A tank holds 50 gal of water, which drains from a leak at the bottom, causing the tank to empty in 20 min. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after t minutes as V(t)=50(1−t20)20≤t≤20 (a) Find V(0) and V(20). (b) What do your answers to part (a) represent? (c) Make a table of values of V(t) for t = 0, 5,10, 15, 20. (d) Find the net change in the volume V as t changes from 0 min to 20 min.

Answer:

  x = 1/3 . . . . math facts or logarithms are involved; take your pick

Step-by-step explanation:

When you are solving for a variable that is in an exponent, logarithms are often useful. Taking the log of both sides of this equation, you have ...

  log(8^x) = log(2)

Using the rules of logarithms, that is ...

  x·log(8) = log(2)

  x = log(2)/log(8) . . . . . divide by the coefficient of x

You can find the value of this on your calculator, and it will tell you the value is  0.333333333333 or as many digits as your calculator displays. That is a clue that the exact answer is probably 1/3.

__

You should recognize that 8 = 2·2·2 = 2^3, so log(8) = 3log(2) and the above solution becomes ...

  x = log(2)/(3log(2)) = 1/3

__

Recognizing that 8 = 2^3, you can make that substitution into the original equation to get ...

  (2^3)^x = 2

  2^(3x) = 2^1

  3x = 1 . . . . . . . matching exponents; equivalent to taking logs base 2

  x = 1/3 . . . . . .  divide by 3

__

All of the above using 2 as a base of exponents is just dancing around the fact that you already know the math fact ...

  8^x = 2 = 8^(1/3)

  x = 1/3 . . . . . equating exponents; equivalent to taking logs base 8

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Answer:

Lol it either means that they don't like a clown such as comic performer who employs slapstick or similar types of physical comedy, often in a mime style.

or a person that's been fooled and has been called a clown.

Answer: The statement is true (see Step-by-step explanation).

Step-by-step explanation:

The slope of the tangent line for all point of the curve is determine by derive the expression abovementioned in the statement:

[tex]y' = 12 \cdot x^{2} + 7[/tex]

The previous expression represents a parabola, whose output will be positive for all [tex]x[/tex] due to the symmetry of [tex]x^{2}[/tex] and the positive coefficients of the polynomial. If the function is evaluated at [tex]x = 0[/tex], where the minimum occurs, it is evident that the smallest value is [tex]y' = 7[/tex] . Therefore, the inexistence of any tangent line with slope 2 associated with that curve is true.

Answer:

11.4 units

Step-by-step explanation:

diagonal² = 6² + 6² - 2(6)(6)cos(145)

diagonal² = 130.9789471888

diagonal = 11.444603409

Answer: there is nothing shown below

Step-by-step explanation:

Answer:

compare your total monthly incoming with your total monthly outgoing. How balanced is your budget at this point? Remember that you estimated some of your expenditures. You can't know for sure until you actually track your expenses for at least a month and have real numbers to work with.

Step-by-step explanation: Basically it helps you decide how you want to manage your budget

Option C is the best choice to make the best financial decision to make.

An upfront cost is an initial sum of money owed in a purchase or business venture.

The monthly payment is the amount paid per month to pay off the loan in the time period of the loan.

this is because it's upfront front cost is less comparable to other two.

This is because it requires high up-front cost as compared to other and there will be problem, mileage restriction.

For third option, the person will get the car outright once the loan is paid and it's cheaper.

option C is the best financial decision to make.

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Answer:

5 cups

Step-by-step explanation:

2 1/4 + 2 3/4= 5 cups of flour

Answer:

The answer is 5

Step-by-step explanation:

A trick you can use here is by understanding what the fraction is indicating, the number 4 indicates a full cup reached and the first 1/4 is 1 part out of 4 parts to be reached. the second fraction is 3 out of 4 and if you add the one on the other fraction then you get 3+1==4 which majes a full cup

and once you ad the other 4 cups you get 5

Answer:

B

Step-by-step explanation:

the reason why b is the answer is because the perimeter is the outside of the rectangle and you are just ummm yea its b. another way is to add all the sides up and you would get your answer.

Answer:

(0.353, -48)

Step-by-step explanation:

dy/dx = 9x² + 28x - 11 = 0

Using the quadratic formula:

x = -3.46 and 0.353

d²y/dx² = 18x + 28

Minima when d²y/dx² is positive

x = 0.35284 or (-14+sqrt(295))/9

y = 3x³ + 14x² - 11x - 46

y = -48.00651351

The relative minimum of a function is (1/2, -65/4), derive the function, set it to zero, solve for x, and find the corresponding y-value.

To find the relative minimum of the function y = 3x^3 + 14x^2 - 11x - 46, we need to first take the derivative of the function, set it equal to zero, and then solve for the critical point.

This will give us the x-coordinate of the minimum. Next, plug this x-value back into the original function to find the corresponding y-coordinate.

The relative minimum of the function y = 3x^3 + 14x^2 - 11x - 46 is at the point (1/2, -65/4).

Answer:

Austin should score between 85 and 100 in the final exam to have an average between 85 and 95, since reaching 96 is mathematically impossible.

Step-by-step explanation:

The problem has to be solved in two parts, the first to obtain an average of 85 and the second to obtain an average of 96.

In the first part with an average of 85 would be as follows.

Let x be the final exam grade.

Weighted average = (85 + 2 * X) / 3

85 = 85/3 + (2/3) * X

Rearranging

X = (3/2) * (85- (85/3))

Resolving

X = 85, therefore you must take 85 in the final test to get an average of 85.

The second part with an average of 96:

96 = 85/3 + (2/3) * X

Rearranging

X = (3/2) * (96- (85/3))

Resolving

X = 101.5, therefore you must take 101.5 in the final test to get an average of 96, therefore it is impossible to have an average of 96, because the highest score is up to 100.

Taking 100 your average would be:

Weighted average = (85 + 2 * 100) / 3 = 95

Then Austin should score between 85 and 100 in the final exam to have an average between 85 and 95, since reaching 96 is mathematically impossible.

Austin should score between [tex]85[/tex] and [tex]100[/tex] in the final exam to have an average between [tex]85[/tex]  and [tex]95[/tex].

Let us consider that  [tex]X[/tex] be the final exam grade.

So that, Weighted average [tex]= \frac{85 + 2 * X}{3}[/tex]

               [tex]85 =\frac{85}{3} + \frac{2}{3} X\\\\\frac{2}{3} X=85-\frac{85}{3} \\\\X=85[/tex]

Thus, you must take 85 in the final test to get an average of 85.

The second part with an average of 96:

[tex]96 = \frac{85}{3} + \frac{2}{3} X\\\\\frac{2}{3}X=96-\frac{85}{3}\\ \\ X=101.5[/tex]

Thus,  it is impossible to have an average of 96, because the highest score is up to 100.

Taking 100 your average would be:

Weighted average [tex]=\frac{85 + 2 * 100}{3} =95[/tex]

Hence,  Austin should score between 85 and 100 in the final exam to have an average between 85 and 95.

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The student made a mistake in setting their proportion, which resulted in the wrong calculation. The correct proportion should have been 4/6 = x/24, where x is the number of oranges needed. Solving this proportion gives us 16 oranges needed for 24 fluid ounces of juice.

The error in the student's work lies in the improper establishment of the proportion. The initial ratio provided was 4 oranges to 6 fluid ounces.

The proportion, therefore, should have been StartFraction 4 over 6 EndFraction = StartFraction x over 24 EndFraction, where 'x' is the number of oranges required for 24 ounces of juice. The student incorrectly used '24' as the numerator rather than the denominator, causing an error in calculation.

To correct this, we can cross-multiply to get 4 * 24 = 6 * x, leading to x = 16 oranges. Therefore, 16 oranges are needed for 24 fluid ounces of juice, which is the correct solution.

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Sample response: The second ratio in the proportion is set up as ounces over oranges. The units should be in the same place in the proportion as the first ratio

Step-by-step explanation:

Answer: They need 18 teachers more.

Step-by-step explanation:

Hi, to solve this problem, first, we have to divide the expected number of children (14,700) by 70.

We obtain that they are expecting 210 groups of 70 children each. Since the school district needs 3 teachers for every 70 children, we have to multiply the number of groups (210) by the number of teachers per group (3).

We obtained the total number of teachers needed for 14,700 children.

Since at this time the district has 612 teachers, to find the teachers missing we have to subtract the actual number of teachers (612) to the number of teachers needed (630).

They need 18 teachers more.

Answer:

a. 45

b. 176

c. 252

Step-by-step explanation:

First take into account the concept of combination and permutation:

In the permutation the order is important and it is signed as follows:

P (n, r) = n! / (n - r)!

In the combination the order is NOT important and is signed as follows:

C (n, r) = n! / r! (n - r)!

Now, to start with part a, which corresponds to a combination because the order here is not important. Thus

 n = 10

r = 2

C (10, 2) = 10! / 2! * (10-2)! = 10! / (2! * 8!) = 45

There are 45 possible scenarios.

Part b, would also be a combination, defined as follows

n = 10

r <= 3

Therefore, several cases must be made:

C (10, 0) = 10! / 0! * (10-0)! = 10! / (0! * 10!) = 1

C (10, 1) = 10! / 1! * (10-1)! = 10! / (1! * 9!) = 10

C (10, 2) = 10! / 2! * (10-2)! = 10! / (2! * 8!) = 45

C (10, 3) = 10! / 3! * (10-3)! = 10! / (2! * 7!) = 120

The sum of all these scenarios would give us the number of possible total scenarios:

1 + 10 + 45 + 120 = 176 possible total scenarios.

part c, also corresponds to a combination, and to be equal it must be divided by two since the coin is thrown 10 times, it would be 10/2 = 5, that is our r = 5

Knowing this, the combination formula is applied:

C (10, 5) = 10! / 5! * (10-5)! = 10! / (2! * 5!) = 252

252 possible scenarios to be the same amount of heads and tails.

Answer:

3 pls pass brainlist

Step-by-step explanation:

because 3*4 is 12

Complete question:

Two bicycle riders start from the same point and ride in opposite directions. Rider x moves twice as fast as rider y and in three hours they are 72 miles apart. Find the rate (speed) of each rider.

Answer:

y = 8 miles/hour

x = 16 miles/hour

Step-by-step explanation:

We can solve this question by first setting up the following equations:

Rider x travels twice as fast as rider y:

x = 2*y                - Equation 1

In three hours they are 72 miles apart:

3*x + 3*y = 72              -Equation 2

Solving equations 1 and 2 simultaneously we get:

y = 8 miles/hour

x = 16 miles/hour

This question is done simply by writing down the question statements as equations. This can be seen being done in Equation 1 and Equation 2 above.

Final answer:

The y-intercept of the line y = 11x + 6 is (0, 6), and the x-intercept is (-6/11, 0). To find these, you set x to 0 to find the y-intercept and set y to 0 to solve for the x-intercept.

Explanation:

To determine the intercepts of the line given by the equation y = 11x + 6, we need to find where the line crosses the x-axis and the y-axis. The y-intercept is found when x = 0; substituting x = 0 into the equation gives us y = 6. Therefore, the y-intercept is at the point (0, 6). To find the x-intercept, we set y = 0 and solve for x; setting y = 0 in our equation results in 0 = 11x + 6. Solving for x gives us x = -6/11. The x-intercept is at the point (-6/11, 0).

Answer: the horizontal distance between the tower and the fire is 732.89 feet

Step-by-step explanation:

Considering the situation, a right angle triangle is formed. The height of the fire watch tower represents the opposite side of the right angle triangle.

The horizontal distance, h between the tower and the fire represents the adjacent side of the right angle triangle.

If the angle of depression to the fire that is 7°, the angle of elevation from of the tower watcher from the fire is also 7° because they are alternate angles.

To determine h, we would apply

the tangent trigonometric ratio.

Tan θ, = opposite side/adjacent side. Therefore,

Tan 7 = 90/h

h = 90/tan7 = 90/0.1228

h = 732.89 feet

Answer:

The answer to your question is below

Step-by-step explanation:

Data

A (4 , -3)

r = ?

sin = ?

cos = ?

tan = ?

Process

1.- Plot the point

This point is in the fourth quadrangle

2.- Calculate r

We have the Opposite side and the Adjacent side

       tan Ф = -3/4

       tan⁻¹ Ф = Ф = 323.1

        Ф = 323.1°

3.- sinФ =

Calculate the hypotenuse

        c² = 4² + (-3)²

        c² = 16 + 9

        c² = 25

        c = 5

 sinФ = -3/5

cos Ф = 4/5

tan Ф = -3/4

The value of r is 5. The exact value of sin is -3/5, cos is 4/5, and tan is -3/4.

The value of r can be found using the Pythagorean theorem, which states that for any point (x, y) on the terminal side of an angle in standard position, the value of r can be calculated as √(x^2 + y^2). In this case, we have (x, y) = (4, -3), so r = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.

The exact value of sin for this angle can be calculated as y/r, so sin = -3/5.

The exact value of cos for this angle can be calculated as x/r, so cos = 4/5.

The exact value of tan for this angle can be calculated as y/x, so tan = -3/4.

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Answer:

[tex] v(t) = 8t +5[/tex]

And that represent the instantaneous velocity at a given time t.

And then we just need to replace t =2 in order to find the instantaneous velocity and we got:

[tex] v(t=2) = 8*2 + 5 = 16+5 = 21[/tex]

Step-by-step explanation:

For this case we have the position function s(t) given by:

[tex] s(t) = 4t^2 + 5t+5[/tex]

And we can calculate the instanteneous velocity with the first derivate respect to the time, like this:

[tex] v(t) = s'(t)= \frac{ds}{dt}[/tex]

And if we take the derivate we got:

[tex] v(t) = 8t +5[/tex]

And that represent the instantaneous velocity at a given time t.

And then we just need to replace t =2 in order to find the instantaneous velocity and we got:

[tex] v(t=2) = 8*2 + 5 = 16+5 = 21[/tex]

Answer: the cost of each pencil is $0.5

the cost of each eraser is $0.15

Step-by-step explanation:

Let x represent the cost of one pencil.

Let y represent the cost of one eraser.

Becky spent $1.45 for 2 pencils and 3 erasers. This means that

2x + 3y = 1.45- - - - - - - - - -1

Luke spent $2.65 for 5 pencils and 1 eraser. This means that

5x + y = 2.65- - - - - - - - - -2

Multiplying equation 1 by 1 and equation 2 by 3, it becomes

2x + 3y = 1.45

15x + 3y = 7.95

Subtracting, it becomes

- 13x = - 6.5

x = - 6.5/- 13

x = 0.5

Substituting x = 0.5 into equation 2, it becomes

5 × 0.5 + y = 2.65

2.5 + y = 2.65

y = 2.65 - 2.5

y = 0.15

The cost of one eraser is [tex]\(\$0.15\).[/tex]

Let's denote the cost of one pencil as p dollars and the cost of one eraser as e dollars.

According to the given information:

1. Becky spent $1.45 for 2 pencils and 3 erasers, so her total cost can be expressed as:

[tex]\[ 2p + 3e = 1.45 \][/tex]

2. Luke spent $2.65 for 5 pencils and 1 eraser, so his total cost can be expressed as:

[tex]\[ 5p + 1e = 2.65 \][/tex]

We can now solve this system of equations to find the cost of one eraser (e ).

From equation 1, we can express p in terms of e :

[tex]\[ 2p = 1.45 - 3e \]\[ p = \frac{1.45 - 3e}{2} \][/tex]

Substitute this expression for p  into equation 2:

[tex]\[ 5\left(\frac{1.45 - 3e}{2}\right) + e = 2.65 \][/tex]

Multiply both sides by 2 to eliminate the fraction:

[tex]\[ 5(1.45 - 3e) + 2e = 5.3 \]\[ 7.25 - 15e + 2e = 5.3 \][/tex]

Combine like terms:

[tex]\[ 7.25 - 13e = 5.3 \][/tex]

Subtract 7.25 from both sides:

[tex]\[ -13e = 5.3 - 7.25 \]\[ -13e = -1.95 \][/tex]

Divide both sides by -13:

[tex]\[ e = \frac{-1.95}{-13} \]\[ e = 0.15 \][/tex]

Therefore, the cost of one eraser is [tex]\(\$0.15\).[/tex]

Answer:

  BG = 3; GE = 6

Step-by-step explanation:

The centroid of a triangle divides the median into two parts in the ratio 1 : 2. That is, the short segment is 1/3 the length of the median, and the long segment is 2/3 the length of the median.

  BG = 1/3·BE = 9/3 = 3

  GE = 2/3·BE = 2/3·9 = 18/3 = 6

Answer: the perimeter of the Pentagon is 12.5 inches

Step-by-step explanation:

Let x represent the length of each side of the Pentagon.

A rectangle has width twice as long as the side of the pentagon. This means that the width of the rectangle, w is 2x

The rectangle has length four times as long as the side of the pentagon. This means that the length of the rectangle, l is 4x

The formula for determining the perimeter of a rectangle is expressed as

Perimeter = 2(l + w)

The perimeter of the rectangle is 30 inches. This means that

2(2x + 4x) = 30

12x = 30

x = 30/12 = 2.5

A Pentagon has 5 sides. This means that the perimeter of the Pentagon is

5 × 2.5 = 12.5 inches

Answer:

Hobson v. Hansen (1967)

Step-by-step explanation:

Hobson v. Hansen (1967) was a federal court case filed by civil rights activist Julius W. Hobson against Superintendent Carl F. Hansen and the District of Columbia's Board of Education on the charge that the current educational system underprivileged blacks and the poor of their right to equal educational opportunities relative to their white and affluent peers, on account of race and socioeconomic status.

Before the landmark 1954 Brown v. Board of Education decision, school systems used standardized tests to maintain segregated programs in public schools. This was in violation of federal laws and highlighted by the Coleman Report, which sparked a debate about desegregation and testing bias.

Prior to the Brown v. Board of Education ruling in 1954, many school systems employed a variety of tactics to circumvent the racial desegregation process. Standardized tests and testing procedures were frequently used to place minority children into segregated programs within public schools, reinforcing educational segregation. This practice was a violation of Title VI of the Civil Rights Act of 1964 and other federal legislations aimed at ensuring equal opportunity in education.

After the Coleman Report in 1966, the debate around desegregation, busing, and cultural bias in standardized testing intensified. New policies like mandated busing were implemented to correct the discriminatory practices and to achieve the goals of desegregation. Despite these efforts, many districts faced challenges in successfully integrating schools, resulting in a variety of voluntary and court-ordered methods to promote equal education.

Answer:

Step-by-step explanation:

The identities you need here are:

[tex]r=\sqrt{x^2+y^2}[/tex]  and   [tex]r^2=x^2+y^2[/tex]

You also need to know that

x = rcosθ  and

y = rsinθ

to get this done.

We have

r = 6 sin θ

Let's first multiply both sides by r (you'll always begin these this way; you'll see why in a second):

r² = 6r sin θ

Now let's replace r² with what it's equal to:

x² + y² = 6r sin θ

Now let's replace r sin θ with what it's equal to:

x² + y² = 6y

That looks like the beginnings of a circle.  Let's get everything on one side because I have a feeling we will be completing the square on this:

[tex]x^2+y^2-6y=0[/tex]

Complete the square on the y-terms by taking half its linear term, squaring it and adding it to both sides.

The y linear term is 6.  Half of 6 is 3, and 3 squared is 9, so we add 9 in on both sides:

[tex]x^2+(y^2-6y+9)=9[/tex]

In the process of completing the square, we created within that set of parenthesis a perfect square binomial:

[tex]x^2+(y-3)^2=9[/tex]

And there's your circle!  Third choice down is the one you want.

Fun, huh?

The polar equation r = 6 sinθ is converted to rectangular form using the sine and cosine functions, eventually leading to a quadratic equation in terms of x and y as (x^2 + y^2)y^2 = 36x^2 + 36y^2..

To convert the polar equation r = 6 sinθ into rectangular form, we use the relationships

x = rcosθ and y = r *sinθ.

Substituting for r from the given equation, we have

y = (6sinθ) * sinθ = 6 sin^2θ. Now, since sin^2θ = 1/2 - 1/2cos(2θ), and  

cos(2θ) = 1 - 2sin^2θ,

we can express this in terms of x and y as

cos(2θ) = 1 - 2(y/6)^2.

Therefore, y =1/2 - 1/2(1 - 2(y/6)^2) simplifies to y^2 = 36(1 - cos^2 θ).

Since cos^2 θ =x^2/r^2 =  x^2/x^2+y^2, we plug this back into the equation to get

y^2 = 36 -36x^2/x^2+y^2.

Multiplying through by (x^2+y^2) we end up with (x^2 + y^2)y^2 = 36x^2 + 36y^2.

Answer:

289/4

Step-by-step explanation:

x² + 17x + 12 = 0

x² + 17x = -12

Take half of the second coefficient, square it, then add the result to both sides.

(17/2)² = 289/4

x² + 17x + 289/4 = -12 + 289/4

(x + 17/2)² = 241/4

The answer is 289/4.

By using quadratic equation formula a[tex]x^{2}[/tex] + bx + c = 0, we get:

⇒ [tex]x^{2}[/tex] + 17x + 12 = 0

⇒ [tex]x^{2}[/tex] + 17x = -12

Take half of the second coefficient, square it, then add the result to both sides.

⇒ (17/2)² = 289/4

⇒ [tex]x^{2}[/tex] + 17x + 289/4 = -12 + 289/4

⇒ (x + 17/2)² = 241/4

Quadratic equations are second-degree algebraic expressions and are of the form a[tex]x^{2}[/tex] + bx + c = 0.

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The relationship between the volume of the cylinder and cone is the volume of the cone is 1.5 times bigger than the volume of the cylinder.

Explanation:

The radius is given by [tex]r=\frac{d}{2} =\frac{8}{2} =4[/tex]

The volume of the cone can be determined using the formula,

[tex]V=\pi r^{2} \frac{h}{3}[/tex]

where [tex]\pi=3.14, r=4, h=18[/tex]

Volume of the cone  [tex]=\pi r^{2} \frac{h}{3}[/tex]

                                  [tex]=3.14(4)^2\frac{(18)}{3}[/tex]

                                  [tex]=301.44 \ cm^3[/tex]

The volume of the cone is [tex]301.44 \ {cm}^{3}[/tex]

The volume of the cylinder can be determined using the formula,

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

where [tex]\pi=3.14, r=4, h=9[/tex]

Volume of the cylinder [tex]=\pi r^{2} h[/tex]

                                      [tex]=3.14(16)(9)[/tex]

                                      [tex]=452.16 \ cm^3[/tex]

Thus, the volume of the cylinder is [tex]452.16 \ {cm}^{3}[/tex]

Hence, the relationship between the volume of the cylinder and cone is the volume of the cone is 1.5 times bigger than the volume of the cylinder.

Answer:

First is to define the language in which to make the statement, I chose the language C, because it is the most common. To initialize the variable, the first thing is to define the type, in this case it is an integer type.

In language C, that is determined by int, then the name of the variable, in this case it would be age. We close with; To finish the order. Then we give the courage that asks us to initiate.

In the end, it would look like this:

int age;

age = 15;

int weight;

weight = 90;

Final answer:

To declare and initialize two integer variables called age and weight, one can use the code 'int age = 15, weight = 90;'.

Explanation:

To declare and initialize two integer variables in most programming languages, you can use a single line of code. For the variables age and weight, the statement might look like this:

int age = 15, weight = 90;

This line of code creates two variables of type integer named age and weight. The variable age is initialized with the value 15, and weight is initialized with the value 90.

The use of the comma in the statement allows for both variables to be declared and initialized in one concise statement.

Answer:

345

Step-by-step explanation:

345