du

Plugging in the values, we have:

aₙ = (1/π) [x (1/nω₀) sin(nω₀x) - ∫ (1/nω₀) sin(nω₀x) dx]

= (1/π) [x (1/nω₀) sin(nω₀x) + (1/nω₀)² cos(nω₀x)] ∣ [0,π]

= (1/π) [(π/nω₀) sin(nω₀π) + (1/nω₀)² cos(nω₀π) - (0/nω₀) sin(nω₀(0)) - (1/nω₀)² cos(nω₀(0))]

= (1/π) [(π/nω₀) sin(nπ) + (1/nω₀)² cos(nπ) - 0 - (1/nω₀)² cos(0)]

= (1/π) [(π/nω₀) sin(nπ) + (1/nω₀)² - (1/nω₀)²]

= (1/π) [(π/nω₀) sin(nπ)]

= (1/n) sin(nπ)

= 0 (since sin(nπ) = 0 for n ≠ 0)

For bₙ:

bₙ = (1/π) ∫[0,π] x sin(nω₀x) dx

Using integration by parts, we have:

u = x (derivative = 1)

dv = sin(nω₀x) dx (integral = (-1/nω₀) cos(nω₀x))

Applying the integration by parts formula, we get:

∫ u dv = uv - ∫ v du

Plugging in the values, we have:

bₙ = (1/π) [x (-1/nω₀) cos(nω₀x) - ∫ (-1/nω₀) cos(nω₀x) dx]

= (1/π) [-x (1/nω₀) cos(nω₀x) + (1/nω₀)² sin(nω₀x)] ∣ [0,π]

= (1/π) [-π (1/nω₀) cos(nω₀π) + (1/nω₀)² sin(nω₀π) - (0 (1/nω₀) cos(nω₀(0)) - (1/nω₀)² sin(nω₀(0)))]

= (1/π) [-π (1/nω₀) cos(nπ) + (1/nω₀)² sin(nπ)]

= (1/π) [-π (1/nω₀) \((-1)^n\) + 0]

= (-1/n) \((-1)^n\)

Now, we can write the complete Fourier series representation:

f(x) = a₀/2 + Σ[aₙcos(nω₀x) + bₙsin(nω₀x)]

Since a₀ = π/2 and aₙ = 0 for n ≠ 0, and bₙ = (-1/n) \((-1)^n\), the Fourier series representation becomes:

f(x) = (π/4) + Σ[(-1/n) \((-1)^n\) sin(nω₀x)]

where Σ represents the summation from n = 1 to ∞.

This is the complete Fourier series representation of the given 2π periodic signal f(x) = x for 0 < x < π.

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The question is -

Determine the Fourier series representation for the 2n periodic signal defined below:

f(x) = x, 0 < x < π

Perimeter of the soccer field is 306 meters.

A shape's perimeter  is defined as the total length of its bounds. The perimeter of a shape is determined by summing all sides and side lengths that enclose the shape. It is measured in linear measurement units such as centimeters, meters, inches, and feet.

Given,

Length of the soccer field = 98 meters

Width of the soccer field = 55 meters wide

Perimeter of rectangle = 2(Length + Width)

Perimeter of soccer field = 2(98 + 55)

Perimeter of soccer field = 2(153)

Perimeter of soccer field = 306 meters

Hence, 306 meters is the perimeter of the soccer field.

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Answer:

A = 142.5

Step-by-step explanation:

Area of triangle = 1/2 bh

A = 1/2 (15)(19)

A = 142.5

Answer: 2 units to the left

Step-by-step explanation:

The correct statement relating the solutions to the system of equations is given as follows:

Infinitely many solutions.

The system of equations in the context of this problem is defined as follows:

From the first equation, we have that:

6y = -3x

y = -0.5x.

Replacing into the second equation, the value of x is given as follows:

x + 2(-0.5)x = 0

x - x = 0

0 = 0.

As 0 = 0 is a statement that is always true, the system has an infinite number of solutions.

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Answer:

y = 3x + 5

Step-by-step explanation:

y = mx + c

8 = 3(1) + c

8 = 3 + c

5 = c

i assume slope is the gradient if it isnt then ._.

Answer:

(-0.8, 2.2)

Step-by-step explanation:

Where the two lines intersect is the solution to the System of Equations.

Answer:

the answer would be (-0.8, 2.2) since it hasn't hit -3 yet and if it were to be -1 it would be in the bottom left

Answer:

False

Step-by-step explanation:

Because it is convex, that is, it has bulges

The reduced price of the piano is $301 if Lou went shopping for a piano at our ST store a piano originally priced at $860 had a price reduction of 65%

What is percentage ?

Percentage can be defined as the product of ratio of given value, total value and hundred.

To find the reduced price of the piano, we need to multiply the original price by the discount percentage (65%) and then subtract the result from the original price.

The amount of the discount is:

65% of $860 = 0.65 x $860 = $559

So, the reduced price of the piano after the discount is:

Reduced price = Original price - Discount

Reduced price = $860 - $559

Reduced price = $301

Therefore, the reduced price of the piano is $301.

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The equation for the plane that contains the line (t) = (2,3t,1,6t,1,-t) and is perpendicular to the plane x+3y-z=63 is given by the vector, the point of intersection of the two planes, {v₁}=(3t,6t,-1) is a vector along the given line and {v₂}=(x⁻²,y⁻¹,z⁻¹) is a vector normal to the given plane.

Therefore, the equation of the plane is given by (x-2)+3(y-1)+(z-1) = 0.

This equation can also be written in standard form as 3x+y-z=59.

The equation of the plane provides the set of all points (x, y, z) that are in the plane. This plane is perpendicular to the given plane and contains the line (t)=(2,3t,1,6t,1,-t).

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