A cocktail has the ratio 2 parts orange juice to 5 parts lemonade.
If you have 7 litres of orange juice to use, how much lemonade do
you need?

The root test for the series ∑ (n / (3n + 9)^(4/n)) is inconclusive, as the limit evaluates to 1. Therefore, we cannot determine whether the series converges or diverges using the root test alone.

To determine whether the series ∑ (n / (3n + 9)^(4/n)) converges or diverges using the root test, we need to evaluate the limit:

lim (n → ∞) |n / (3n + 9)^(4/n)|.

Using the properties of limits, we can rewrite the expression inside the absolute value as:

lim (n → ∞) (n^(1/n)) / (3 + 9/n)^(4/n).

Since the limit involves both exponentials and fractions, it is not immediately apparent whether it converges to a specific value or not. To simplify the expression, we can take the natural logarithm of the limit and apply L'Hôpital's rule:

ln lim (n → ∞) (n^(1/n)) / (3 + 9/n)^(4/n).

Taking the natural logarithm allows us to convert the exponentiation into multiplication, which simplifies the expression. Applying L'Hôpital's rule, we differentiate the numerator and denominator with respect to n:

ln lim (n → ∞) [(1/n^2) * n^(1/n)] / [(4/n^2) * (3 + 9/n)^(4/n - 1)].

Simplifying further, we obtain:

ln lim (n → ∞) [n^(1/n-2) / (3 + 9/n)^(4/n - 1)].

Now, we can evaluate the limit as n approaches infinity. By analyzing the exponents in the numerator and denominator, we see that as n becomes larger, the terms n^(1/n-2) and (3 + 9/n)^(4/n - 1) both tend to 1. Therefore, the limit simplifies to:

ln (1/1) = 0.

Since the natural logarithm of the limit is 0, we can conclude that the original limit is equal to 1.

According to the root test, if the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.

In this case, the limit is equal to 1, which means that the root test is inconclusive. We cannot determine whether the series converges or diverges based on the root test alone. Additional tests or methods would be required to reach a conclusion.

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The effective interest rate of the loan would be

In this case, we are given 4 options:

The formula of compounded interest rate is:

\(A = P (1+\frac{r}{n})^{nt}\)

Where:

A = amount, P = principal amount, r = interest rate, n = number of times interest rate compounded, t = time

Let’s assume the principal amount is $100 for 1 year

Loan L =

\(A = 100 (1+\frac{0.08254}{356})^{356x1}\)

A = $108.603

Loan M =

\(A = 100 (1+\frac{0.08474}{52})^{52x1}\)

A = $108.834

Loan N =

\(A = 100 (1+\frac{0.08533}{12})^{12x1}\)

A = $108.875

Loan O =

\(A = 100 (1+\frac{0.08604}{1})^{1x1}\)

A = $108.604

Therefore, the best effective interest rate for Craig is Loan L with nominal rate of 8.254% compounded daily.

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Answer:

-9/6

Step-by-step explanation:

Answer:

-3/2    -1/2     -1.5

good luck!!!!

Answer:

all real values

Step-by-step explanation:

This is a line and the range of this line is all real values

Answer:

26 i believe 66 - 40 = 26 hope this helps

Answer:

The zeros are at 0 and -2/3

Answer:

234

Step-by-step explanation:

Answer:

25 ft

Step-by-step explanation:

Got it right on EDG2020

The ball will reach the maximum height in a second. Then the maximum height the ball reaches will be 25 ft.

The rate of change of a function with respect to the variable is called differentiation. It can be increasing or decreasing.

Lori tosses a ball into the air. The equation models the height, y, in feet, of the ball after x seconds.

\(\rm y = -16(x - 1)^2 + 25\)

Differentiate the equation with respect to x, then we have

\(\begin{aligned} y' &= 0\\\\\dfrac{d}{dx}[-16(x - 1)^2 + 25] &= 0\\\\ -32(x-1) &= 0\\\\x &= 1 \end{aligned}\)

Then the maximum height the ball reaches will be

\(\rm y = -16(1 - 1)^2 + 25\\\\y = -16*0^2 + 25\\\\y = 25\)

Then the maximum height the ball reaches will be 25 ft.

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Answer:

so the length of square b is 50%.

Answer: slope = -3 and y intercept = 5

Step-by-step explanation: this equation is written in SLOPE INTERCEPT form, which looks like this: y = mx + b. m = slope, b = y intercept. Just substitute values in place of variables and solve

Answer:

the slope is 3 and the y-intercept is −5

Step-by-step explanation:

y=mx + c with m = slope = 3 c = y- intercept =−5

Answer:

25x-5

Step-by-step explanation:

The area of a rhombus with the given diagonals measure is 48 square units.

Given that, the length of two diagonals are 8 units and 12 units.

We know that, Area of rhombus = 1/2 × (product of the lengths of the diagonals)

Here, Area = 1/2 × 8 × 12

= 48 square units

Therefore, the area of a rhombus with the given diagonals measure is 48 square units.

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The number of tickets sold during the 12 games is 83,100 tickets

The college stadium has 6,925 seats for the baseball games this season.

During the entire 12 baseball games, al the seats were filled up and tickets were sold out.

Therefore the number of tickets sold in the entire 12 games can be calculated by multiplying  6,925 by 12

= 6925 × 12

= 83,100

Hence 83,100 tickets were sold in the entire 12 games

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The equation of the perpendicular bisection that passes through the point (2,6) and has a slope -3/2 is  y-6 = -3(x-2)/2 and this can be determined by using the point-slope form of the line.

Given :

Endpoints  -- (8,10) and (-4,2)

The following steps can be used in order to determine the equation of the perpendicular bisector:

Step 1 - First, determine the slope of the line that passes through points (8,10) and (-4,2).

m = 2 - 10 / -4 - 8

m = 2 /3

Step 2 - So, the slope of the perpendicular bisector is given by:

m.m' = -1

m' = -3/2

Step 3 - Now, determine the midpoint of the line that passes through the points (8,10) and (-4,2).

x = (8 - 4)/2 = 2

y = 6

Step 4 - So, the equation of the perpendicular bisection that passes through the point (2,6) and has a slope -3/2 is:

y-6 = -3(x-2)/2

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Answer:

(x - 5)² = 1

Step-by-step explanation:

x² - 10x + 24 = 0

look for a number that can be added to give -10 and be multiplied to give 24 which is -6 and -4,

i.e -6 + (-4) = - 6 - 4 = -10

(-6) × (-4) = 24

x² - 6x - 4x + 24 = 0

factorize

x(x - 6) -4(x - 6) =0

(x - 4)(x - 6) = 0

(x - 4) = 0

x = 4

x

(x - 6) = 0

x = 6

(x - 5)² = 1

take root of both sides

√(x - 5) = √1

x - 5 = 1

x = 1 + 5 = 6

The value of x is either 3 or -5 based on the distance formula.

What is a co-ordinate system?

In pure mathematics, a coordinate system could be a system that uses one or additional numbers, or coordinates, to uniquely confirm the position of the points or different geometric components on a manifold like euclidean space.

Main body:

according to question

Given points A(-1,4) and B(x,7)

Also AB = 5 cm

Formula of distance = \(\sqrt{(y1-y2)^{2}+(x1 -x2)^{2} }\)

here by using points ,

5 = \(\sqrt{(x+1)^{2} +(7-4)^{2} }\)

taking square on both side ,'

25 = \((x+1)^{2} +3^{2}\)

25-9 = (x+1)²

16 = (x+1)²

taking square root on both sides,

x+1= ±4

x = 4-1 = 3                                  or              x = -4-1 = -5

Hence value of x is either 3 or -5.

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Answer:

232,26

Step-by-step explanation:

because 37×3.14×2=232,26

The acceleration of the object moving at a constant speed in a circular path of radius r at a rate of 1 revolution per second is given by option C, 2π^2r^2.

The object experiences centripetal acceleration towards the center of the circle. Although its speed remains constant, the direction of its velocity continuously changes, resulting in acceleration. The magnitude of centripetal acceleration can be calculated using the formula a = (v^2) / r, where v is the linear velocity and r is the radius of the circular path. In this case, the linear velocity is the circumference of the circle (2πr) divided by the time (1 second), squared, and divided by the radius (r), resulting in 2π^2r^2.

The acceleration of an object moving in a circular path is directed toward the center of the circle and is known as centripetal acceleration. In this scenario, the object moves at a constant speed of 1 revolution per second, which means it completes a full circular path in 1 second. The linear velocity can be calculated by dividing the circumference of the circle (2πr) by the time taken to complete one revolution (1 second). Since the speed is constant, there is no tangential acceleration. However, the object experiences centripetal acceleration due to the continuously changing direction of its velocity. The magnitude of the centripetal acceleration is given by the formula a = (v^2) / r, where v is the linear velocity and r is the radius. Plugging in the values, we get a = ((2πr) / 1)^2 / r = 4π^2r^2 / r = 4π^2r, which corresponds to option D, 4π^2r.

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The average change per hour is: -4 F/h

So the temperature decreases by 4 degrees Fahrenheit each hour.

To get the average change in degrees per hour, we need to take the quotient between the change in temperature and the number of hours in which this happens.

We know that initially the temperature is 0F

The final temperature is -12F

and this happens on an interval of 3 hours, then the average change per hour is:

(-12F - 0F)/3h = -12F/3h = -4 F/h

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Answer:

Step-by-step explanation:

Its A

Answer:

\( \pi\: \& \: \sqrt 5\)

Step-by-step explanation:

\(\pi \: and \: \sqrt{5} \: are \: irrational\)

Answer:

Step-by-step explanation:

The ratio of white loaves to brown loaves is 6:4

20 loaves per day divide 10 parts(6+4)

20/10=2  One part is 2 loaves

In one day, the baker makes 6x2=12 white loaves and 4x2=8 brown loaves

240 divided by 8 = 30 days.

one solution of the equation is approximately 1.8925469 radians.

The equation is:

tan(x) + 3 = 0

Subtracting 3 from both sides, we get:

tan(x) = -3

Taking the inverse tangent of both sides, we get:

x = arctan(-3)

However, the tangent function is periodic with period π, which means that there are infinitely many solutions to this equation. In general, the solutions are given by:

x = arctan(-3) + nπ, where n is an arbitrary integer.

Using a calculator to approximate arctan(-3), we get:

arctan(-3) ≈ -1.2490458

Therefore, the general solution to the equation is:

x ≈ -1.2490458 + nπ, where n is an arbitrary integer.

If we substitute n = 1, we get:

x ≈ -1.2490458 + π

Using a calculator to approximate this value, we get:

x ≈ 1.8925469

So one solution of the equation is approximately 1.8925469 radians.

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The if `a^2+b^2=0`, then `a=0` and `b=0`. And, if `a=0` and `b=0`, then `a^2+b^2=0`. Hence, we have proved that `a^2+b^2=0` if and only if `a=0` and `b=0`.

Given that `a^2+b^2=0`, show that `a=0` and `b=0`.To show that `a=0` and `b=0` if and only if `a^2+b^2=0`.Let us assume that `a=0` and `b=0`. Substituting these values in `a^2+b^2=0`, we get `0+0=0`. Therefore, `a^2+b^2=0` is satisfied when `a=0` and `b=0`.Now, let us assume that `a^2+b^2=0`.

This implies that `a^2 = -b^2`. Since `a^2` is always non-negative, `-b^2` must be non-negative as well, which can only happen when `b=0`. Substituting `b=0` in the given equation, we get `a^2+0=0`, which implies that `a=0`.Therefore, if `a^2+b^2=0`, then `a=0` and `b=0`. And, if `a=0` and `b=0`, then `a^2+b^2=0`. Hence, we have proved that `a^2+b^2=0` if and only if `a=0` and `b=0`.

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Exactly 195 more blue marble are there than red marble in the jar

Total marble in a jar = 520

The ratio of blue marble to red marble is 5:3.

To find the number of red marble and blue marble,

Let the number of blue marble is 5x.

Let the number of red marble is 3x.

We add the number of red marble and blue marble. Then we equal the total number equal to 520. Then after solving we determine the value of x. Then we put the value of x in the number of red marble and the number of blue marble. This way we can determine the value of total number of blue and red marble.

Now adding the red and blue marble and equating equal to 520.

5x + 3x = 520

8x = 520

Divide by 8 on both side, we get

x = 65

Now putting the value of x in the number of blue marble and red marble.

The number of blue marble = 5x = 5×65 = 325

The number of blue marble = 2x = 2×65 = 130

Now determining exactly how many more blue marble are there than red marble in the jar.

More Blue Marble = The number of blue marble - The number of blue marble

More Blue Marble = 325 - 130

More Blue Marble = 195

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The expression of 0.785 in the form p/q is 157/200

The ratio is defined as a way of dividing a variable by another variable.

Given the decimal number 0.785, to convert to fraction, we will place divide 785 by 1000 (since the decimal is on thousandth)

0.785 = 785/1000

Express in its simplest form

785/1000 = 157/200

Hence 0.785 written in the form p/q is 157/200 where p = 157 and q = 200.

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The fractional coverage from Company A  is 40 % and the fractional coverage for Company B is 60 %.

Company A pays $340,000 and Company B pays $510,000 of the total loss.

Fractional coverage from Company A = Coverage from Company A / Total Coverage

= $ 500, 000 / ( 500, 000 + 750, 000 )

= $ 500, 000 / $ 1, 250, 000

= 0. 4

= 40%

Fractional coverage from Company B = Coverage from Company B / Total Coverage

= $ 750, 000 / $ 1, 250,000

= 0. 6

= 60 %

Amount paid by Company A = Fractional coverage from Company A x Total Loss

= 0.4 x $ 850,000

= $ 340, 000

Amount paid by Company B = Fractional coverage from Company B x Total Loss

= 0. 6 x  $ 850,000

= $ 510,000

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Answer:

6.75

Step-by-step explanation:

The term 6.75 is within absolute value bars. These bars force any term inside of them to be positive. Since 6.75 is already positive, the bars have no effect.

The term 6.75 is already simplified.