Select the statement that correctly describes the solution to this system of equations. 4x+2y=6

4x+2y=4


A. There is no solutions

B. There are infinitely many solutions

C. There is exactly one solution at (4,2)

D. There is exactly one solution at (6,4)

The limit of the sequence a_n is 0. The sequence a_n = (cos n)/[tex]7^n[/tex] oscillates between -1/[tex]7^n[/tex] and 1/[tex]7^n[/tex] since the cosine function is bounded between -1 and 1. Therefore, by the squeeze theorem, the limit of the sequence is 0 as n approaches infinity.

The cosine function oscillates between -1 and 1, so we have:

-1/[tex]7^n[/tex] ≤ cos(n)/7^n ≤ 1/[tex]7^n[/tex]

Dividing each term by [tex]7^n[/tex], we obtain:

-1/[tex]7^n[/tex] ≤ a_n ≤ 1/[tex]7^n[/tex]

By the squeeze theorem, since -1/[tex]7^n[/tex] and 1/[tex]7^n[/tex] both approach zero as n approaches infinity, we have:

lim a_n = 0

Therefore, the limit of the sequence a_n is 0.

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If a rectangular piece of cardboard with dimensions 5 inches by 8 inches is used to make the curved side of a cylinder-shaped container, the greatest volume the cylinder can hold is 80/π cubic inches.

To find the greatest volume the cylinder can hold, we need to determine the dimensions of the cylinder that can be made from the given cardboard.

First, we need to calculate the circumference of the cylinder using the length of the cardboard, which will be the height of the cylinder. The length of the cardboard is 8 inches, so the circumference of the cylinder will be 8 inches.

The circumference of a cylinder is given by the formula C = 2πr, where r is the radius of the cylinder.

Therefore, 8 = 2πr, or r = 4/π inches.

Next, we need to determine the length of the curved side of the cylinder, which is given by the formula L = 2πr.

So, L = 2π(4/π) = 8 inches.

Finally, we can calculate the volume of the cylinder using the formula V = πr²h, where h is the height of the cylinder, which is 5 inches.

V = π(4/π)²(5) = 80/π cubic inches.

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Substituting these values into the expression for f(θ), we get:
f(θ) = -sin(θ) - cos(θ) + 5θ + 4

To find f, we need to integrate f''(θ) twice.

First, we integrate sin(θ) + cos(θ) with respect to θ to get f'(θ):

f'(θ) = -cos(θ) + sin(θ) + C1
where C1 is the constant of integration.
Next, we integrate f'(θ) with respect to θ to get f(θ):

f(θ) = -sin(θ) - cos(θ) + C1θ + C2

where C2 is the constant of integration.

Using the initial conditions given, we can solve for C1 and C2:
[tex]f(0) = -1 - 1 + C2 = 2[/tex]
C2 = 4
f'(0) = -1 + 0 + C1 = 4

C1 = 5

Substituting these values into the expression for f(θ), we get:
f(θ) = -sin(θ) - cos(θ) + 5θ + 4

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The height of the funnel is 11.31, under the condition that one funnel has a base with a diameter of 8 in. And a slant height of 12 in.

Here we have to apply the Pythagorean theorem to evaluate the height of the funnel. The Pythagorean theorem projects that the square of the hypotenuse (the slant height) is equal to the sum of the squares of the other two sides (the radius and height).

Now, we have a cone that has a base diameter of 8 inches which says that the radius is 4 inches. The slant height is 12 inches. Then the height is
h² + r² = l²
h² + 4² = 12²
h² = 144 - 16
h² = 128
h = √(128)
h ≈ 11.31

Hence, 11.31 inches is the approximate height of the funnel after rounding to the nearest hundredth.
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Polar coordinates for rectangular coordinates if x<0: r=√(x²+y²) and θ=tan⁻¹(y/x)+π if y≥0 or θ=tan⁻¹(y/x)−π if y<0, For x≥0: r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.

The polar coordinates of a point with rectangular coordinates (x,y) depend on the sign of x.

If x<0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2. If x≥0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2.

If x<0, t

hen r=√(x²+y²) and

θ=tan⁻¹(y/x)+π if y≥0

or θ=tan⁻¹(y/x)−π if y<0.

The value of r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line segment from the origin to the point.

If x≥0, then r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.

In this case, θ is the angle between the positive x-axis and the line segment from the origin to the point, measured counterclockwise.

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Derivative of y w.r.t t is dy/dt = 16 at x = -1.

We need to find dy/dt at x = -1, given y = -2x² + 4 and dx/dt = 4.

Step 1: Differentiate y with respect to x.
Since y = -2x² + 4, we have:

dy/dx = -4x

Step 2: Substitute x = -1 into the dy/dx equation.
When x = -1, we get:

dy/dx = -4(-1) = 4

Step 3: Use the Chain Rule to find dy/dt.
The Chain Rule states that dy/dt = (dy/dx)(dx/dt). We know dy/dx = 4 and dx/dt = 4, so we have:

dy/dt = (4)(4) = 16

Thus, dy/dt = 16 at x = -1.

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Answer:

11.86 units

Step-by-step explanation:

Volume = length x width x height

We know that the volume is 210 and the width is 5 3/5 (or 5.6) and the height is 3 1/3 (or 3.33).

Substituting these values into the formula, we get:

210 = length x 5.6 x 3.33

To solve for the length, we can divide both sides of the equation by (5.6 x 3.33):

length = 210 / (5.6 x 3.33)

Simplifying this expression, we get:

length ≈ 11.86

Therefore, the length of the large box is approximately 11.86 units.

The measure of the smaller acute angle of the triangle to the nearest tenth of a degree is 28.3 degrees.

Let's denote the smaller acute angle of the triangle as θ. We can use the tangent function to find the measure of this angle:

tan(θ) = opposite/adjacent

In this case, the opposite side is the length of the short leg (17) and the adjacent side is the length of the long leg (32). So we have:

tan(θ) = 17/32

Using a calculator, we can take the inverse tangent (tan^-1) of both sides to solve for θ:

θ = tan^-1(17/32) ≈ 28.3 degrees

So the measure of the smaller acute angle of the triangle is approximately 28.3 degrees.

Here's a diagram to illustrate the triangle:

            / l

         /    l

  17 /       l opposite

    /         l

  /  θ       l

/_______l

    adjacent

      32

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The definite integral

[tex]-e^(2z)/u + 16e^z/u - 64ln(u)/u + C[/tex]

To evaluate this definite integral, we need to find the antiderivative of the integrand and evaluate it at the limits of

integration.

Let's start by using u-substitution:

Let [tex]u = e^z+8z[/tex]

Then [tex]du/dz = e^z+8[/tex]

And [tex]dz = 1/e^z+8 du[/tex]

Substituting this into the integral, we get:

[tex]∫(e^z) + 8/ (e^z+8z)^2 dz[/tex]

= [tex]∫(1/u^2)(e^z+8)^2 du[/tex]

= [tex]∫(1/u^2)(e^(2z)+16e^z+64) du[/tex]

= [tex]-e^(2z)/u + 16e^z/u - 64ln(u)/u + C[/tex]

Now we need to evaluate this antiderivative at the limits of integration.

Let's assume the limits are a and b:

= [tex][-e^(2b)/(e^b+8b) + 16e^b/(e^b+8b) - 64ln(e^b+8b)/(e^b+8b)] - [-e^(2a)/(e^a+8a) + 16e^a/(e^a+8a) - 64ln(e^a+8a)/(e^a+8a)][/tex]

Simplifying this expression is not easy, but it can be done with some algebraic manipulation.

Therefore, The definite integral

[tex]-e^(2z)/u + 16e^z/u - 64ln(u)/u + C[/tex]

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Gail's observations about the box is correct. According to the question the edge of a cube-shaped box is 1 yard.

Now convert the edge length to feet as : 1 yard = 3 feet.

On checking the observations:

1.  According to Janae the area of each face is 9 square feet. Calculating the surface area of one face of the cube is given by:

Area = (edge length)² = (3 feet)² = 9 square feet.

Now, Total surface area = 6*9square feet= 54 square feet.

Hence Observations of Janae is not correct.

2. According to Archie perimeter of each face of the box is 3 feet. Calculating the perimeter of each face of the cube is given by:

Perimeter= 4 × (edge length)=4×1 yard= 4×3feet=12 feet

Hence Archie observation is not correct also.

3. According to Gail the volume of the box is 1 cubic yard.

The volume of a cube is given by:

volume= (edge length)³ = (1 yard)³ = 1 cubic yard.

Therefore Gail's observation is correct.

From all the above observations it can be concluded that only Gail's observation is correct.

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Answer: Pairs (0, 14) and (10, 0

Step-by-step explanation:

Answer:  x=20  y=43

Step-by-step explanation:

That little symbol in <1 means that the angle is a right angle, which is = to 90°  so

<1 = 90°

133-y = 90        solve for y by subtracting 133 from both sides

-y = -43             divide by -1 on both sides

y=43

Because all 3 angles make a line, which is 180, and you know <1 = 90   then <2+<3=90 as well.

<2+<3=90

22 + x + 48 =90     simplify

70 + x =90              subtract 70 from both sides

x=20

Answer:

C. The distributive property

6.2 × 10⁴ in decimal notation is simply 62,000.

In scientific notation, a number is expressed as the product of a decimal number between 1 and 10 and a power of 10.

Now, let's talk about converting a number in scientific notation to decimal notation. Decimal notation simply means expressing a number in the standard way, using digits and a decimal point. To convert a number in scientific notation to decimal notation, we just need to evaluate the product of the decimal number and the power of 10.

In the case of 6.2 × 10⁴, the decimal number is 6.2, and the power of 10 is 4. To evaluate the product, we simply move the decimal point in 6.2 four places to the right, since the power of 10 is positive. This gives us:

6.2 × 10⁴ = 62,000

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Complete Question:

Convert to decimal notation: 6.2 × 10⁴

Mrs. Vilakazi will make a profit of r233,88 if she bakes 204 scones and sells 171.

The profit is the difference between the total revenue and the total cost.

The total cost is the cost per scone multiplied by the number of scones baked:

total cost = r0,53/scone × 204 scones = r108,12

The total revenue is the selling price per scone multiplied by the number of scones sold:

total revenue = r2,00/scone × 171 scones = r342,00

Therefore, the profit is:

profit = total revenue - total cost

profit = r342,00 - r108,12

profit = r233,88

Mrs. Vilakazi will make a profit of r233,88 if she bakes 204 scones and sells 171.

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The gross income for selling 348 bushels of apples at a price of $16.50 per bushel is $5,742.

The gross income is the total revenue earned from the sales of a product or service, before any expenses or deductions are taken out. To calculate gross income, we multiply the quantity of goods sold by the price per unit.

In this case, we are given that 348 bushels of apples were sold at a price of $16.50 per bushel. Multiplying these two values together gives us the total revenue earned from the sale of these apples, which is the gross income.

To calculate the gross income, we can use the formula:

Gross income = Quantity sold x Price per unit

Plugging in the given values, we get:

Gross income = 348 x $16.50

Simplifying the calculation, we get:

Gross income = $5,742

Therefore, the gross income for selling 348 bushels of apples at a price of $16.50 per bushel is $5,742.

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The correlation between the number of points scored by the basketball team and the weeks of the season is likely negative.

However, without seeing the actual data, it is difficult to determine the exact correlation. As for the coach's statement, it could be either correct or incorrect depending on the actual data. Based on the given information, the basketball coach's observation suggests a negative correlation between the number of weeks into the season and the points scored by his team. Without the actual data, it is impossible to determine if his observation is correct or incorrect.

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The correct answer is option 2. Yes, the mean of the sample proportions is 0.6, which is the same as the population proportion.

To determine if the sample proportion is an unbiased estimator, we need to check if the mean of the sample proportions equals the population proportion. In this case, the population proportion of officers who are younger than 18 is given as 0.6. The sample proportions for all possible samples of size 2 are calculated and given in the table.
To calculate the mean of the sample proportions, we add up all the proportions in the table and divide by the total number of samples, which is 9.
Mean of sample proportions = (0 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 1 + 1 + 1) / 9 = 0.6
Since the mean of the sample proportions equals the population proportion, we can say that the sample proportion is an unbiased estimator.

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The magnitude and direction of the vector u = <-4, 7> are |u| ≈ √65 and θ ≈ 119.74°.

To find the magnitude and direction of the vector u = <-4, 7>, we will use the following steps:

1. Calculate the magnitude using the Pythagorean theorem.
2. Calculate the direction using the arctangent function.

Step 1: Calculate the magnitude.
Magnitude (|u|) = √((-4)^2 + (7)^2) = √(16 + 49) = √65

Step 2: Calculate the direction (angle θ).
θ = arctan(opposite/adjacent) = arctan(7/-4) ≈ -60.26° (in degrees)
Since the vector is in the second quadrant, we need to add 180°.
θ = -60.26° + 180° ≈ 119.74°

So, the magnitude and direction of the vector u = <-4, 7> are |u| ≈ √65 and θ ≈ 119.74°.

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For the function "h(x) = |2x| - 8", the domain is (-∞, ∞) and the range is  [-8, ∞).

The function h(x) = |2x| - 8 is defined for all real numbers x, so the domain of h(x) is the set of all real-numbers, or (-∞, ∞).

To find the range of the function, we determine set of all possible output values of function. Since the function involves the absolute value of 2x, the output can never be less than -8.

When "2x" is positive, |2x| = 2x. When 2x is negative, |2x| = -2x. This means that the function h(x) will have two branches depending on whether 2x is positive or negative.

⇒ When 2x is positive, h(x) = |2x| - 8 = 2x - 8. This branch of the function will have all non-negative values.

⇒ When 2x is negative, h(x) = |2x| - 8 = -2x - 8. This branch of the function will have all non-positive values.

Combining the two , we get the range of the function h(x) as [-8, ∞).

Therefore, the domain of h(x) is (-∞, ∞) and the range of h(x) is [-8, ∞).

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Half way between 4/5 and 14/15 is 13/15.

To find the halfway point between 4/5 and 14/15, we need to calculate the average of the two fractions. Here's the process:

1. Make sure the fractions have a common denominator. In this case, the least common denominator (LCD) for 5 and 15 is 15.
2. Convert the fractions to equivalent fractions with the common denominator: 4/5 becomes 12/15 (multiply both numerator and denominator by 3), while 14/15 stays the same.
3. Add the two equivalent fractions together: 12/15 + 14/15 = 26/15.
4. Divide the sum by 2 to find the halfway point: (26/15) ÷ 2. To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number: 26/15 × 1/2 = 26/30.
5. Simplify the resulting fraction: 26/30 can be simplified by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2 in this case. Thus, 26 ÷ 2 = 13, and 30 ÷ 2 = 15. The simplified fraction is 13/15.

So, the halfway point between 4/5 and 14/15 in its simplest form is 13/15.

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Bill sold 140 bottles of cherry soda and 70 bottles of grape soda.

Let's assume that x is the number of bottles of cherry soda sold and y is the number of bottles of grape soda sold. We can set up a system of equations to represent the given information:

x + y = 210 (equation 1: the total number of bottles sold is 210)

1.15x + 0.99y = 230.30 (equation 2: the total cost of the sodas is $230.30)

We can use the first equation to solve for y in terms of x:

y = 210 - x

Substituting this expression for y into the second equation, we get:

1.15x + 0.99(210 - x) = 230.30

Simplifying and solving for x, we get:

1.15x + 207.9 - 0.99x = 230.30

0.16x = 22.4

x = 140

So Bill sold 140 bottles of cherry soda. Substituting this value into equation 1, we get:

140 + y = 210

y = 70

Therefore, Bill sold 140 bottles of cherry soda and 70 bottles of grape soda.

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The temperature at 11:00 A.M. yesterday was 78°F.

To find the temperature at 11:00 A.M., we need to subtract 9 from the temperature at 4:00 P.M. because the temperature rose 9 degrees between those times. So, we can use the following equation:

Temperature at 11:00 A.M. = Temperature at 4:00 P.M. - 9

We know that the temperature at 4:00 P.M. was 87°F, so we can substitute that into the equation and solve for the temperature at 11:00 A.M.:

Temperature at 11:00 A.M. = 87 - 9

Temperature at 11:00 A.M. = 78°F

Therefore, the temperature at 11:00 A.M. yesterday was 78°F.

It's important to remember that when solving word problems like this, we need to pay close attention to the details and make sure we understand what the question is asking us to find. In this case, we needed to use the information given about the temperature rising 9 degrees between 11:00 A.M. and 4:00 P.M. to find the temperature at 11:00 A.M. By using the equation provided and substituting in the known values, we were able to solve for the unknown temperature at 11:00 A.M.

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Density of liquid b = 0.4 g/cm³.

Let the volume of liquid B added be V cm³.

The total volume of the mixture = Volume of A + Volume of B = 150 + V cm³

Using the formula:

Density = Mass/Volume

Density of C = (Mass of C) / (Volume of C)

1.1 = 220 / (150 + V)

Solving for V, we get:

V = 100 cm³

Therefore, the volume of liquid B added is 100 cm³.

The total mass of the mixture = Mass of A + Mass of B = (Density of A x Volume of A) + (Density of B x Volume of B)

220 = (1.2 x 150) + (Density of B x 100)

Solving for Density of B, we get:

Density of B = (220 - 180) / 100 = 0.4 g/cm³

Therefore, the density of liquid B is 0.4 g/cm³.

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None of the bills have an experimental probability of 3, as all probabilities are between 0 and 1.

Based on the table, the experimental probability for each bill being drawn from the bag next can be calculated by dividing the frequency of each bill by the total number of draws (100). Using this formula, we can calculate the experimental probabilities for each bill:

1. For the $1 bill: Experimental probability = [tex]\frac{(Frequency of  $1 bill)}{Total draws} = \frac{8}{100} = 0.28[/tex]
2. For the $10 bill: Experimental probability =[tex]\frac{(Frequency of  $10 bill)}{Total draws} = \frac{56}{100} = 0.56[/tex]
3. For the $20 bill: Experimental probability =[tex]\frac{(Frequency of  $20 bill)}{Total draws} = \frac{2}{100} = 0.02[/tex]

None of the bills have an experimental probability of 3, as all probabilities are between 0 and 1.

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Answer:

f(-2) = 9

Step-by-step explanation:

f(x) = -3x + 3                       Solve for f(-2)

f(-2) = -3(-2) + 3

f(-2) = 6 + 3

f(-2) = 9

The instant when the radar gun is used, the rate at which the distance between the two cars is changing is g'(t) = 7968t + 368/5 kilometers per hour.

Let's break down the problem. We have two cars, one traveling south at 112 km/h and another traveling west at 96 km/h. The police car is stationed at an intersection and the two cars are at different points relative to the intersection. The first car is 4/5 kilometer north of the intersection while the second car is 2/5 kilometer east of the intersection.

Let's call this distance "d". Using the Pythagorean theorem, we can write:

d² = (4/5)² + (2/5)² d² = 16/25 + 4/25 d² = 20/25 d = sqrt(20)/5 d = 2sqrt(5)/5 kilometers

Now, we need to find the rate at which the distance between the two cars is changing. This is equivalent to finding the derivative of the distance with respect to time. Let's call this rate "r".

To find "r", we need to use the chain rule. The distance between the two cars is a function of time, so we can write:

d = f(t)

where t is time. We can then write:

r = d'(t) = f'(t)

where d'(t) and f'(t) denote the derivatives of d and f with respect to time, respectively.

To find f'(t), we need to express d in terms of t. We know that the first car is traveling at a constant speed of 112 km/h due south. Let's call the position of the first car "x" and the time "t". Then we have:

x = -112t

The negative sign indicates that the car is moving south. Similarly, we can express the position of the second car in terms of time. Let's call the position of the second car "y". Then we have:

y = 96t

The positive sign indicates that the car is moving west.

Now, we can use these expressions to find the distance between the two cars as a function of time. Let's call this function "g(t)". Then we have:

g(t) = √((x + 4/5)² + (y - 2/5)²) g(t) = √((-112t + 4/5)² + (96t - 2/5)²)

To find g'(t), we need to use the chain rule. We have:

g'(t) = (1/2)(x + 4/5)'(x + 4/5)'' + (y - 2/5)'x(y - 2/5)''

where the primes denote derivatives with respect to time. We can simplify this expression by noting that x' = -112 and y' = 96. We also have x'' = y'' = 0, since the speeds of the two cars are constant.

Substituting these values, we get:

g'(t) = -112x(-112t + 4/5)/√((-112t + 4/5)² + (96t - 2/5)²) + 96x(96t - 2/5)/√((-112t + 4/5)² + (96t - 2/5)²)

Simplifying this expression, we get:

g'(t) = (-112x(-112t + 4/5) + 96x(96t - 2/5))/√((-112t + 4/5)² + (96t - 2/5)²)

We can further simplify this expression by multiplying out the terms in the numerator:

g'(t) = (-12544t + 560/5 + 9216t - 192/5)/√((-112t + 4/5)² + (96t - 2/5)²)

g'(t) = (7968t + 368/5)/√((-112t + 4/5)² + (96t - 2/5)²)

g'(t) = 7968t + 368/5

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Complete Question:

A car is traveling at 112 km/h due south at a point 4/5 kilometer north of an intersection_ police, the car Is traveling at 96 km/h due west to at point 2/5 kilometer due cust of the same intersection. At that instant; the radar in the police car measures the rate at which the distance between the two cars [ changing: What does the radar gun register?

Proportionately, if 8 cm represents​ 10%, the length of a line segment that is​ 100% is 80 cm.

Proportion is the ratio of two quantities equated to each other.

Proportion also represents the portion or part of a whole.

Proportions can be represented using decimals, fractions, or percentages, like ratios.

The percentage of 8 cm length = 10%

The whole length = 100%

Proportionately, 100% = 80 cm (8 ÷ 10%) or (8 x 100 ÷ 10)

Thus, we can conclude that 100% of the line segment will be 80 cm if 8 cm is 10%.

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If 8 cm represents​ 10%, what is the length of a line segment that is​ 100%? Explain.