Determine the eccentricity, the type of conic, and the directrix for
r=6/1+2cos theta

Answer:

Answer:2.4

Step-by-step explanation:

The correct answer will be 2.4

Hope this helped

Answer:

x=0.27

Step-by-step explanation:

We are given the equation;

[tex]1.2*10^{4x}-4.2=9.9[/tex]

The first step is to add 4.2 on both sides of the equation;

[tex]1.2*10^{4x}=14.1[/tex]

The next step will be to divide both sides of the equation by 1.2;

[tex]10^{4x}=11.75[/tex]

Next we take natural logs on both sides of the equation;

[tex](4x)ln10=ln11.75[/tex]

Finally, we divide both sides by 4*ln10 and simplify to determine x;

[tex]x=\frac{ln11.75}{4ln10}=0.27[/tex]

Answer:

Step-by-step explanation:

Volume = length * width * height.

Volume = 20 * 10 * 5

The tub can hold 1000 cubic feet of water.

The amount of water tub hold is 28316.8 liters.

Space is used by every three-dimensional object. The volume of this space is used as a measurement. The volume of an object in three-dimensional space is the amount of space it occupies within its boundaries. The capacity of the object is another name for it.

An object's volume can help us figure out how much water is needed to fill it, like how much water is needed to fill a bottle, aquarium, or water tank.

Given shape of bath tub is rectangular prism

which is  20 feet long, 10 feet wide, and 5 feet deep

length = 20 feet

width = 10 feet

height = 5 feet

to find the capacity of tub we need to calculate volume of tub

volume for rectangular prism = l x b x h

V = 20 x 10 x 5 = 1000 cubic feet

and 1 cubic feet = 28.3168 liter

1000 cubic feet = 28316.8 liter

Hence, the capacity of tub is 28316.8 liters.

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Answer:

B

Step-by-step explanation:

C and D dont make since to the problem

so i was left was A and B and i can't realy how i got my answer but i came down B

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this one you can solve easily.

take each equation and put number of miles in that , calculate cost and compare the table.

for example

taking first equation

c = 0.15m + 5.75

noe.from table take m = 10, after putting this we should get cost equal to 5.75 as mentioned in table.

c = 0.15×10 + 5.75 = 7.25 ≠ 5.75

so ignore this and take next equation

c = 0.15m + 4.25

m =10

c = 0.15×10 + 4.25 = 5.75

which matches with our table. putting other values of m from table gives us the right cost mentioned in table . so answer is option B

you can verify that no other equation satisfies the table data.

Answer:

 the answer is B (5 square root 2, 315°), (-5 square root 2, 135°)

Step-by-step explanation:

1) Let A be the point (x, y) = (5, - 5)  

=> x = 5 and y = - 5  

r = √(x² + y²) = √(25 + 25) = √50 = ± 5√2  

tan Θ = - 5/5 = - 1  

=> Θ = (i) 315º or - 45º ; (ii) 135º or - 225  

Hence, the Polar Coordinates of A are (i) (5√2, 315º) (ii) (- 5√2, 135º)

Two pairs of polar coordinates for the point is option b,

Here we assume that A be the point (x, y) = (5, - 5)  

So,

x = 5 and y = - 5  

Now

[tex]r = \sqrt(x^2 + y^2) = \sqrt(25 + 25) = \sqrt50 = \pm 5\sqrt2[/tex]

tan Θ = - 5/5 = - 1  

Now

(i) 315º or - 45º ; (ii) 135º or - 225  

So, the polar coordinates should be (5 square root 2, 315°), (-5 square root 2, 135°)

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Answer:

6.95 feet

Step-by-step explanation:

The shape of the cot is rectangular. A diagonal of the rectangle divides the rectangle into two Congruent Right Angled triangles. The length and width of the rectangle become the legs of the right triangle and the diagonal is the hypotenuse of the right triangle.

In order to find the length of the hypotenuse which is the diagonal in this case we can use the Pythagoras Theorem. According to the theorem, square of hypotenuse is equal to the sum of square of its legs. So for the given case, the formula will be:

[tex]\textrm{(Diagonal)}^{2}=\textrm{(Length)}^{2}+\textrm{(Width)}^{2}\\\\ \textrm{(Diagonal)}^{2}=6^{2}+3.5^{2}\\\\ \textrm{(Diagonal)}^{2}=48.25\\\\ \textrm{(Diagonal)}=\sqrt{48.25}=6.95[/tex]

Thus, rounded of to nearest hundredth foot, the diagonal distance from corner to corner is 6.95 feet

Answer:

C

Step-by-step explanation:

f(x)=x-16 is just a straight line with a slope of one at a y intercept of -16. Therefore, x can hit all numbers in the x axis making the domain x is in the element of all real numbers.

Answer:

all real numbers

Step-by-step explanation:

literally the domain can be anything but the range is limited because of the vertical line check

The Ice-Cream Palace needed to buy 34 more pints of milk to reach their requirement for the daily special, as they only had 14 pints and required 48 pints in total.

To solve this question, we need to convert all the quantities to the same unit. In this case, we'll use pints, since it's the smallest unit given. We know that 1 gallon equals 4 quarts, and 1 quart equals 2 pints. Therefore, 1 gallon equals 8 pints.

Ice-Cream Palace needed 6 gallons of milk, which is 6 * 8 = 48 pints. They already had 6 1/2 quarts of skim milk and 1 pint of whole milk. The 6 1/2 quarts equals 6.5 * 2 = 13 pints.

So, in total, Ice-Cream Palace had 13 pints + 1 pint = 14 pints of milk. This means they still needed to buy 48 - 14 = 34 pints of milk for their daily special.

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The Ice-Cream Palace does not need to buy any more pints of milk because they already have more milk than they need.

To find the number of pints of milk the Ice-Cream Palace still needed to buy, we first need to convert the given quantities to the same unit. Since we need to find the number of pints, we'll convert the 6 gallons and 6 1/2 quarts to pints.

1 gallon = 4 quarts = 8 pints

So, 6 gallons = 6 x 8 = 48 pints

1 quart = 2 pints

So, 6 1/2 quarts = 6 x 2 + 1 x 2 = 12 + 2 =14 pints

1 pint is already given as 1 pint.

Now, to find the number of pints of milk still needed to buy, we subtract the total amount of milk they already have (48 pints + 14 pints + 1 pint) from the amount they need (6 gallons = 48 pints).

48 pints (needed) - (48 pints + 14 pints + 1 pint) (already have) = 48 pints - 63 pints = -15 pints

Since the result is negative, it means they have more milk than they need. So, they don't need to buy any more pints of milk.

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75÷12=6.25

which means Steve needs 7 bags of apples total

Answer:

11.3cm = RC

Step-by-step explanation:

We can find the radius by using Pythagorean theorem and finding RC

a^2 + b^2 = c^2

RQ^2 + QC^2 = RC^2

We are given QC = 8

QC is the perpendicular bisector of PR so QR is 1/2 PR

QR = 1/2 (16) = 8

RQ^2 + QC^2 = RC^2

8^2 + 8^2 = RC^2

64+64 = RC^2

128 = RC^2

Take the square root of each side

sqrt(128) = sqrt(RC^2)

11.3137 = RC

Rounding to the nearest tenth

11.3cm = RC

Final answer:

After solving the given system of equations, the value of y is determined to be 6.4, but this answer does not match any of the provided choices. There appears to be an error in the options given or the original equations.

Explanation:

To find the value of y in the solution to the system of equations, we first need to solve the system. We have two equations:

x + y = 1

2x – 3y = –30

From the first equation, we can express x in terms of y:

x = 1 - y

Next, we substitute this expression for x into the second equation:

2(1 - y) - 3y = -30

Expanding this equation:

2 - 2y - 3y = -30

Combining like terms:

-5y = -32

Divide both sides by -5 to find the value of y:

y = 32 / 5

Now, we can see that none of the options provided in the question (-8, -3, 3, 8) match the correct value we found, which is 6.4. It seems there might be an error in the options provided. However, if the question intended for y to be an integer, then we should re-examine the arithmetic or consider that there might be a typo in the original equations or options given.

Answer:

see explanation

Step-by-step explanation:

Using the trigonometric identities

• 1 + cot² x = csc²x and csc x = [tex]\frac{1}{sinx}[/tex]

• sin²x + cos²x = 1 ⇒ sin²x = 1 - cos²x

Consider the left side

sin²Θ( 1 + cot²Θ )

= sin²Θ × csc²Θ

= sin²Θ × 1 / sin²Θ = 1 = right side ⇔ verified

-----------------------------------------------------------------

Consider the left side

cos²Θ - sin²Θ

= cos²Θ - (1 - cos²Θ)

= cos²Θ - 1 + cos²Θ

= 2cos²Θ - 1 = right side ⇒ verified

Answer:

continuously

Step-by-step explanation:

The more compounding you have, the greater the yield.  Obviously of the three, compounding continuously is the largest.  In math, we used the mathematical constant "e" to compute continuous compounding.

There are 1000 meters in 1 kilometer so you can set up a proportion to find the number of meters. 1000m / 1km = x / 40075 km cross multiply and you get the answer of 40,075,000 meters. To change to scientific notation, move the decimal point next to the 4 (becuase 4.0075 is less than 10) and you get the answer of 4.0075 x 10^7 meters.

Answer: option a.

Step-by-step explanation:

 To solve the given exercise and write the expression as a single logarithm, you must keep on mind the following properties:

[tex]log(a)+log(b)=log(ab)\\m*log(a)=log(a)^m[/tex]

Therefore, by applying the properties shown above, you can rewrite the expression given, as following:

[tex]3logx+4log(x-2)=logx^3+log(x-2)^4=logx^3(x-2)^4[/tex]

Then, the answer is the option a.

Answer:

6

Step-by-step explanation:

48 tiles divided by 8 ft is 6 ft

1) look for parallel lines for example the bottom one is 6 and 3, from here you will know the size is 2x. So what you do is 10 = 2(2x -5)

10 = 4x-10

20 = 4x

x = 5

2) (i cant see, the image is not clear :()

Answer:

4.780 °

144''

Step-by-step explanation:

Given that,

door's level is 12 feet off the ground

the ramp can not have an incline surpassing a ratio of 1:12

An incline surpassing a ratio of 1:12 , means that every 1" of vertical rise requires at least 12" of ramp length

So,

1' rise = 12' length

12' = 12'x12

   = 144''

now we know the length and height of ramp so we can use trigonometry identities to find the angle

sinФ = height / length

sinФ = 12 / 144

sinФ =  sin^-1(12/144)

Ф       =  4.780 °

Answer:

the difference between bobby and the pool deck is 35 feet

Step-by-step explanation:

The difference between the pool deck and Bobby's position is 35 feet.

To find the difference between the pool deck and Bobby's position, we subtract Bobby's depth from the pool deck's height. The pool deck is 15 feet above sea level, while Bobby is diving 50 feet below sea level. We can compare these two numbers by subtracting 50 from 15.

15-50=35

The difference between the pool deck and Bobby's position is 35 feet.

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(a) Wild guess:

[tex]f(x)=\dfrac1{(6+x)^2[/tex]

Recall the power series

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

With some manipulation, we can write

[tex]\displaystyle\frac1{6+x}=\frac16\frac1{1-\left(-\frac x6\right)}=\frac16\sum_{n=0}^\infty\left(-\frac x6\right)^n=\sum_{n=0}^\infty\frac{(-x)^n}{6^{n+1}}[/tex]

Take the derivative and we get

[tex]\displaystyle-\frac1{(6+x)^2}=-\sum_{n=0}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=1}^\infty\frac{n(-x)^{n-1}}{6^{n+1}}[/tex]

[tex]\displaystyle=-\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

so we have

[tex]\displaystyle\frac1{(6+x)^2}=\sum_{n=0}^\infty\frac{(n+1)(-x)^n}{6^{n+2}}[/tex]

By the ratio test, this series converges if

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(n+2)(-x)^{n+1}}{6^{n+3}}}{\frac{(n+1)(-x)^n}{6^{n+2}}}\right|=\left|\frac x6\right|\lim_{n\to\infty}\frac{n+2}{n+1}=\left|\frac x6\right|<1[/tex]

or [tex]|x|<6[/tex], so that the radius of convergence is [tex]R=6[/tex].

(b). If we take the second derivative, we get

[tex]\displaystyle\frac2{(6+x)^3}=\sum_{n=0}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=1}^\infty\frac{n(n+1)(-x)^{n-1}}{6^{n+2}}[/tex]

[tex]\displaystyle=\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

[tex]\displaystyle\frac1{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^n}{6^{n+3}}[/tex]

Apply the ratio test again and we get [tex]R=6[/tex].

(c) Multiply the previous series by [tex]x^2[/tex] and we get

[tex]\displaystyle\frac{x^2}{(6+x)^3}=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-x)^nx^2}{6^{n+3}}[/tex]

[tex]\displaystyle=\frac12\sum_{n=0}^\infty\frac{(n+1)(n+2)(-1)^nx^{n+2}}{6^{n+3}}[/tex]

The ratio test yet again tells us [tex]R=6[/tex].

Answer:

197 million square miles

Step-by-step explanation:

Remark

What the equator is telling you is that the circumference around the earth is approximately 24902 miles. So before you can find the surface area, you need to find the radius of that circumference.

Equations

C = 2*pi*r

Area = 4pi*r^2

Solution

Radius

C = 24902

pi = 3.14

r = ?

24902 = 2 * pi * r

r = 24902 / (2 * pi)

r = 3965.29

==========

Surface Area

Area = 4 * pi * r^2

Area = 4 * 3.14 * 3965.29^2

Area = 4 * 3.14 * 15,723,498

Area = 197 000 000 square miles

Final answer:

The approximate surface area of the Earth, an oblate spheroid, is calculated using the mean radius derived from the average of the equatorial and polar radii, resulting in an estimated surface area of around 197 million square miles.

Explanation:

Calculating Earth's Surface Area

To approximate the surface area of the Earth, we will use the formula for the surface area of a sphere, which is 4πr². Since the Earth is not a perfect sphere but rather an oblate spheroid, we will use the mean radius. The equatorial radius is approximately 3963.296 miles, and the polar radius is 3949.790 miles. Thus, the mean radius would be the average of these two measurements.

First, we calculate the mean radius:

(3963.296 + 3949.790) / 2 = 3956.543 miles

Now, plug the mean radius into the formula for the surface area of a sphere:

Surface Area = 4π(3956.543)² ≈ 197,000,000 square miles

This calculation provides an approximation of the Earth's surface area, taking into account its oblate spheroid shape.

Answer:

option B

x = 1 + 3√6  or x = 1 - 3√6

Step-by-step explanation:

Given in the question an equation,

3(x-1)² - 162 = 0

rearrange the x terms to the left and constant to the right

3(x-1)² = 162

(x-1)² = 162/3

(x-1)² = 54

Take square root on both sides

√(x-1)² = √54

x - 1 = ±3√6

x = ±3√6 + 1

So we have two values for x

x = 3√6 + 1    OR  x = -3√6 + 1

Answer:

b.x = 1+3√6, 1-3√6

Step-by-step explanation:

We have given a quadratic equation.

3(x-1)²-162  = 0

We have to find the solution of given equation by taking the square root of both sides.

Simplifying above equation, we have

3(x-1)² = 162

Dividing above equation by 3, we have

(x-1)² =  54

Taking square root to both sides of equation, we have

x-1 = ±√54

x = ±√54+1

x = ±√(9×6)+1

x = ±3√6+1

x = 1+3√6, 1-3√6  which is the solution of given equation.

By writing the quotient in lowest terms, 3 and 1/8 divided by 1 and 2/3 equals 5 and 5/24.

To divide 3 and 1/8 by 1 and 2/3, we can follow these steps:

Step 1: Convert the mixed numbers to improper fractions.

3 and 1/8 = (3 * 8 + 1) / 8 = 25 / 8

1 and 2/3 = (1 * 3 + 2) / 3 = 5 / 3

Step 2: Invert the divisor (the second fraction) and multiply.

25/8 ÷ 3/5 = 25/8 * 5/3

Step 3: Simplify the fractions if possible.

The numerator of 25/8 and the denominator of 5/3 have a common factor of 5.

25/8 * 5/3 = (5 * 25) / (8 * 3) = 125/24

Step 4: Express the improper fraction as a mixed number (if necessary).

125/24 can be expressed as 5 and 5/24.

Therefore, 3 and 1/8 divided by 1 and 2/3 equals 5 and 5/24.

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The solution to [tex]\(3\dfrac{1}{8} \div 1\dfrac23\) is \(\frac{15}{8}\),[/tex] expressed as a fraction in its simplest form after converting the mixed numbers to improper fractions and performing division.

Let's solve [tex]\(3\dfrac{1}{8} \div 1\dfrac23\)[/tex]

convert the mixed numbers into improper fractions:

[tex]\(3\dfrac{1}{8} = \frac{3 \times 8 + 1}{8} = \frac{24 + 1}{8} = \frac{25}{8}\)[/tex]

[tex]\(1\dfrac23 = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}\)[/tex]

Now, we have:

[tex]\(\frac{25}{8} \div \frac{5}{3}\)[/tex]

To divide by a fraction, we multiply by its reciprocal:

[tex]\(\frac{25}{8} \times \frac{3}{5}\)[/tex]

Multiply the numerators and denominators:

Numerator:[tex]\(25 \times 3 = 75\)[/tex]

Denominator: [tex]\(8 \times 5 = 40\)[/tex]

Therefore, [tex]\(3\dfrac{1}{8} \div 1\dfrac23 = \frac{75}{40}\)[/tex]

Reduce the fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor, which is 5:

[tex]\(\frac{75}{40} = \frac{75 \div 5}{40 \div 5} = \frac{15}{8}\)[/tex]

Hence,[tex]\(3\dfrac{1}{8} \div 1\dfrac23 = \frac{15}{8}\).[/tex]

Answer:

[tex]A1=28.3\ m^{2}[/tex],[tex]A2=78.5\ m^{2}[/tex],[tex]A3=132.7\ m^{2}[/tex]

Step-by-step explanation:

we know that

The area of a circle is equal to

[tex]A=\pi r^{2}[/tex]

Part 1) Find the area of the circle with diameter 6 m

we have

[tex]r=6/2=3\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A1=(3.14)(3)^{2}=28.3\ m^{2}[/tex]

Part 2) Find the area of the circle with diameter 10 m

we have

[tex]r=10/2=5\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A2=(3.14)(5)^{2}=78.5\ m^{2}[/tex]

Part 3) Find the area of the circle with diameter 13 m

we have

[tex]r=13/2=6.5\ m[/tex] -----> the radius is half the diameter

substitute the values

[tex]A3=(3.14)(6.5)^{2}=132.7\ m^{2}[/tex]

The areas of the circles with diameters 6m, 10m, and 13m are approximately A₁=28.3 m², A₂ =78.5 m², and A₃=132.7 m², respectively.

To determine the area of each circle, we use the formula for the area of a circle: A = πr², where A is the area and r is the radius of the circle. First, we need to find the radii of the circles, which are half of their diameters.

Therefore, the areas of the circles are approximately 28.3 m², 78.5 m², and 132.7 m² respectively.

Answer:18.375

Step-by-step explanation:8x/7+9=30

We simplify the equation to the form, which is simple to understand  

8x/7+9=30

Simplifying:

+1.14285714286x+9=30

We move all terms containing x to the left and all other terms to the right.  

+1.14285714286x=+30-9

We simplify left and right side of the equation.  

+1.14285714286x=+21

We divide both sides of the equation by 1.14285714286 to get x.  

x=18.375

Brainliest please :)

Answer: x=18 3/8

Step-by-step explanation:

I got it right on odyssey ware

Answer: option c.

Step-by-step explanation:

To solve the exercise you must apply the formula given in the problem, which is the following:

[tex]y=33,430e^{0.0397t}[/tex]

The problem asks for the projected population of Bakersfield in 2010.

Therefore, keeping on mind that  t is th number of years since 1950, you have that:

[tex]t=2010-1950\\t=60[/tex]

Substitute the value of t into the formula.

Therefore, you obtain:

 [tex]y=33,430e^{0.0397(60)}=361,931[/tex]

Answer:

Vertices at (-7, 5) and (-1, 5).

Foci at (-9, 5) and (1,5).

Step-by-step explanation:

(x + 4)²/9 - (y - 5)²/16 = 1

The standard form for the equation of a hyperbola with centre (h, k) is

(x - h²)/a² - (y - k)²/b² = 1

Your hyperbola opens left/right, because it is of the form x - y.

Comparing terms, we find that

h = -4, k = 5, a = 3, y = 4

In the general equation, the coordinates of the vertices are at (h ± a, k).

Thus, the vertices of your parabola are at (-7, 5) and (-1, 5).

The foci are at a distance c from the centre, with coordinates (h ± c, k), where c² = a² + b².

c² = 9 + 16 = 25, so c = 5.

The coordinates of the foci are (-9, 5) and (1, 5).

The Figure below shows the graph of the hyperbola with its vertices and foci.

Answer:

Step-by-step explanation:

56 ft is the answer

Answer:

Step-by-step explanation:

A graphing program can help a lot in this case. Even without the program, it seems clear from a graph that only the first three points will lie on the same line.

The slope of the line can be found from the first two points as ...

  ∆y/∆x = (7/6 -1/6)/(-2 -1) = (6/6)/-3 = -1/3

Then the point-slope form of the equation can be written as

  y -1/6 = -1/3(x -1)

Multiplying by 6 gives ...

  6y -1 = -2x +2

Adding 2x+1 puts the equation into standard form:

  2x + 6y = 3

Answer:

The answer is (b)

Step-by-step explanation:

* Lets check how to find the inverse of the matrix,

 its dimensions is 2 × 2

* To know if the inverse of the matrix exist find the determinant

- If its not equal 0, then it exist

* How to find the determinant

- It is the difference between the multiplication of

 the diagonals of the matrix

Ex: If the matrix is [tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex]

     its determinant = ad - bc

- After that lets swap the positions of a and d, put negatives

 in front of b and c, and divide everything by the determinant

- The inverse will be [tex]\left[\begin{array}{ccc}\frac{d}{ad-bc} &\frac{-b}{ad-bc}\\\frac{-c}{ad-bc} &\frac{a}{ad-bc}\end{array}\right][/tex]

* Lets do that with our problem

∵ The determinant = (9 × 9) - (-2 × -10) = 81 - 20 = 61

- The determinant ≠ 0, then the inverse is exist

∴ The inverse is [tex]\frac{1}{61}\left[\begin{array}{ccc}9&2\\10&9\end{array}\right][/tex]=

   [tex]\left[\begin{array}{ccc}\frac{9}{61}&\frac{2}{61}\\\frac{10}{61} &\frac{9}{61}\end{array}\right][/tex]

* The answer is (b)