For time 0≤t≤10 , water is flowing into a small tub at a rate given by the function F defined by F(t)=arctan(π/2−t/10). For time 5≤t≤10 , water is leaking from the tub at a rate given by the function L defined by L(t)=0. 03(20t−t^2−75). Both F(t) and L(t) are measured in cubic feet per minute, and t is measured in minutes. The volume of water in the tub, in cubic feet, at time t minutes is given by W(t).


(a) At time t=3 , there are 2. 5 cubic feet of water in the tub. Write an equation for the locally linear approximation of W at t=3 , and use it to approximate the volume of water in the tub at time t=3. 5.

(b) Find W′′(8). Using correct units, interpret the meaning of W′′(8) in the context of the problem.

(c) Is there a time t , for 5
(d) The tub is in the shape of a rectangular box that is 0. 5 foot wide, 4 feet long, and 3 feet deep. What is the rate of change of the depth of the water in the tub at time t=6 ?

Step-by-step explanation:

The average rate of change of a function on an interval [x, x + h] is given by the formula:

(F(x + h) - F(x)) / h

For the function F(x) = 6x^2 + 5, the average rate of change on the interval [x, x + h] is:

(F(x + h) - F(x)) / h = (6(x + h)^2 + 5 - (6x^2 + 5)) / h = (6(x^2 + 2xh + h^2) - 6x^2 + 5) / h = (6(2xh + h^2)) / h = (12xh + 6h^2) / h

So the average rate of change of the function on the interval [x, x + h] is (12xh + 6h^2) / h for a real number h.

Part A: We are 99% confident that the mean age of all instructors in College A is between 42.79128 and 51.2088 years old.

Part B: We are 99% confident that the mean age of all instructors in College A is College C by between 37.36368 and 54.63632 years old.

Part C: We are 99% confident that the mean age of all instructors in College B is College C by between 46.24672 and 57.75328 years old.

In this question, they are asking to calculate the 99% confidence intervals for the mean age of instructors in three different colleges (A, B, and C). They want to compare the mean age of instructors in College A to College B, College A to College C, and College B to College C. The numeric answers should be rounded to 5 decimal places.

To calculate these confidence intervals, you need to find the sample mean and standard deviation of the data for each college. Then, you can use the formula for a two-sample t-test to calculate the confidence intervals. The formula is (sample mean A - sample mean B) ± t-critical value * (standard deviation of A/√nA + standard deviation of B/√nB), where nA and nB are the sample sizes for Colleges A and B, respectively.

Part A: We are 99% confident that the mean age of all instructors in College A is College B by between 20.81733 and 23.72867.

To calculate the 99% confidence interval, we must first calculate the mean and standard deviation of the sample data. The mean age of all instructors in College A is 22.273 and the standard deviation is 1.936.

Next, we must use the t-distribution to calculate the 99% confidence interval. The critical value is 2.228. The confidence interval is calculated as follows:

Lower Bound = 22.273 - (2.228 x 1.936) = 20.81733

Upper Bound = 22.273 + (2.228 x 1.936) = 23.72867

Therefore, we are 99% confident that the mean age of all instructors in College A is College B by between 20.81733 and 23.72867.

Part B: We are 99% confident that the mean age of all instructors in College A is College C by between 17.73173 and 25.71367.

To calculate the 99% confidence interval, we must first calculate the mean and standard deviation

Part C: We are 99% confident that the mean age of all instructors in College B is College C by between 46.24672 and 57.75328 years old.

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The probability that a randomly chosen college student exercises in the morning or afternoon is B. 0. 39.

The probability that a college student chosen at random will either exercise in the morning or afternoon can be found by the formula :

= Probability that student exercises in the morning + Probability that student exercises in the afternoon

Probability that student exercises in the morning = 0. 25

Probability that student exercises in the afternoon = 0. 14

The probability would therefore be:

= 0. 25 + 0. 14

= 0. 39

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A. Obtuse Triangle

B. Acute Triangle

C. Right Triangle

D. Right Triangle

What is meant by an obtuse triangle?

Obtuse-angled triangles or obtuse triangles are triangles with any one of its angles being an obtuse angle or greater than 90°. Only 180° is equal to the internal angles of an acute triangle.

Given triangles are A. 15,16,17. B 20,18,7. C 17,144,145. D 24,32,40.

If the sum of squares of least two sides is greater than the square of the largest side then it is an obtuse triangle.

If the sum of squares of least two sides is less than the square of the largest side then it is an acute triangle.

If the sum of squares of least two sides is equal to the square of the largest side then it is a right triangle.

A. side lengths are 15,16,17

[tex]15^{2}+16^{2} =481\\17^{2} =289\\481 > 289[/tex]

Therefore this triangle is an obtuse triangle.

B. side lengths are 7,18,20

[tex]7^{2} +18^{2} =373\\20^{2}=400 \\373 < 400[/tex]

Therefore this triangle is an acute triangle.

C. side lengths are  17,144,145

[tex]17^{2} +144^{2} =21025\\145^{2}=21025 \\21025=21025[/tex]

Therefore this triangle is a right triangle.

D. side lengths are  24,32,40

[tex]24^{2} +32^{2} =1600\\40^{2}=1600 \\1600=1600[/tex]

Therefore this triangle is a right triangle.

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Answer:

Step-by-step explanation:

The slope of a line can be found using the formula:

slope = (y2 - y1) / (x2 - x1)

Where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, we can use points A(4, 5) and B(9, 7):

slope = (7 - 5) / (9 - 4) = 2 / 5

So, the slope of line AB is 2/5.

Answer:

Step-by-step explanation:

The cost of renting a car for "d" days can be represented by the following function:

c(d) = 75d + 30

where "c" is the total cost of the car rental and "d" is the number of days the car is rented for. The first term, 75d, represents the daily rental fee of $75 per day. The second term, 30, represents the flat fee for the full tank of gas that can be returned either full or close to empty.

Answer:

The answer is 100, see Step-by-step explanation:

Step-by-step explanation:

If anyone looks at the verified answer, its wrong and so are the comments under it. In the answer above it says "Sci-fi = 100-24-22-20-14=20" that is correct but the answer is not. 20 percent of 500 is 100 not 72 or 20, I don't know how anyone got those answers.

The probability of Rosh winning the tournament is 50%, which means the probability of one of the other three players winning is 50% divided by 3, or approximately 16.67% each. Therefore, the probability that Craig wins the tournament is 16.67%.

Answer:

51°

Step-by-step explanation:

180°-129°=51°

I hope it helps you

Answer:

Step-by-step explanation:

Here's a step-by-step explanation of the reflections:

Point R is located at (1, 1) on the coordinate plane.

Reflection over the y-axis: This reflection changes the sign of the x-coordinate. In other words, if the point (x, y) is reflected over the y-axis, the reflected point will be ( -x, y). So, for point R at (1, 1), the reflection over the y-axis creates R' at (-1, 1).

Reflection over the x-axis: This reflection changes the sign of the y-coordinate. In other words, if the point (x, y) is reflected over the x-axis, the reflected point will be (x, -y). So, for point R' at (-1, 1), the reflection over the x-axis creates R'' at (-1, -1).

So the final location of R'' is at the ordered pair (-1, -1).

The evaluation of the cubic function represented by the graph can be presented as follows;

a. The equation of the function, S = a·t³ + b·t² + c·t + d, indicates that the equation is a cubic equation

b. i. d = -5

ii. The three equations are;

a + b + c = 8...(1)

8·a + 4·b + 2·c = 4...(2)

27·a + 9·b + 3·c = 0...(3)

iii. a = 2, b = -12, and c = 18

c. The student is expected to be in debt for 2 years, 4 months and 3 days

A cubic function is a degree three polynomial that can be expressed in the form; f(x) = a·x³ + b·x² + c·x + d.

The specified function can be presented as follows;

S = a·t³ + b·t² + c·t + d

a. A feature of the graph that indicates that a cubic equation is the appropriate model, is the largest power of the input time variable t in the function is 3, which is the same as the power of a cubic function.

b. i. The value of d can be found by plugging in t = 0, in the function; S = a·t³ + b·t² + c·t + d

From the table in the question, we get;

At t = 0, S = -5

Therefore;

S = a×0³ + b×0² + c×0 + d = -5

d = -5

The value of d = -5

b. ii. The simultaneous equations can be obtained by plugging in the values of t from the table into the function for S as follows;

S = a·t³ + b·t² + c·t + d

S = a·t³ + b·t² + c·t - 5

When t = 1, we get;

S = a·1³ + b·1² + c·1 - 5 = 3

a + b + c - 5 = 3

a + b + c = 3 + 5 = 8

When t = 2, we get;

S = a·2³ + b·2² + c·2 - 5 = 3

8·a + 4·b + 2·c - 5 = -1

8·a + 4·b + 2·c = -1 + 5 = 4

When t = 3, we get;

S = a·3³ + b·3² + c·3 - 5 = 3

27·a + 9·b + 3·c - 5 = -5

27·a + 9·b + 3·c = -5 + 5 = 0

iii. Solving the system of equations using a graphing calculator, indicates that we get;

c. Plugging in the values into the specified equation, we get;

S = 2·t³ - 12·t² + 18·t - 5

The solution of the above equation can be found as follows;

2·t³ - 12·t² + 18·t - 5 = 0

Based on an online tool, the Newton Raphson method indicates that a solution of the equation is; t ≈ 0.35821, which indicates that a factor of the equation, 2·t³ - 12·t² + 18·t - 5 = 0 is; (x - 0.35821)

(2·t³ - 12·t² + 18·t - 5) ÷ (x - 0.35821) ≈ 2·t² - 11.28356·t + 13.95804

The other solutions are therefore;

t ≈ 1.83, and t ≈ 3.81

The student is expected to be in debt at time t = 0, and from the graph, cross the x-axis out of depth first at time, t ≈ 0.35821 of a year

The student will be in dept again between t ≈ 1.83 and t ≈ 3.81

The total time the student is expected to be in debt is therefore;

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The measure of angle A is given as follows:

<A = 60º.

The three trigonometric ratios are defined as follows:

For the angle A, we have that:

Applying the sine ratio, the angle measure is given as follows:

sin(A) = 13/15

A = arcsin(13/15)

A = 60º.

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Answer:

...here is the answer...

The  total manufacturing costs for each factory are to be the same is M = 3318.52.

Let's denote the number of pairs of jeans produced at the Medney factory as M, and the number of pairs of jeans produced at the Selbourne factory as S.

Total cost of production at the Medney factory = Total cost of production at the Selbourne factory

28000 + 30M = 35200 + 24S

Total number of pairs of jeans produced = 6000

M + S = 6000

We can solve for M and S by using substitution or elimination. Here, we will use substitution:

M + S = 6000 (from equation 2)

M = 6000 - S (subtracting S from both sides)

28000 + 30M = 35200 + 24S (from equation 1)

Substituting M = 6000 - S:

28000 + 30(6000 - S) = 35200 + 24S

180000 - 30S = 35200 + 24S

180000 = 54S + 35200

54S = 144800

S = 2681.48

So the number of pairs of jeans produced at the Selbourne factory is approximately 2681.48. To find the number of pairs of jeans produced at the Medney factory, we can substitute this value of S back into equation 2:

M + S = 6000

M + 2681.48 = 6000

M = 3318.52

So the number of pairs of jeans produced at the Medney factory is approximately 3318.52.

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The value of y from x= 0 to x = 3 is: y = 5(2)⁰ = 5, y = 5(2)¹ = 10, y = 5(2)² = 20, y = 5(2)³ = 40. The most rapid rate of change is: y = 7ˣ. The constant 1.20 is greater than 1 thus, the function is exponential growth.

The formula for an exponential function is f (x) = axe, where x is a variable and an is a constant that serves as the function's base and must be bigger than 0. The transcendental number e, or roughly 2.71828, is the most often used exponential function basis.

The given function is:

y = 5(2)ˣ

The value of y from x= 0 to x = 3 is:

y = 5(2)⁰ = 5

y = 5(2)¹ = 10

y = 5(2)² = 20

y = 5(2)³ = 40

The most rapid rate of change as x increases beyond 0 is:

y = 7ˣ

This is observed as the graph increases rapidly with change in value of x.

3. y = (1.20)ˣ

The constant 1.20 is greater than 1 thus, the function is exponential growth.

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It would take Caro-line 3 hours to sketch 9 cartoon str-ips.

What is a ratio?

A ratio in mathematics demonstrates how many times one number is present in another. For instance, if a dish of the fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six.

We are given that, Caro-line can sketch 6 car-toon str-ips in two hours.

And we are asked that how much time it would require for Caro-line to sketch 9 str-ips

Now let the time required be x

Hence, the ratio becomes

6/2=9/x

x= (9*2)/6

x= 3

Hence, It would take Caroline 3 hours to sketch 9 cartoon str-ips

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The height of each elevator at this time is, -2.5 m, negative means below ground level.

The distance covered by the object is equal to the product of the speed at which the object is moving and time taken for covering the distance.

Distance = Time × Speed

To find how long it would take each elevator to reach ground level, we can set h = 0 in the given expressions for their travel times:

Elevator A:  

= 0.8h + 16

= 0.8(0) + 16

= 16 seconds

Elevator B:  

= -0.8h + 12

= -0.8(0) + 12
= 12 seconds

Therefore, elevator A would take 16 seconds and elevator B would take 12 seconds to reach ground level at this time.

To find at what height the elevators pass each other, we can set their travel times equal to each other and solve for h:

0.8h + 16 = -0.8h + 12

1.6h = -4

h = -2.5

Now, we can substitute value of h in the expression

Elevator A:

= 0.8(-2.5) + 16

= 14 seconds

Elevator B:

= -0.8(-2.5) + 12

= 14 seconds

Therefore, the elevators would pass each other 14 seconds after they both start moving towards each other.

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The complete question:

Elevators A building has two elevators that both go above and below ground.

At a certain time of day, the travel time it takes elevator A to reach height h in meters is 0.8h+16 seconds.

The travel time it takes elevator B to reach height h in meters is -0.8h+12 seconds.

How long would it take each elevator to reach ground level at this time? If the two elevators travel toward one another, at what height do they pass each other? How long would it take?

Answer:

Note that 0.01 is the decimal form of 1%, so 0.01 multiplied by 7 yields 7%. So, if we were to find 1% of 250, which is 2.5, we can multiply the value by 7, which is 17.5.

Therefore, 7% of 250 is 17.5

The degree of the polynomial is; 6.

The leading coefficient of the polynomial is; 1.

The maximum number of real zeroes is; 6.

As evident in the task content; the given polynomial is;

f (x) = 2x⁵ - x³ + x⁶ - 4.

By observation, the term with the highest power of x is; x⁶.

Therefore, the degree is; 6.

The leading coefficient is the coefficient of x⁶ which is; 1.

The maximum number of real zeroes possible is dependent on the degree of the polynomial and is; 6.

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WXYZ must be a square, all the sides must have a length of 5.

The vertices of the quadrilateral are:

W(-1, 1), X(3, 4), Y(6,0), and Z(2, -3)

Distance can be calculated using the formula derived from Pythagoras theorem. In coordinate geometry, the distance formula is:

[tex]\sqrt{[(x_2 -x_1)^2 + (y_2 - y_1)^2]}[/tex]

We have to prove that WXYZ is a square.

Using the distance formula :

[tex]WX = \sqrt{(-1-3)^2+(1-4)^2}\\ \\WX = \sqrt{16 + 9}\\ \\WX = \sqrt{25}\\ \\WX = 5[/tex]

Since WXYZ must be a square, all the sides must have a length of 5.

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The given question is incomplete, complete question is:

Select the correct answer from each drop-down menu.

Given: W(-1, 1), X(3, 4), Y(6,0), and Z(2, -3) are the vertices of quadrilateral WXYZ.

Prove: WXYZ is a square.

Using the distance formula, I found that

…

A. all four sides have a length of 5

B. all four sides have different lengths

C. only 2 sides have the same length

D. all four sides have a length of 10

The coordinates of point C in the given right triangle ABC is (2, 2√3-2)

A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.

Given that, triangle ABC is a 30°-60°-90° triangle, where two vertices are A(-4, -2) and B(4, -2).

Now, AB=4-(-4)=8

AC=AB×sin60°= 8×√3/4√3

In right triangle ACD

AD=AC×cos30°

= 4√3×√3/2

= 6

Xc=6-4=2

Yc=AC×sin30°-2

= 4×3×1/2-2

= 2√3-2

The coordinates of point C(2, 2√3-2)

Therefore, coordinates of C(2, 2√3-2).

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No, it is impossible to construct a triangle PQR such that QR=3 cm, PQ=6 cm, PR=3 cm as the dimensions defy the triangle inequality theorem.

It follows from the triangle inequality theorem that the sum of any two side lengths of a triangle must be greater than the third side length and also, the difference of two side lengths of a triangle must be less than the third side lengths.

However, in this case, since QR + PR = PQ, it can be inferred that the two side lengths cannot possibly be used to construct a triangle.

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Answer:

Step-by-step explanation:

This is a standard quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -5 and c = 6. To solve for x, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Where ± represents the positive and negative square root, and the expression under the square root is called the discriminant (b^2 - 4ac).

So, substituting the values for a, b, and c, we have:

x = (-(-5) ± √((-5)^2 - 4 * 1 * 6)) / 2 * 1

x = (5 ± √(25 - 24)) / 2

x = (5 ± √1) / 2

Since the square root of 1 is 1, the two solutions are:

x = (5 + 1) / 2 = 3

x = (5 - 1) / 2 = 2

So, the solutions to the equation x^2 - 5x + 6 = 0 are x = 2 and x = 3.

26 is the value of a from the given figure.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

We need to find the value of a from the figure.

∠D is 46 degrees.

AD is 28, AT is 37

CosD=Adjacent side/ Hypotenuse

Cos46=a/37

0.695=a/37

a=0.697×37

a=26

Hence, the value of a is 26

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Answer:

2489

Step-by-step explanation:

it would be 30+ 1,000+29+30+400+1,000+ 2489

I really hope this helps

Proverbs 4:6-7

Do not forsake wisdom, and she will protect you; love her, and she will watch over you. The beginning of wisdom is this: Get wisdom. Though it cost all you have, get understanding. Cherish her, and she will exalt you; embrace her, and she will honor you. (just a little encouragement to keep studying hard!)

Answer:

2489

Step-by-step explanation:

Answer:

∠ a = 30° , ∠ b = 90°

Step-by-step explanation:

∠ a and 150° are a linear pair and sum to 180° , that is

∠ a + 150° = 180° ( subtract 150° from both sides )

∠ a = 30°

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

the sum of the 3 angles in a triangle = 180° , that is

∠ b + ∠ a + 60° = 180°

∠ b + 30° + 60° = 180°

∠ b + 90° = 180° ( subtract 90° from both sides )

∠ b = 90°

The equation of inverse of f is f-1(x) = √(x/2)

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

f(x) = 2x^2

Express as an equation

so, we have the following representation

y = 2x^2

Swap x and y

so, we have the following representation

x = 2y^2

Divide by 2

y^2 = x/2

Take the square roots

y = √(x/2)

Hence, the inverse is f-1(x) = √(x/2)

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Answer:yes

Step-by-step explanation: