Determine the maximum r-value of the polar equation r = 3 + 3 cos theta

Answer:

\(\textsf{36)} \quad f(x)=\dfrac{7x}{x-12}\)

\(\textsf{37)} \quad g(x)=\dfrac{1}{x+2}\)

\(\textsf{38)} \quad h(x)=\dfrac{x-2}{(x-2)(x+3)}\)

Step-by-step explanation:

Given that the function has a horizontal asymptote at y = 7, the degree of the numerator must be equal to the degree of the denominator.

Also, 7 will be the result of dividing the highest degree term of the numerator by the highest degree term of the denominator.

Given that the function has a vertical asymptote at x = 12, the denominator must equal zero when x = 12.

Therefore, a rational function that meets these conditions is:

\(f(x)=\dfrac{7x}{x-12}\)

Given limits:

\(\displaystyle \lim_{x \to\infty} k(x)=0\)

\(\displaystyle \lim_{x \to-2^+} k(x)=\infty\)

The first limit means that as x approaches positive infinity, the function equals zero. Therefore, the rational function has a horizontal asymptote at y = 0. So the degree of the numerator should be less than the degree of the denominator.

The second limit means that as x approaches -2 from the positive side, the function equals positive infinity.  Therefore, the rational function has a vertical asymptote at x = -2.

Therefore, a rational function that meets these conditions is:

\(g(x)=\dfrac{1}{x+2}\)

A removable discontinuity (hole) exists on the graph of a rational function at any input value that causes both the numerator and denominator of the function to be equal.

Therefore, a rational function that meets these conditions is:

\(h(x)=\dfrac{x-2}{(x-2)(x+3)}\)

For this rational function the hole is at (2, ¹/₅), as when x = 2, the numerator and denominator are both zero.

The functions can be represented as y = 7x/(x - 7), y = 1/(x + 2) and y = (x - 3)/[(x - 3)(x + 1)]

Conditions #1

From the question, we have the following parameters that can be used in our computation:

Vertical asymptote, x = 12

Horizontal asymptote, y = 7

A function with the following features

Vertical asymptote, x = a

Horizontal asymptote, y = b

Can be represented as

y = bx/x - a

So, we have

y = 7x/(x - 12)

Conditions #2

Here, we have the following limits

lim x⇒∝ k(x) = 0 and lim x⇒2⁺ k(x) = ∝

This means that:

Vertical asymptote, x = -2

Using the rule in (1), we have

y = 1/(x + 2)

Condition #3

Here, we have

Function with a hole

This implies that the denominator and the numerator have a common factor

An instance of this is

y = (x - 3)/[(x - 3)(x + 1)]

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Answer: y=54(lb)x+170(lb)

Step-by-step explanation: that's the function for it but I tried. You need to include how much weight the elevator can hold though

The coordinate of point B is (√3, 1 ).

The coordinate of point C is (3√3/2,  3/2).

The coordinate of points of B and C is calculated by applying the following formula as follows;

The given coordinate of point A;

A = ( √3/2, 1/2)

Fromm the given coordinate of point A, we can see that;

The coordinate of point B is calculated as;

point B; x-axis = (√3/2 x 2) = √3

point B; y axis = 1

B = (√3, 1 )

The coordinate of point C is calculated as;

point C; x-axis = (√3/2 x 3) = 3√3/2

point C; y axis = 3/2

C = (3√3/2,  3/2)

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Answer:

-2 I think

Step-by-step explanation:

Answer:

4 ...................

A function that represents a reflection of \(f(x) = \frac{3}{8} (4)^x\) across the y-axis include the following: D. \(g(x) = \frac{3}{8} (4)^{-x}\).

In Mathematics and Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).

This ultimately implies that, a reflection over or across the y-axis or line x = 0 would maintain the same y-coordinate while the sign of the x-coordinate changes from positive to negative or negative to positive.

By applying a reflection over the y-axis to the parent exponential function, we would have the following transformed exponential function:

(x, y)                                              →                 (-x, y).

\(f(x) = \frac{3}{8} (4)^x\)                               →              \(g(x) = \frac{3}{8} (4)^{-x}\)

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Answer:

evaluation : 38.5 38.5   is the answer because you simplify this term

Step-by-step explanation:

Answer: |a|=−4

Step-by-step explanation:Divide each term in −|a|=4by −1 and simplify.

Explanation:

A = set of odd numbers = {1,3,5}

B = set of factors of 12 = {1,2,3,4,6}

A U B = union of set A and set B

A U B = {1,3,5} union {1,2,3,4,6}

A U B = {1,3,5,       1,2,3,4,6}

A U B = {1,2,3,4,5,6}

The set union operation combines two sets into one bigger set. Duplicates are tossed out.

There are 6 elements in the set A U B = {1,2,3,4,5,6} out of 6 faces of the number cube.

Therefore, the probability event  A U B happens is 6/6 = 1 = 100%; i.e. it is guaranteed to happen. Each face of the number cube is either odd, a factor of 12, or both.

Side notes:

There is no proportional relationship between the two types of yarn as the cost per yard is different for each type.

To determine if there is a proportional relationship between the Number of yards and cost of each type of yarn, we calculate the cost per yard for each type of yarn.

Yarn 1 :

Cost of 4 yards = $100

Cost per yard = ($100 ÷ 4) = $25

Hence, Yarn 1 costs $25 per yard

Yarn 2 :

Cost of 5 yards = $150

Cost per yard = ($150 ÷ 5) = $30

Hence, Yarn 2 costs $30 per yard

Since, the cost per yard of each type of yarn is different, then we can conclude that there is no proportional relationship between the two types of yarn.

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Answer:

see below

Step-by-step explanation:

5/6 + ( -1 1/6)

Changing the mixed number

5/6 + - (6/6 +1/6)

5/6 + ( -7/6)

-2/6

Simplify

-1/3

Since these are divided into 6 pieces, go two pieces to the left of zero.

Answer:

Here  x=2−3–√ ................…………………………………….....(1)

Hence  12−3√=12−3√⋅2+3√2+3√=2+3–√ .....................(2)

From (1) and (2) we get  x+1x=4 .......……………………………....(3)

and  x−1x=−23–√ ...............…………………….………....(4)

From (3) and (4) we get  x2−1x2=(x+1x).(x−1x)=−83–√.

Hoped I helped

The graph of y = 3x² + 7x + m will have two x-intercepts if m < 49/12.

The given function is,

y = 3x² + 7x + m

Having two x-intercepts means that the value of y should be 0.

So, we can write,

3x² + 7x + m = 0

Now, this has become a quadratic equation and it will have two zeroes according to the question,

As we know, the condition of quadratic to have two different values of x is,

0 < √(b²-4ac)

Where,

a = 3

b = 7

c =m

Putting all the values,

√(7²-4(3)(m)) > 0

Squaring both sides,

7²-4(3)(m) > 0

49-12m > 0

49/12 > m.

So, the graph will have two intercepts if m < 49/12.

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Solution:

Step-1: Find the volume of the rectangular prism.

Step-2: Find the volume of the small cube.

Step-3: Divide.

Thus, 560 cubes are needed to fill the rectangular prism.

Answer:

A: 24°

B: 22°

C: 134°

Step-by-step explanation:

A triangle will always add up to 180°

A + B + C = 180°

(2x + 4) + (2x + 2) + (13x + 4) = 180

2x + 2x + 13x + 4 + 2 + 4 = 180

17x + 10 = 180

17x = 180 - 10

17x = 170

17x/17 = 170/17

x = 10

A: 2x + 4

B: 2x + 2

C: 13x + 4

A: 24°

2x + 4

2(10) + 4

20 + 4

24°

B: 22°

2x + 2

2(10) + 2

20 + 2

22°

C: 134°

13x + 4

13(10) + 4

130 + 4

134°

Answer:

yes

Step-by-step explanation:

Answer:

i hope this helpes

Step-by-step explanation:

a) A group of 16 pays $40 + ($2 x (16 - 10)) = $40 + $6 = $46.

b) The cost function can be represented as C(x) = 40 + 2(x - 10) for x > 10 and C(x) = 40 for x <= 10, where x is the number of people in the group and C(x) is the cost for the group.

c) The domain of the cost function is all non-negative real numbers less than or equal to 50 (0 <= x <= 50), since the maximum group size is 50. The range of the cost function is all non-negative real numbers greater than or equal to 40 (C(x) >= 40), since the minimum charge for a group is $40.

Answer:

a) $52

b) C(x) = 40 + 2x, if x > 10 and x ≤ 50

C(x) = 40, if x ≤ 10

c) Domain = {x ≥ 1 and x ≤ 50}

Range = { from 40 to 140}

Base on the system, the equation has infinite number of solution.

A system of Equations is when we have two or more linear equations working together.

Therefore, the system of equation can be solved as follows:

p - 6q + 4r = 2

2p + 4q - 8r = 16

p - 2q = 5

Using equation(iii)

p = 5 + 2q

substitute the value of p in equation (i) and equation(ii)

Therefore,

5 + 2q - 6q + 4r = 2

-4q + 4r = -3

2(5 + 2q) + 4q - 8r = 16

10 + 4q + 4q - 8r = 16

8q - 8r = 6

Therefore, combine the new equations to find q and r.

-4q + 4r = -3

8q - 8r = 6

Multiply equation(1) by 2

- 8q + 8r = -6

8q - 8r = 6

add the equations

0 = 0

Therefore, equation have infinite number of solution.

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The limit of the function As x → - ∞, f(x) → 2.

The limit of a function is the value the function tends to as the independent variable tends to a given value.

Given the graph of the function above, to find the limit of the function As x → -∞, f(x) →? We proceed as follows

Looking at the graph, we see that f(x) has a horizontal asymptote at y = 2. Now, we see that As x → -∞, f(x) approaches 2.

So, As x → - ∞, f(x) → 2.

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Volume of the cylinder =  6838.92 cubic inches

We have to given that;

Sam needs to fill a cylindrical container with punch.

And, The container has a radius of 11 inches and a height of 18 inches.

Since, To determine the amount of cat litter the cylinder can hold, we need to calculate the volume of the cylinder.

Volume of a cylinder = πr²h

Where,

r=radius

h= height

Then,

π=22/7 or 3.14 (constant value)

r=11 inches

h=18 inches

Volume of a cylinder is,

= 3.14 × (11)² × 18

= 6838.92 cubic inches

Hence, Volume of the cylinder =  6838.92 cubic inches

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Answer:

Let the dryer cost x.

Then, the washer costs (x+87)

Together, they cost  (x) + (x+87) = 787

        2x+87 = 787

           2x = 700           [subtract 97 from both sides]

            x = 350             [divide both sides by 2]

The dryer cost $350.

Check:   The washer cost $437.  Together, they cost $787  [good to check, right?]

Hope this helps...

Answer:

474

Step-by-step explanation:

Always remeber less then means subtract ok

The statement about the linear equation 3a + 1/3(6a −9) = 1/2(10a − 6) that is true is: "The equation has an infinite number of solutions." (Option B)

A linear equation is defined as an equation with a maximum of one degree. A nonlinear equation is one with a degree greater than or equal to two.

On the graph, a linear equation looks like a straight line. On the graph, a nonlinear equation creates a curve.

To justify the above answer, we state the linear equation above:


3a + 1/3(6a −9) = 1/2(10a − 6) ; Simplified by opening the brackets, we have

3a + 2a − 3 = 5a − 3; collecting like terms on the left side, we have

5a -3 = 5a -3

Because 5a -3 = 5a -3, the expression holds true regardless of what integer or value a represents. Hence, it has an infinite number of solutions.

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Answer:

increases by 30

Step-by-step explanation: its right

The equation of a line perpendicular to the line \(m\) is  \(y=-\frac{3}{2}x+5\)

Slope of two perpendicular lines

Two perpendicular lines' slopes are negative reciprocals of each other.

For example, if a line is perpendicular to a line that has a slope \(m\), then the slope of the line is \(-\frac{1}{m}\).

The equation for the line  \(m\) is \(y=-\frac{2}{3}x+5\)

The slope of a line \(m\) is \(-\frac{2}{3}\)

The slope of a line that is perpendicular to the line m is \(-\frac{1}{\frac{2}{3} }\)

That is \(-\frac{3}{2}\)

Thus, the equation of a line perpendicular to the given line is  \(y=-\frac{3}{2}x+5\)

Hence, option (C) is correct.

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Answer:

a) The 95% confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day is (1.49, 1.57).

d) The nutritionist is 95% confident that the mean amount of time spent eating or drinking per day for any individual is between 1.49 and 1.57 hours.

(c and b can not be concluded from the confidence interval)

Step-by-step explanation:

We have to calculate a 95% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=1.53.

The sample size is N=1082.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

\(s_M=\dfrac{s}{\sqrt{N}}=\dfrac{0.71}{\sqrt{1082}}=\dfrac{0.71}{32.89}=0.022\)

The degrees of freedom for this sample size are:

\(df=n-1=1082-1=1081\)

The t-value for a 95% confidence interval and 1081 degrees of freedom is t=1.962.

The margin of error (MOE) can be calculated as:

\(MOE=t\cdot s_M=1.962 \cdot 0.022=0.042\)

Then, the lower and upper bounds of the confidence interval are:

\(LL=M-t \cdot s_M = 1.53-0.042=1.49\\\\UL=M+t \cdot s_M = 1.53+0.042=1.57\)\(LL=M-t \cdot s_M = 1.53-0.042=1.49\\\\UL=M+t \cdot s_M = 1.53+0.042=1.57\)

The 95% confidence interval for the mean is (1.49, 1.57).

Answer:

d ≥ c

Step-by-step explanation:

Answer:

dogs ≥ cats

Step-by-step explanation:

The area of the square is given as 100 square unit

You should be aware that the square has all its sides equal

The perpendicular from opposite vertices represent distance

The given vertices are

(2,2) and (10,8)

Using the formula for distance between two points

d=√(10-2)²+(8-2)²

d=√8²+6²

d = √64+36

d=√100

This implies that d=10

The area of a square is given as s²

Area = 10²

Atrea = 100 square units

In conclusion, the area of the square is 100 square units

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Translated question:

The vertices of a square ABCD are A(2,2) and B(10,8), Find the area of the square

Son solving the provided question we can say that - the fraction is (2 + 1/x ) / ( 7-1/x ) = 5/13

Any number of equal portions, or fractions, can be used to represent a whole. Fractions in standard English indicate how many units of a certain size there are. 8, 3/4. A whole includes fractions. The ratio of the numerator to the denominator is how numbers are expressed in mathematics. Each of these is an integer in simple fractions. In the numerator or denominator of a complex fraction is a fraction. True fractions have numerators that are less than their denominators. A fraction is a sum that constitutes a portion of a total. By breaking the entire up into smaller bits, you can evaluate it. Half of a full number or item, for instance, is represented as 12.

here, the fraction is

(2 + 1/x ) / ( 7-1/x )

2x+1/7x-1

x = 2

5/13

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Answer:

y-intercept: Put x = 0 in the equation y = 2x - 6 and solve for x. The y-intercept is -6.