Rewrite each equation in vertex form by completing the square. Then, give the key features revealed by both equations.
Question 7.

Answer:

Step-by-step explanation:

[tex]\frac{1}{5}[/tex] ∠ 7x

[tex]\frac{1/5}{7}[/tex] ∠ x

1/35 ∠x

x ≥ [tex]\frac{1}{35}[/tex]

The missing dimension of the cone is its height, which is approximately 2.5 m when rounded to the nearest tenth.

We can use the formula for the volume of a cone, which is:

Volume = (1/3)πr²h

where r is the radius of the base and h is the height of the cone.

We are given the volume of the cone as 13.4 m³ and the radius as 3.2 m. Substituting these values into the formula, we get:

13.4 = (1/3)π(3.2)²h

Multiplying both sides by 3 and dividing by π(3.2)², we get:

h = 3 × 13.4 / π(3.2)²

h ≈ 2.5 m

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The first inequality -4 ≤ x ≤ 3 represents the values of x that fall between the two vertical lines, while the second inequality 1 ≤ y ≤ 6 represents the values of y that fall between the two horizontal lines.

An inequality is a mathematical statement that compares two quantities or expressions using inequality symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "!=" (not equal to).

Inequalities can involve variables or constants, and can be expressed in one variable or multiple variables. The solution to an inequality is the set of values that satisfy the inequality.

For example, the inequality 2x + 3 > 7 is true for values of x that are greater than 2, since if we substitute x = 2, we get 2(2) + 3 = 7, which is not greater than 7. On the other hand, if we substitute x = 3, we get 2(3) + 3 = 9, which is greater than 7, so the inequality is true for x > 2.

Inequalities have many applications in mathematics and other fields, such as economics, physics, and engineering. They are used to represent constraints in optimization problems, to model relationships between variables, and to describe ranges of possible values for a quantity or variable.

To determine the double inequalities that define the shaded region, we need to find the equations of the two boundary lines that form the sides of the shaded region.

The two vertical lines are x=-4 and x=3. The two horizontal lines are y=1 and y=6.

The shaded region is enclosed by these four lines, so the double inequalities that define it are:

-4 ≤ x ≤ 3 and 1 ≤ y ≤ 6

The first inequality -4 ≤ x ≤ 3 represents the values of x that fall between the two vertical lines, while the second inequality 1 ≤ y ≤ 6 represents the values of y that fall between the two horizontal lines. Together, they define the rectangular shaded region with vertices (-4,1), (-4,6), (3,6), and (3,1).

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Answer:

i don't know i haven't done integers in a long time

Step-by-step explanation:

Answer:

The answer should be 20,365.714 but I am not sure

Step-by-step explanation:

The dimensions of the box that will maximize the enclosed volume are l = √(45/2) m and h = (22.5 - l²)/2 m.

To determine the dimensions of the box that will maximize the enclosed volume if we have 45 m² of material to build a box with a square base and no top, we can use the following steps:

Step 1: Write the formula for the volume of the box.V = l²h

Step 2: Write the formula for the surface area of the box.S = 2l² + 4lh

Step 3: Substitute S = 45 m² and simplify.2l² + 4lh = 45 m²l² + 2lh = 22.5 m²h = (22.5 - l²)/2

Step 4: Substitute the value of h in the formula for the volume and simplify.V = l²[(22.5 - l²)/2]V = (22.5l² - l⁴)/2

Step 5: Take the derivative of V with respect to l.dV/dl = 45l - 2l³

Step 6: Set dV/dl = 0 and solve for l.45l - 2l³ = 0l(2l² - 45) = 0l = 0 or l = √(45/2)

Step 7: Determine if the critical point is a maximum or a minimum by taking the second derivative of V with respect to l.d²V/dl² = 45 - 6l²d²V/dl² > 0 for l = √(45/2)

Therefore, the dimensions of the box that will maximize the enclosed volume are l = √(45/2) m and h = (22.5 - l²)/2 m.

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P(a number greater than 5 and an even number) = 1/6 or 0.167

Part 1 of 3 (a) To find the probability of getting an even number, divide the number of favorable outcomes (rolling a 2, 4, or 6) by the total possible outcomes (rolling any number between 1 and 6). There are 3 even numbers and 6 total possible outcomes.

P(an even number) = [tex]3/6 = 1/2 or 0.500[/tex]

Part 2 of 3 (b) To find the probability of getting a number less than or equal to 4, count the favorable outcomes (rolling a 1, 2, 3, or 4) and divide by the total possible outcomes (6). There are 4 favorable outcomes and 6 total possible outcomes.

P(a number less than or equal to 4) =[tex] 4/6 = 2/3 or 0.667[/tex]

Part 3 of 3 (c) To find the probability of getting a number greater than 5 and an even number, count the favorable outcomes (rolling a 6) and divide by the total possible outcomes (6). There is 1 favorable outcome and 6 total possible outcomes.

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Diameter = 16 mm

Radius = Diameter/2

= 16/2

= 8 mm

Total height of figure = 32 mm

Height of cylinder = 19 mm

Height of cone = 32 - 19

= 13 mm

Volume of cylinder = πr²h

Substituting the values of r and h in above formula,,

[tex] \longrightarrow \sf \pi \times {(8)}^{2} \times 19 \\ \\ \longrightarrow \sf \pi \times 64 \times 19 \\ \\ \longrightarrow \sf 1216 \pi \\ \\ \longrightarrow \sf 3820.17 \: mm^{3}\\ \\ [/tex]

Hence, Volume of cylinder is 3820.17 mm³

Now,

Volume of cone = 1/3 πr²h

[tex] \longrightarrow \sf \dfrac{1}{3} \times \pi \times {(8)}^{2}\times 13 \\ \\ \longrightarrow \sf \dfrac{1}{3} \times \pi \times 64 \times 13 \\ \\ \longrightarrow \sf 871.26~ mm^3 \\ \\ [/tex]

Hence, Volume of the cone is 871.26 mm³

Volume of composite figure = Volume of cylinder + Volume of cone.

= 3820.17 + 871.26

= 4691.43 mm³

Therefore, Volume of the composite figure is 4691.43 mm³.

The median of 20 is the most accurate to use, since the data is skewed.

A histogram is a graphical representation of data that shows the frequency distribution of a set of continuous data. It is a bar graph-like structure where the bars represent the frequency or proportion of data in specific intervals or ranges of values. The x-axis represents the range of values, and the y-axis represents the frequency or proportion of data in each range. Histograms are commonly used in statistical analysis to represent the distribution of data, especially when dealing with large datasets.

Here,

The data is not evenly distributed, as there are many donations at the lower end and fewer donations at the higher end. Therefore, the median, which represents the middle value in the data set, is a better measure of center than the mean, which can be influenced by extreme values. The median donation in this data set is $20.

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a) The power series representation of f(x) = 1/5 + x f(x) centered at x = 0 is f(x) = sigma^infinity_n = 0 ((x/5)^n)

b) The interval of convergence is (-5, 5)

To find the power series representation of f(x), we can use the formula for the geometric series

1 / (1 - r) = sigma^infinity_n = 0 (r^n)

where r is a constant.

In this case, we have

f(x) = 1/5 + x f(x)

We can solve for f(x) to get

f(x) = 1/5 / (1 - x)

Using the formula for the geometric series with r = x/5, we have

f(x) = sigma^infinity_n = 0 ((x/5)^n)

Multiplying both sides by 5, we get

5f(x) = sigma^infinity_n = 0 (x^n

So the power series representation of f(x) centered at x = 0 is

f(x) = sigma^infinity_n = 0 ((x/5)^n)

To determine the interval of convergence, we can use the ratio test

| (x/5)^(n+1) | / | (x/5)^n | = |x/5|

The series converges if the limit of |x/5| as n approaches infinity is less than 1. This is true when |x| < 5, so the interval of convergence is (-5, 5).

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The value of the angle x is 30 degrees

Trigonometric identities are identified as those equalities of a function holding that all of the values in the given function, equation or expression are true.

There are distinct trigonometric identities. They are enumerated thus;

Note that the sine identity is represented as;

sin θ = opposite/ hypotenuse

Opposite of angle x = 48cm

Hypotenuse = 96cm

Substitute the values

sinx =48/96

sin x = 0. 5000

Take sine inverse

x = 30 degrees

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the surface area of the given triangular prism with sides of 4in, 5in, 6in, 6in, 8in, and 7in is 146 square inches.

To calculate the surface area of a triangular prism, you need to find the area of each of the faces and add them up.

First, let's find the area of the two triangular faces. To do this, we need to find the base and height of each triangle. Since the prism is isosceles, the base of each triangle is 6 inches (the length of one of the sides of the equilateral triangle). The height of each triangle can be found using the Pythagorean theorem. We have two sides of the triangle: 4 inches and 5 inches. Using the Pythagorean theorem, we can find the height:

[tex]h^2 = 5^2 - 4^2\\h^2 = 25 - 16\\h^2 = 9\\h = 3[/tex]

So the height of each triangular face is 3 inches. Now we can find the area of each triangular face:

Area of one triangular face = (1/2) x base x height

= (1/2) x 6 x 3

= 9 square inches

Since there are two triangular faces, the total area of the triangular faces is:

Total area of triangular faces = 2 x 9 = 18 square inches

Next, let's find the area of the three rectangular faces. We have two rectangles with sides of 6 inches by 8 inches, and one rectangle with sides of 4 inches by 8 inches. The area of each rectangular face is:

Area of rectangular face = length x width

So the area of the rectangular faces are:

Area of rectangular face 1 = 6 x 8 = 48 square inches

Area of rectangular face 2 = 6 x 8 = 48 square inches

Area of rectangular face 3 = 4 x 8 = 32 square inches

Therefore, the total surface area of the triangular prism is:

Total surface area = 18 + 48 + 48 + 32 = 146 square inches

So the surface area of the given triangular prism with sides of 4in, 5in, 6in, 6in, 8in, and 7in is 146 square inches.

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To solve this problem, we need to convert the hexadecimal numbers (20cba)16 and (a02)16 to decimal form, add them together, and then convert the result back to hexadecimal form.

(20cba)16 = 2x16^4 + 12x16^3 + 11x16^2 + 10x16^1 = 131402

(a02)16 = 10x16^2 + 0x16^1 + 2x16^0 = 256

Adding the two decimal numbers together gives us:

131402 + 256 = 131658

To convert this decimal number back to hexadecimal form, we can use the repeated division method.

131658 / 16 = 8228 remainder 10 (A)

8228 / 16 = 514 remainders 4 (4)

514 / 16 = 32 remainder 2 (2)

32 / 16 = 2 remainder 0

2 / 16 = 0 remainder 2

Therefore, (20cba)16 + (a02)16 = (131658)10 = (2002A)16.

To find their product, we can multiply the two decimal numbers together and then convert the result to hexadecimal form.

131402 x 256 = 33559552

Converting this decimal number to hexadecimal form gives us:

33559552 / 16 = 2097472 remainder 0

2097472 / 16 = 131092 remainder 0

131092 / 16 = 8193 remainder 4 (4)

8193 / 16 = 512 remainders 1 (1)

512 / 16 = 32 remainder 0

32 / 16 = 2 remainder 0

2 / 16 = 0 remainder 2

Therefore, the product of (20cba)16 and (a02)16 is (33559552)10 = (2011004)16.

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The inverse function for the given function F(x)=1/x² -3x +1  is given by

f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.

Function f(x) is equals to,

F(x)=1/x² -3x +1

Inverse of a function, we need to swap the positions of the x and y variables and then solve for y.

Let's start with the original function

f(x) = 1/x^2 - 3x + 1

Now we will swap x and y,

⇒ x = 1/y^2 - 3y + 1

Next, Solve for y in terms of x

⇒ x = 1/y^2 - 3y + 1

⇒ x - 1 = 1/y^2 - 3y

⇒ 1/(x - 1) = y^2 - 3y

⇒ 1/(x - 1) = y(y - 3)

⇒ y(y - 3) = 1/(x - 1)

⇒ y^2 - 3y - 1/(x - 1) = 0

Using the quadratic formula, solve for y to get inverse function we have,

y = (3 ± √(9 + 4/(x - 1))) / 2

Therefore, the inverse function of f(x) is equal to f⁻¹(x) =(3 ± √(9 + 4/(x - 1))) /2.

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The above question is incomplete, the complete question is:

F(x)=1/x squared -3x +1 then Find the inverse

The speed of the particle is 8 m/s, the force of resistance is [tex]0.3(8)^2[/tex], or 19.2 N.

A particle of mass 1.2 kg is moving with a speed of 8 m/s in a straight line on a horizontal table. A resistance force is applied to the particle in the direction of the motion. The magnitude of the force is proportional to the square of the speed, such that [tex]F=0.3v^2[/tex]

The force of resistance is an opposing force that acts to reduce the speed of the particle. As the particle moves faster, the resistance force increases. The force is proportional to the square of the speed, meaning that if the speed doubles, the force is multiplied by four. The force is also in the same direction as the motion, meaning that it will reduce the speed of the particle.

The equation for the force of resistance is [tex]F=0.3v^2[/tex], where v is the speed of the particle. Therefore, if the speed of the particle is 8 m/s, the force of resistance is [tex]0.3(8)^2,[/tex] or 19.2 N. This means that the force of resistance acting on the particle is 19.2 N.

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There are a total of 32 different possible footwear choices that we can make.

Given, The number of pairs of socks = 4

The number of pairs of shoes = 4

We are to find out the number of possible footwear choices we can make if we can wear any combination of socks and shoes, including mismatched pairs.

So, We can wear any pair of socks with any pair of shoes including a mismatch.

Thus, for each pair of socks, there are 4 possible pairs of shoes.

And for each pair of shoes, there are 4 possible pairs of socks.

Therefore, we can form,

Total number of possible footwear choices = 4 pairs of socks * 4 pairs of shoes * 2 (considering the case of mismatched pairs) = 32 pairs.

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If the price of a new car is $24,000, the total owed for tax is $1,518.50.

Tax is the money that the government collects from people and organizations for providing public services and infrastructure.

The total owed for tax on a $24,000 car purchase is calculated by multiplying the sales tax rate of 6.25% by the price of the car.

The result of this calculation is the sales tax portion of the total amount owed.

The title fee of $18.50 is added to this amount to get the total amount due for tax on the purchase of the car.

Mathematically, the equation for the total owed for tax is:

Total Tax = (6.25% x $24,000) + $18.50

Total Tax = ($1,500 + $18.50)

Total Tax = $1,518.50

Therefore, if the price of a new car is $24,000, the total owed for tax is $1,518.50.

This amount includes the sales tax portion of 6.25% of the car's price and a title fee of $18.50. This tax amount must be paid before the buyer can take possession of the car.

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the probability of winning exactly 2 of the 5 times is 0.2048, or approximately 20.48%

Probability refers to the measure of the likelihood or chance of a particular event occurring. It is represented by a number between 0 and 1, with 0 denoting an impossibility and 1 denoting a certainty.

The formula for the binomial distribution can be used to resolve this issue:

P(X=k) = (n choose k) × pᵇ×(1-p)ⁿ⁻ˣ

where:

The number of trials, n, is five in this instance.

bis the number of successes we want (in this case, b = 2)

p is the probability of success on each trial (in this case, p = 0.20)

So, plugging in the values:

P(X=2) = (5 choose 2) ×0.20² × (1-0.20)⁵⁻²

= 10 × 0.04×0.512

= 0.2048

Therefore, the probability of winning exactly 2 of the 5 times is 0.2048, or approximately 20.48%.

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Answer:

16/5

Step-by-step explanation:

We can see that the new triangle is larger than the original triangle so we know the scale factor is greater than 1.

The scale factor can be found by calculating the ratio of the length of any two corresponding sides. The length of the bottom side of the original triangle is 5. The length of the bottom side of the new triangle is 16. We can calculate the ratio which is 16:5 or 16/5.

Alexis has 12 quarters and 8 dimes.

Let's use d to represent the number of dimes and q to represent the number of quarters. We know that Alexis has a total of 20 coins, so d + q = 20.

We also know that the value of these coins is $3.80. Since dimes are worth $0.10 and quarters are worth $0.25, we can write an equation for the total value in cents:

10d + 25q = 380

To make things easier, let's divide both sides of the equation by 5:

2d + 5q = 76

Now we can use the first equation to solve for d in terms of q:

d + q = 20

d = 20 - q

Substituting this into the second equation gives:

2(20 - q) + 5q = 76

Expanding the parentheses and simplifying, we get:

40 - 2q + 5q = 76

3q = 36

q = 12

Therefore, Alexis has 12 quarters. We can check this by plugging q back into the first equation to find that she has 8 dimes as well:

d + q = 20

d + 12 = 20

d = 8

The total value of 12 quarters and 8 dimes is:

12 quarters x $0.25 per quarter + 8 dimes x $0.10 per dime = $3.00 + $0.80 = $3.80


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Answer:

N = N0 x 2^n

Step-by-step explanation:

The formula for calculating the number of bacteria cells after n generations is N = N0 x 2^n, where N is the total number of cells after n generations, N0 is the initial number of cells, and 2^n represents the number of times the population doubles after n generations. Assuming an initial population of 100 cells, there will be approximately 102,400 cells after 10 generations.

Answer:

The explicit formula for the number of bacteria cells after each generation can be written as:

N = N0 * r^n

Where:

N is the number of bacteria cells after n generations

N0 is the initial number of bacteria cells (at n=0)

r is the growth rate (how many new cells are produced per existing cell)

Assuming that each bacteria cell doubles in number with each generation (i.e. r=2), the formula can be simplified to:

N = N0 * 2^n

To find the number of cells after 10 generations, we can substitute n=10 into the formula:

N = N0 * 2^10

Since we don't have a specific value for N0, we can't find the exact number of cells after 10 generations. However, we can make some assumptions. For example, if we assume that there are initially 100 bacteria cells (N0=100), we can calculate:

N = 100 * 2^10 = 102,400

So, if each bacteria cell doubles in number with each generation and there were initially 100 cells, there will be 102,400 cells after 10 generations.

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As per the combination concept, there are 735,471 ways to choose 16 shows from a set of 24.

To find the number of ways to choose 16 shows from a set of 24, we can use the formula for combinations, which is:

ⁿCₓ = n! / x!(n-x)!

Where n is the total number of objects in the set (in this case, 24), and x is the number of objects we want to choose (in this case, 16). The exclamation mark (!) denotes the factorial function, which means multiplying the number by all positive integers less than itself.

Plugging in the numbers, we get:

²⁴C₁₆ = 24! / 16!(24-16)! = 735471

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Answer:

Yes, I can find the absolute value of -3 2/3.

The absolute value of a number is the distance of the number from zero on the number line. It is always a positive number.

To find the absolute value of -3 2/3, we need to ignore the negative sign and just look at the magnitude of the number.

So, the absolute value of -3 2/3 is:

|-3 2/3| = 3 2/3

Therefore, the absolute value of -3 2/3 is 3 2/3.

Step-by-step explanation:

y=mx+b

let(-4,-1)

x. y

then lets fill it with the formula

-4=-45(-1)+b

-4=45+b

-b=45+4

-b=49

b=-49

Answer:

area = 127[tex]units^{2}[/tex]

perimeter = 46 units

Step-by-step explanation:

count them

Bytaking the quotient between the probabilities we can see that Getting a 17 is three times more likely to get a 18.

If you toss 3 dices with 6 sides each, then the total number of combinations is:

6*6*6 = 216

Now, the outcomes that add up to 18 are:

dice 1, dice 2, dice 3, sum:

6            6         6       , 18

The outcomes that add up to 17 are:

dice 1, dice 2, dice 3, sum:

5           6         6       , 17

6            5         6       , 17

6            6         5       , 17

So the probabilities are:

P(18) = 1/216

P(17) = 3/16

The quotient gives:

P(17)/P(18) = 3

Getting a 17 is 3 times more likely to get a 18.

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Answer:

m = 50 + 6

m= 56

lol easy ques

This one is easy. All you have to do is add 6 to both sides to get the value of m

m-6=50

m=50+6

m=56

The largest area a farmer can fence in is 612.5 m².

The given question is based on finding the largest area a farmer can fence in for building a rectangular enclosure.

The farmer is given with 70 meters of fencing for fencing 3 sides of a rectangular enclosure.

The enclosure's fourth side is the wall of the farmer's barn.

To find the largest area fenced in, the farmer must try to use all 70 meters of fencing to enclose as large an area as possible.

First, let's assume that the rectangular enclosure's width is x.

So, we can find the length by dividing the remaining length of fencing after subtracting the width of the fence from the total length of fencing.

The remaining length of fencing is 70 - 2x.

After calculating the length, the area of the rectangular enclosure can be calculated by multiplying length and width.

The formula for calculating the area of the rectangle is A = l x w.

A = (70 - 2x)x

  = 70x - 2x²

To find the maximum area fenced in, differentiate A with respect to x and equate it to zero.

dA/dx = 70 - 4x

          =0x = 17.5 meters.

After finding x, the length of the enclosure can be calculated by subtracting the width from the total length of fencing.

l = 70 - 2x = 70 - 2(17.5) = 35 meters.

Therefore, the largest area fenced in will be A = l x

w = 35 x 17.5 = 612.5 m².

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