Suppose a particle moves along a continuous function such that its position is given by f(t)=1/7 t^3-4t-12 where f is the position at time t, then determines the value of r such that f(r)=0.

There are 9 tables with 4 chairs in the restaurant.

Let's establish the variables:

Let x be the number of tables with 4 chairs

Let y be the number of tables with 5 chairs

Let z be the number of tables with 6 chairs

We know that there are a total of 19 tables, therefore:

x + y + z = 19 (equation 1)

We also know that the total number of people that can be accommodated in tables with 4 or 5 chairs is 64, therefore:

4x + 5y = 64 (equation 2)

We want to find the value of x, so we need to eliminate y from the equations above. We can do this by multiplying equation 2 by 4, and then subtracting it from equation 1:

x + y + z - 16x - 20y = 19 - 256

Simplifying:

-15x - 19y = -237

Dividing both sides by -19:

x = 9

Therefore, there are 9 tables with 4 chairs.

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Translated Question: Clear the selection Question 2: In a restaurant for 94 people there are 19 tables that can seat 4.5 or 6 people. If we know that the total number of tables with 4 or 5 chairs can accommodate 64 people, how many tables have 4 chairs?

The number of dolphins that will be there in the year 2100 is 1003, under the condition that in the Gulf of Mexico has been decreasing at a rate of 4% every 10
years.

Here the population of dolphins in the Gulf of Mexico in 2020 was 4,670.The rate of decrease is 4% every 10 years.
Therefore, the population would  decrease by 4% every 10 years.
We want to evaluate the population in 2100, which is  80 years from now, which is eight 10-year periods.
Now, we have to calculate the population after eight 10-year periods.
Each period would decrease the population by 4%.
Hence, the population after eight periods is
4670 × (1 - 0.04)⁸
= 4670 × (0.96)⁸
= 1003

Then, if things progress like this, the population of dolphins in the Gulf of Mexico in the year 2100 will be close to 1000.
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The value of sin(θ) = √15/4

The value of csc(θ) = [tex]\sqrt[4]{\frac{15}{15} }[/tex]

The value of sec(θ) = 4

The value tan(θ) = ±√15

The value of cot(θ) = ±√15/15

We can use the identity sec²(theta) - 1 = tan²(theta) to find the value of tan(theta).

Given sec(θ) = 4, we have:

sec²(θ) = 4² = 16

Then, using the identity:

tan²(θ) = sec²(θ) - 1 = 16 - 1 = 15

Taking the square root of both sides, we get:

tan(θ) = ±√(15)

Since sec(θ) is positive, we know that cos(theta), which is the reciprocal of sec(θ), is also positive. This tells us that θ is in the first or fourth quadrant, where sin(θ) is also positive.

Therefore:

sin(θ) = √(1 - cos²θ))

= √(1 - (1/16))

= √15/16)

= √(15))/4

Using the reciprocal identities, we can find the values of csc(θ) and cot(θ):

csc(θ) = 1/sin(θ)

= 4√(15)

[tex]\sqrt[4]{\frac{15}{15} }[/tex]

cot(θ)

= 1/tan(θ)

= ±√(15)/15

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The probability that cards #1, #2, and #3 are still in order on the top of the stack is 0.03%. Therefore, the correct option is D.

To find the probability, we need to calculate the number of ways in which the first 3 cards can remain in order on the top of the stack, and divide it by the total number of ways the cards can be arranged.

The number of ways in which the first 3 cards can remain in order is 3! (3 factorial), because there are 3 cards and they can be arranged in 3! = 6 ways.

The total number of ways the cards can be arranged is 10!, because there are 10 cards and they can be arranged in 10! = 3,628,800 ways.

So, the probability is:

3! / 10! = 6 / 3,628,800 = 0.000166 = 0.0166%

We can convert it to a percentage by multiplying by 100:

0.0166 x 100 = 1.66%

However, this is the probability that the first 3 cards are in a specific order, not necessarily the original order. Since the question asks for the probability that the original order is maintained, we need to divide the probability by 3!, which gives:

0.0166 / 3! = 0.000277 = 0.0277%

This is closest to answer choice D) 0.03%.

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The model approximates the volume, V, in liters, of air in the lungs during a five-second respiratory cycle using the equation V = -0.0374t + 3 + 0.1525t^2 + 0.1729t.

The given equation represents a mathematical model for estimating the volume of air in the lungs during a respiratory cycle. It is a quadratic equation with three terms: -0.0374t, 0.1525t^2, and 0.1729t.

The term -0.0374t represents the linear decrease in volume over time, indicating that the volume decreases by 0.0374 liters for every second of the respiratory cycle.

The term 0.1525t^2 represents the quadratic relationship between volume and time squared, indicating that the rate of change of volume with respect to time is influenced by the square of time.

The term 0.1729t represents the linear increase in volume over time, indicating that the volume increases by 0.1729 liters for every second of the respiratory cycle.

Overall, this model provides an approximation of the volume of air in the lungs during a five-second respiratory cycle, taking into account both linear and quadratic relationships with time.

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The height of the right circular cone ,candle is approximately 6.8 inches.

To find the height of the third candle, which is a right circular cone with a volume of 16 cubic inches and a radius of 1.5 inches, we will use the formula for the volume of a cone: V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height.

1. Substitute the given values into the formula: 16 = (1/3)π(1.5)^2h
2. Simplify the equation: 16 = (1.5^2 * π * h) / 3
3. Solve for h:
  a. Multiply both sides by 3: 48 = 1.5^2 * π * h
  b. Divide by π: 48/π = 1.5^2 * h
  c. Divide by 1.5^2: (48/π) / 1.5^2 = h
4. Calculate the height, and round to the nearest tenth: h ≈ 6.8 inches

The height of the candle is approximately 6.8 inches.

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Mr. Ali filled 6 of the 400 ml juice bottles and 8 of the 1 liter juice bottles.

First, convert 15.8 liters to milliliters:

15.8 L = 15,800 mL

Let x be the number of 400 ml juice bottles filled, and let y be the number of 1 liter juice bottles filled.

The total amount of juice filled can be represented as:

400 ml/bottle * x + 1000 ml/bottle * y = 15,800 ml

Simplifying, we get:

4x + 10y = 158

We also know that there are 3200 ml of juice left:

400 ml/bottle * (x - 3200/400) + 1000 ml/bottle * y = 0

Simplifying, we get:

x + 2.5y = 28

We now have two equations with two variables. Solving for x and y, we get:

x = 6

y = 8

Therefore, Mr. Ali filled 6 of the 400 ml juice bottles and 8 of the 1 liter juice bottles.

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Step-by-step explanation:

1st i think u divide 20 and 6 and then round tht to the tenth place because we already know our answer is going to be a decimal bc 6 cant go into 20.

Out of the 100 students, 20 like only cola b. Therefore, the probability that a randomly chosen student likes only cola b and does not like cola a is 0.2 or 20%, which is the answer. The answer is option B.

The probability that the student likes cola b and does not like cola a can be calculated as follows

Let's start by finding the number of students who like only cola b. We know that 50 students like cola b in total, but 30 of those students also like cola a. Therefore, the number of students who like only cola b is

50 - 30 = 20

So, out of 100 students, 20 like only cola b. Therefore, the probability that a randomly chosen student likes only cola b is

P(likes only cola b) = 20/100 = 0.2

Therefore, the probability that the student likes cola b and does not like cola a is 0.2.

The answer is option B: 0.2.

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--The given question is incomplete, the complete question is given

" In a class of 100 students, 60 students likes cola a, 50 students likes cola b  and 30 students likes both.  

From class a student is randomly chosen from the group.

what is the probability that the student likes cola b and does not like cola a?

0.1

0.2

0.3

0.4 "--

it would take approximately 2,749 hours for a rocket traveling at 3.35×10⁴ kilometers per hour to travel from Earth to Mercury.

what is approximately ?

Approximately means "about" or "roughly". It is used to indicate that a number or value is not exact, but rather an estimate or approximation. When a value is given as approximately a certain number

In the given question,

To calculate the time it would take a rocket traveling at 3.35×10⁴ kilometers per hour to travel from Earth to Mercury, we need to divide the distance between Earth and Mercury by the speed of the rocket:

Time = Distance / Speed

Distance = 9.21×10⁷kilometers

Speed = 3.35×10⁴ kilometers per hour

Time = 9.21×10⁷ km / (3.35×10⁴ km/h)

Time = 2,748.66 hours

Rounding this value to the nearest whole number of hours gives:

Time = 2,749 hours

Therefore, it would take approximately 2,749 hours for a rocket traveling at 3.35×10⁴ kilometers per hour to travel from Earth to Mercury.

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Answer:

< 3 = 3x + 105°

Step-by-step explanation:

There is remot angle theory which is the exterior angle is congrent to the other non adjecent angle in triangle.

so <1 + <EDF = <3

(3x + 15 ) ° + 90° = <3

3x°+ 105° = <3

< 3 = 3x + 105° .... so the measur of angle 3 interms of x is 3x + 105°

Answer:

C. Triangles A and C.

Step-by-step explanation:

In triangles A and C, the ratios of corresponding sides are equal, and the corresponding angles are congruent.

This situation represents exponential decay because the depth of the lake decreases over time.

Exponential decay is a mathematical term used to describe the process of decreasing over time at a constant rate where the amount decreases by a constant percentage at regular intervals. It is a type of exponential function where the base is less than 1.

In other words, the quantity is decreasing by a fixed percentage at regular intervals.

The rate of decay, r, is equal to 0.98 because the depth decreases by 2% per year.

So the depth of the lake each year is 0.98 times the depth in the previous year. It will take between 5 and 6 years for the depth of the lake to reach 26.7 meters.

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To find the point representing the product p of x and y on the number line, locate x and y on the number line and find their product. To find the point representing the quotient q of x and y on the number line, locate x and the reciprocal of y (1/y) on the number line, and find their product.

If p is the product of x and y, the point on the number line that represents p can be found by locating x and y on the number line and then finding their product. For example, if x is at 2 and y is at -3, then their product p is (-6) and is located at the point on the number line that corresponds to -6 which corresponds to point P.

To find the point representing q on the number line if q is the quotient of x and y, we can use the fact that the quotient is the same as the product of x and the reciprocal of y.

In other words, q = x / y = x * (1/y). Therefore, if we locate x and 1/y on the number line, their product gives us the point representing q. For example, if x is at 4 and y is at -2, then 1/y is -1/2 and is located at -2 on the number line. The product of 4 and -1/2 is -2, which corresponds to the point on the number line that represents q.

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We denote triangle ABC, angle A measures 90°, angle B measures 30° and angle C measures 60°.

We apply the cosine of 30 degrees, becuase m(∡A) = 90° and find the hypotenuse of triangle ABC:

cos = (adjacent side) / (hypotenuse)

⇔ cos B = AB/BC ⇔

⇔ cos 30° = 8√3/v ⇔

⇔ √3/2 = 8√3/v ⇔

⇔ √3 • v = 2 • 8√3 ⇔

⇔ v√3 = 16√3 ⇔

⇔ v = 16√3 ÷ √3 ⇔

⇔ v = 16 millimeters

Hope that helps! Good luck! :)

No journal entry is required for the land since it is not depreciated.

To record depreciation expense for buildings and equipment for the year ended December 31, 2021, we need to calculate the depreciation amounts for each asset based on their respective methods.

For the building, we will use the double-declining-balance method. The annual depreciation expense is calculated as (2 / 20) x $442,000 = $44,200. Since depreciation has already been recorded for 2019 and 2020, the accumulated depreciation balance for the building as of December 31, 2020 is $83,980. Therefore, the 2021 depreciation expense for the building is $44,200 - $83,980 = $(-39,780). We record this as follows:

Building Depreciation Expense: $39,780
 Accumulated Depreciation - Building: $39,780

For the equipment, we will use the straight-line method. The annual depreciation expense is calculated as ($245,000 - $13,000) / 10 = $23,200. Since depreciation has already been recorded for 2019 and 2020, the accumulated depreciation balance for the equipment as of December 31, 2020 is $46,400. Therefore, the 2021 depreciation expense for the equipment is $23,200, and we record it as follows:

Equipment Depreciation Expense: $23,200
 Accumulated Depreciation - Equipment: $23,200

No journal entry is required for the land since it is not depreciated.
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Considering the definition of probability, the probability that a randomly selected attraction at this amusement park would be a drink stand is 1/10.

Probability establishes a relationship between the number of favorable events and the total number of possible events.

The probability of any event A is defined as the ratio between the number of favorable cases (number of cases in which event A may or may not occur) and the total number of possible cases:

P(A)= number of favorable cases÷ number of possible cases

In this case, you know:

Replacing in the definition of probability:

P(A)= 2÷ 20

Solving:

P(A)= 1/10

Finally, the probability in this case is 1/10.

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The given data is an example of a nonlinear function. Therefore, the answer is A.

The given data consists of two sets of numbers, X and Y, where each value of X has a corresponding value of Y. We can observe that the points do not lie on a straight line. Instead, the plotted points form a curved shape, which indicates that the relationship between X and Y is not a linear function.

A linear function is a function where the relationship between the input variable (X) and output variable (Y) is a straight line. In this case, we can observe that as the value of X increases, the value of Y increases at an increasing rate, which means the relationship between X and Y is not linear.

In particular, the relationship between X and Y is a quadratic function since the values of Y are the squares of the corresponding values of X.

Therefore, the answer is A.

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To use the Evaluation Theorem to compute the definite integrals, follow these steps:

Step 1: Identify the function and the interval
In this case, we have three separate integrals to evaluate:
(a) ∫(e^3 - e) dx
(b) ∫0 dx
(c) ∫295/6 dx

Step 2: Find the antiderivative of the function
(a) The antiderivative of (e^3 - e) is (e^3x/3 - ex) + C.
(b) The antiderivative of 0 is simply C, where C is the constant of integration.
(c) The antiderivative of 295/6 is (295/6)x + C.

Step 3: Evaluate the antiderivative at the given interval
(a) Since no specific interval is given, we cannot evaluate the integral using the Evaluation Theorem.
(b) Since no specific interval is given, we cannot evaluate the integral using the Evaluation Theorem.
(c) Since no specific interval is given, we cannot evaluate the integral using the Evaluation Theorem.

What are definite Intregal's: Definite integral is the area under a curve between two fixed limits.we can say that the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. Unfortunately, without specific intervals, we cannot use the Evaluation Theorem to compute the definite integrals. Please provide the intervals for each integral, and we can help you compute them.

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The equation to find the number of hours, t, until Patty and Carol are at the same location is: 70t + 50t = 300.


1. Patty and Carol start driving at the same time, towards each other on the same highway.
2. The distance between their cities is 300 miles.
3. Patty drives at an average speed of 70 mph, so in t hours she covers 70t miles.
4. Carol drives at an average speed of 50 mph, so in t hours she covers 50t miles.
5. As they drive towards each other, the sum of the distances they cover should equal the total distance between their cities.
6. Therefore, combining the distances covered by Patty and Carol, we get: 70t (Patty's distance) + 50t (Carol's distance) = 300 (total distance).

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To test the claim that the outcomes of rolling the modified die are not equally likely, we can use a chi-square goodness-of-fit test. We will use a significance level of 0.05.

The null hypothesis is that the outcomes are equally likely. The alternative hypothesis is that the outcomes are not equally likely.

First, we need to calculate the expected frequencies assuming that the outcomes are equally likely.

Since the die has six sides, each outcome has a probability of 1/6. Therefore, the expected frequency for each outcome is 200/6 = 33.33.

To calculate the test statistic [tex]x^2[/tex], we can use the formula:

[tex]x^2 = Σ (observed frequency - expected frequency)^2 / expected frequency[/tex]

where Σ is the sum over all outcomes.

Using the observed and expected frequencies given in the problem, we get:

[tex]x^2 = (27 - 33.33)^2 / 33.33 + (31 - 33.33)^2 / 33.33 + (42 - 33.33)^2 / 33.33 + (40 - 33.33)^2 / 33.33 + (28 - 33.33)^2 / 33.33 + (32 - 33.33)^2 / 33.33[/tex]

[tex]x^2 = 3.02[/tex]

The degrees of freedom for this test is 6 - 1 = 5 (since there are 6 sides on the die).

Using a chi-square distribution table (or calculator), we can find the critical value for a significance level of 0.05 and 5 degrees of freedom to be 11.070.

Since the test statistic x^2 = 3.02 is less than the critical value of 11.070, we fail to reject the null hypothesis.

Therefore, we do not have enough evidence to conclude that the outcomes of rolling the modified die are not equally likely.

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_____________________________

_____________________________

Hose will take about 63.7 minutes to fill the pool and the the depth of the water is increasing at a rate of about 0.0079 feet per minute.

First, let's find the volume of the pool. The pool is in the shape of a cylinder with a height of 4 feet and a diameter of 16 feet, so its radius is half of the diameter, or 8 feet. The volume of a cylinder is given by

V = πr^2h

Plugging in the values, we get

V = π(8 ft)^2(4 ft)

V = 256π cubic feet

Next, let's convert the flow rate to cubic feet per minute. One gallon is equal to 0.1337 cubic feet, so the flow rate is

30 gallons/min x 0.1337 ft^3/gallon = 4.011 ft^3/min

Finally, we can use the formula

time = volume/flow rate

Plugging in the values, we get

time = 256π ft^3 / 4.011 ft^3/min

time ≈ 63.7 minutes

So it will take about 63.7 minutes to fill the pool.

Let's use the formula for the volume of a cylinder again to relate the volume of the water in the pool to its depth

V = πr^2h

We can solve this formula for h

h = V/πr^2

Taking the derivative of both sides with respect to time, we get

dh/dt = d/dt (V/πr^2)

The radius of the pool does not change, so we can treat it as a constant and take it out of the derivative

dh/dt = (1/πr^2) dV/dt

We know the flow rate is constant at 4.011 cubic feet per minute, so the rate of change of the volume of water in the pool is

dV/dt = 4.011

Plugging in the values, we get

dh/dt = (1/π(8 ft)^2) (4.011 ft^3/min)

dh/dt ≈ 0.0079 ft/min

So the depth of the water is increasing at a rate of about 0.0079 feet per minute.

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This function is not differentiable at all points in [tex]R^2[/tex]. To see this, consider the points on the x-axis, where y = 0. At these points, the function is not differentiable because it has a sharp corner.

To show that the function f(x, y) = |y| is differentiable at (0, 0), we need to show that there exists a linear transformation L such that:

[tex]lim (h,k) - > (0,0) [f(0+h,0+k) - f(0,0) - L(h,k)] / \sqrt{(h^2 + k^2)} = 0[/tex]

where f(0,0) = 0 since |0| = 0.

We have:

f(0+h,0+k) - f(0,0) = |k|

Now we need to find L(h,k), which is a linear transformation of (h,k) that approximates f(0+h,0+k) - f(0,0) near (0,0). We can take:

L(h,k) = 0

Since L is a constant function, it is a linear transformation. Also, we have:

f(0+h,0+k) - f(0,0) - L(h,k) = |k|

So we have:

[tex]lim (h,k) - > (0,0) [f(0+h,0+k) - f(0,0) - L(h,k)] / \sqrt{(h^2 + k^2) } = lim (h,k) - > (0,0) |k| / \sqrt{(h^2 + k^2)}[/tex]

Using the squeeze theorem, we can show that this limit is equal to 0, since[tex]|k| < = \sqrt{(h^2 + k^2)}[/tex] for all (h,k) and[tex]lim (h,k) - > (0,0)\sqrt{ (h^2 + k^2) } = 0.[/tex]

Therefore, f(x, y) = |y| is differentiable at (0,0).

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Answer:

-7 feet

Step-by-step explanation:

-3-4 = -7, so it is -7 feet

It would take 1.53 years for the price of pizza to triple with a growth rate of 1.05.

To find the number of years it takes for the price of pizza to triple with a growth rate of 1.05, we need to use the formula for exponential growth:

A = P(1 + r)^t

Where:

A = final amount (triple the original price, or 3*$8 = $24)

P = initial amount ($8)

r = growth rate (1.05)

t = time in years

Substituting the values into the formula, we get:

$24 = $8(1 + 1.05)^t

Simplifying:

3 = (1 + 1.05)^t

Taking the logarithm of both sides with base 10:

log(3) = t*log(1 + 1.05)

t = log(3) / log(1 + 1.05)

Using a calculator, we get:

t ≈ 1.53

Therefore, it would take approximately 1.53 years for the price of pizza to triple with a growth rate of 1.05.

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To the nearest integer, the sum of the first 36 terms of the given series is 3,402.

Given series is 7, 12, 17,,,. we have to find the sum of the first 36 terms of the series.

We can observe that the series is an arithmetic sequence.

Here, [tex]a_{1}=7[/tex]

d = 12 - 7 = 5

and n = 36

We know that the formula for the nth term of A.P. is

[tex]a_{n}=a_{1}+(n-1)d[/tex]

[tex]a_{36}=7+(36-1)5[/tex]

= 7 + 35*5

= 7 + 175

[tex]a_{36}=182[/tex]

We know the sum of n terms in A.P. is

[tex]S_{n}=\frac{n}{2}(a_{n}+a_{1})[/tex]

[tex]S_{36}=\frac{36}{2}(7+182)[/tex]

= 18(189)

= 3,402

Hence, the sum of the first 36 terms of the given series is 3,402.

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The logarithmic expression log7 18 is equivalent to log 18 / log 7 using the change-base formula.

The change-base formula states that the logarithm of a number to a certain base can be converted to the logarithm of the same number to a different base by dividing the logarithm of the number to the first base by the logarithm of the number to the second base.

In this case, we want to convert log7 18 to a logarithm with base 10. Therefore, using the change-base formula, we can write:

log7 18 = log 18 / log 7

Using a calculator, we can evaluate the right-hand side of the equation to get:

log7 18 = 1.2553 / 0.8451

log7 18 = 1.4845 (rounded to four decimal places)

Therefore, the logarithmic expression log7 18 is equivalent to log 18 / log 7, which is approximately equal to 1.4845.

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Answer:

[tex]A = 68.75 \text{ square inches}[/tex]

Step-by-step explanation:

First, we need to identify the trapezoid's dimensions:

base 1 = 16

base 2 = 11.5

height = 5

Then, we can plug these values into the trapezoid area formula:

[tex]A = \dfrac{b_1+b_2}{2} \cdot h[/tex]

[tex]A = \dfrac{16 + 11.5}{2} \cdot 5[/tex]

[tex]A = \dfrac{27.5}{2} \cdot 5[/tex]

[tex]A = \dfrac{137.5}{2}[/tex]

[tex]\boxed{A = 68.75 \text{ square inches}}[/tex]