Solve using proper methods. Show work. (25 POINTS)

Initially a tank contains 10,000 liters of liquid at the time t = 0 minutes a tap is opened, a liquid then follows out of the tank. The volume of the liquid V liters, which remains in the tank after t minutes is given by V = 10,000(0.933)^t

a) Find the value of V after 5 minutes.
b) Find how long, to the nearest second, it takes for half of the initial amount of liquid to follow out of the tank.
c) The tank is regarded as effectively empty when 95% of the liquid has flowed out. Show that it takes almost three quarters of an hour for this to happen.
d) (i) Find the value of 10,000 - V when t = 0.001 minutes
(ii) Hence or otherwise, estimate the initial flow rate of the liquid. Give your answer in liters per minute, correct to two significant figures.

Answer:

Its the First choice.

Step-by-step explanation:

(3 m-2 n)^-3  / 6m n^-2

= (1/27) m^6 n^-3 / 6 m n^-2

= m^5 n ^-3 / 27*6  n^-2

= m^5 n^2 / 162 n^3

= m^4 / 162 n  (answer).

The correct answer is option (A) m5 /162n

Let us solve step by step

See the attached figure for solution

Answer:

42 correct; if all were answered, that means 7 were wrong. 7/49 reduces to 1/7; one seventh were wrong.

QUESTION 1

The given logarithm is

[tex]\log_7\sqrt{a^3b^9}[/tex]

We rewrite to obtain;

[tex]\log_7(a^3b^9)^{\frac{1}{2}}[/tex]

Use the power rule of logarithm ; [tex]\log_a(m^n)=n\log_a(m)[/tex]

[tex]\frac{1}{2}\log_7(a^3b^9)[/tex]

Use the product rule; [tex]\log_a(mn)=\log_a(m)+\log_a(n)[/tex]

[tex]\frac{1}{2}[\log_7(a^3)+\log_7(b^9)][/tex]

Use the power rule of logarithms again;

[tex]\frac{1}{2}[3\log_7(a)+9\log_7(b)][/tex]

Or

[tex]\frac{3}{2}\log_7(a)+\frac{9}{2}\log_7(b)][/tex]

QUESTION 2

Given;

[tex]\log_6(\frac{x^5}{y^9})[/tex]

Apply the quotient rule of logarithm; [tex]\log_a(m)-\log_a(n)=\log_a(\frac{m}{n} )[/tex]

[tex]\log_6(\frac{x^5}{y^9})=\log_6(x^5)-\log_6(y^9)[/tex]

Apply the power rule to get;

[tex]\log_6(\frac{x^5}{y^9})=5\log_6(x)-9\log_6(y)[/tex]

QUESTION 3

Given;

[tex]\log_8(x^3y^7)[/tex]

Use the product rule to get;

[tex]=\log_8(x^3)+\log_8(y^7)[/tex]

Use the power rule now;

[tex]=3\log_8(x)+7\log_8(y)[/tex]

im pretty sure it’s D

The k value in the equation y = f(x) + k causes a vertical shift of the graph. If k is positive, the graph shifts up; if k is negative, it shifts down. Thus, the correct answer is B.

In the function transformation y = f(x) + k, the term k represents a vertical shift of the graph. Specifically:

Thus, the correct answer to the question is:

B. Shifts the graph up k units.

To summarize, the value k in the equation y = f(x) + k causes a vertical shift upwards if it's positive and downwards if it's negative.

Answer:

a) 110,880

b) 8.61x10^37

c) 7.18x10^19

Step-by-step explanation:

To find the amount of options, use the following combination equation.

n!/[r! * (n - r)]

In this equation, n represents the amount of possible options and r represents the amount being chosen at a time. We can now calculate out the answer for each possibility.

a) In this case, n would equal 12 and r would equal 6.

n!/[r! * (n - r)]

12!/[6! * (12 - 6)]

479001600/[720*6]

479001600/4320

110,880

b) In this case, n would equal 36 and r would equal 6

n!/[r! * (n - r)]

36!/[6! * (12 - 6)]

3.72x10^41/[720*6]

3.72x10^41/4320

8.61x10^37

c) For this one, n would equal 24 and r would equal 6. We would also then have to divide the answer at the end by 2.

n!/[r! * (n - r)]

12!/[6! * (12 - 6)]

6.20x10^23/[720*6]

6.20x10^23/4320

1.44x10^20 ÷ 2 = 7.18x10^19

This question deals with calculating combinations of croissants chosen under various conditions, which exemplifies the use of combinations with repetition in mathematics. Solutions depend heavily on the specifics of each part, many of which require customized approaches to account for constraints like minimum selections or caps on certain items.

This question involves calculating different combinations of items from a set, which in this context are types of croissants being chosen in various quantities and conditions. Solving these problems requires understanding permutations and combinations, specifically combinations with repetition, as the order in which the croissants are selected does not matter but repeats are allowed.

We utilize the formula for combinations with repetition: C(n+r-1, r), where n is the number of types of items (in this case, croissants) and r is the number of items to choose.

For example, in part a) choosing a dozen croissants from 6 types would involve finding the combination with repetition of these 6 types over 12 selections.

However, specific formulas need to be applied to cater to the unique conditions stated in parts c) through f). Each of these parts imposes different constraints (e.g., minimum numbers of specific items, caps on how many of certain items can be chosen) that complicate the calculation and often require a piecewise approach or the inclusion/exclusion principle to accurately count the number of valid combinations.

Due to the complex and varied nature of each part of this question, and without specific formulas for parts c) through f), a general overview is provided instead of detailed solutions.

Each condition requires a tailored approach that involves breaking down the problem into smaller, manageable parts, and sometimes subtracting the cases that do not meet the given criteria from the total possible combinations.

It is located in the 2nd quadrant.

(x,y) - 1st quadrant

(-x,y) - 2nd quadrant

(-x,-y) - 3rd quadrant

(x,-y) - 4th quadrant

Answer:

Second quadrant

Step-by-step explanation:

That coordenates are located in the second quadrant of the cartesian plane

Best regards

Answer:

Dependant Variable

Step-by-step explanation:

Answer:

Step-by-step explanation:

The response variable is the dependent variable.

61/3 - 42/3 = 19/3 feet

Answer:

(2,0) is not a solution of the system. The point does not belong to any of the graphs.

Step-by-step explanation:

To easily solve this question, we can graph both equations in a graphing calculator, and verify the intersection point, which is equal to the solution of the system of equations.

Plotting the graphs

y = 3x + 2

g = |x-1| +1

We obtain the intersection point

(0,2)

Solution of the system of equations

Answer:

56.1 in^2

Step-by-step explanation:

When given sides "a" and "b" and the angle α between them, the applicable formula for the area of the triangle is ...

A = (1/2)ab·sin(α)

Substituting the given values, we find the area to be ...

A = (1/2)(14.2 in)(8 in)sin(99°) ≈ 56.1 in^2

Answer:

56.1

Step-by-step explanation:

Diagram

Let's look at the diagram for a moment.

The 8 in line has been extended.

The dashed line is the height associated with the 8 inch line

The dashed line and the 8 inch line's extension meets at right angles.

Discussion

Normally you would take the 8 inch line and the height and divide their product by 2. You don't have the height. So you have to somehow get an expression for it.

The 81o angle is the supplement of the 99o angle. If you take the sine of both of them the sines are equal.

h = sin(81) * 14.2 because

sin(81) = opposite / hypotenuse.

opposite = hypotenuse * sin(81)

opposite = height = hypotenuse* sin(81)

height = 14.2 * sin(81)

Now we can find the area

Area

Area = height * base

base = 8

height = 14.2 * sin(81)

Area = 1/2 * 8 * 14.2 * sin(81)

Area = 56.1

ANSWER: 15.5 units^2

Answer:

coordinates

Step-by-step explanation:

The coordinates are the pairs of number giving a location of a point on the coordinate grid (example (4, 12) or (-3, 4) explain where the location point should be)

The pair of numbers giving the location of a point are called its coordinates.

Two perpendicular number lines define a Cartesian plane, which is named after Rene Descartes, a mathematician who codified its use in mathematics: the horizontal x-axis and the vertical y-axis, respectively. Any point in the plane can be described by an ordered pair of numbers using these axes.

In every direction, the Cartesian plane has an infinite length.

Given

A pair of numbers that use the horizontal and vertical distances between the two reference axes to describe a point's position on a coordinate plane. typically represented by the x- and y-values in the form (x, y).

Coordinates are the numbers who gives the location of point.

Hence, the pairs are called coordinates.

Learn more about Cartesian Plane;

https://brainly.com/question/28574364

#SPJ5

okay so you multiply .15 by 120 which is 18. then you multiply .10 by 120 which is 12. so you do 120 subtracted by 12 and 18 which the answer is 90

Final answer:

The shoes will cost $91.80 after both discounts have been applied.

Explanation:

To find the cost of the shoes, we need to apply two discounts.

First, we calculate the discount amount of 15% off $120 by multiplying $120 by 0.15, which gives us $18.

Then, we subtract this discount from the original price:

$120 - $18 = $102.

Next, we calculate the additional discount of 10% off $102 by multiplying $102 by 0.10, which gives us $10.20.

We subtract this discount from the discounted price:

$102 - $10.20 = $91.80.

So, the shoes will cost $91.80 after both discounts have been applied.

well, 1/6 of 3/4 is simply their product, thus

[tex]\bf \cfrac{3}{4}\cdot \cfrac{1}{6}\implies \cfrac{3}{6}\cdot \cfrac{1}{4}\implies \cfrac{1}{2}\cdot \cfrac{1}{4}\implies \cfrac{1}{8}[/tex]

Answer:

  6.3  (x, y) = (3, 1)

  6.4  (x, y) = (-4, 2)

Step-by-step explanation:

An augmented matrix in this context is an array of the coefficients in the equation. For example, the equation x -2y = 3 can be represented by the row of numbers 1 -2 3. When writing the matrix by hand, we usually just separate the elements in a row using blank space. Some calculators require a comma or other character between matrix elements. The entire array is enclosed in square brackets.

An augmented matrix may have a vertical line separating the square matrix of coefficients from the rightmost column of constants.

Many graphing calculators have functions available for entering and solving linear equations in this form.

__

6.3 The augmented matrix is ...

[tex]\left[\begin{array}{cc|c}2&3&9\\1&-4&-1\end{array}\right][/tex]

This array can be used to solve the system of equations using "row operations." The idea is that any row can be multiplied by any number, and any row can be added to any other row (multiplied by some number or not). So, the same techniques used to solve a system by elimination can be used here.

We can, for example, subtract twice the second row from the first. Then we have ...

[tex]\left[\begin{array}{cc|c}2-2(1)&3-2(-4)&9-2(-1)\\1&-4&-1\end{array}\right]\\\\\left[\begin{array}{cc|c}0&11&11\\1&-4&-1\end{array}\right] \quad\text{simplified}\\\\\left[\begin{array}{cc|c}0&1&1\\1&-4&-1\end{array}\right] \quad\text{first row divided by 11}\\\\\left[\begin{array}{cc|c}0&1&1\\1+4(0)&-4+4(1)&-1+4(1)\end{array}\right] \quad\text{4 times row 1 added to row 2}\\\\\left[\begin{array}{cc|c}0&1&1\\1&0&3\end{array}\right] \quad\text{simplified}[/tex]

When we turn this last matrix back into equations, we get ...

  0x +y = 1

  1x + 0y = 3

That is, the solution is (x, y) = (3, 1).

__

6.4 The augmented matrix is ...

[tex]\left[\begin{array}{cc|c}4&5&-6\\3&2&-8\end{array}\right][/tex]

Let's start the solution of this one by dividing the first row by 4.

[tex]\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\3&2&-8\end{array}\right] \\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\3-3(1)&2-3(\frac{5}{4})&-8-3(-\frac{3}{2})\end{array}\right] \quad\text{row 2 minus 3 times row 1}\\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\0&-\frac{7}{4}&-\frac{7}{2}\end{array}\right] \quad\text{simplify}\\\\\left[\begin{array}{ccc}1&\frac{5}{4}&-\frac{3}{2}\\0&1&2\end{array}\right] \quad\text{multiply row 2 by -4/7}[/tex]

We can eliminate the y-term in the first row by subtracting 5/4 times the second row. This will put the matrix in the form that shows us the solution directly.

[tex]\left[\begin{array}{ccc}1-\frac{5}{4}(0)&\frac{5}{4}-\frac{5}{4}(1)&-\frac{3}{2}-\frac{5}{4}(2)\\0&1&2\end{array}\right] \quad\text{row 1 minus 5/4 times row 2}\\\\\left[\begin{array}{ccc}1&0&-4\\0&1&2\end{array}\right][/tex]

This tells us ...

  x = -4

  y = 2

So the solution is (x, y) = (-4, 2).

An augmented matrix organizes the coefficients of a system of linear equations. After setting up the matrix with the coefficients from the equations, methods like Gaussian elimination or Gauss-Jordan elimination can be used to solve the matrix and consequently, solve the system.

An augmented matrix is a compact way of keeping track of the coefficients in a system of linear equations. Here's how to form it and solve the system:

Gaussian elimination involves using elementary row operations to manipulate the matrix to a form where the solution is easier to find, often called reduced row echelon form. For the Gauss-Jordan method, the main idea is to get zeros below (and above when possible) a leading 1, and make sure that leading 1 is the only nonzero entry in its column.

https://brainly.com/question/35542952

#SPJ2

Answer:

The area of the circle is [tex]530.93\ units^{2}[/tex]

Step-by-step explanation:

we know that

The area of a circle is equal to

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

we have

[tex]r=26/2=13\ units[/tex] ----> the radius is half the diameter

assume

[tex]\pi=3.1416[/tex]

substitute

[tex]A=(3.1416)(13)^{2}=530.93\ units^{2}[/tex]

Answer:

[tex]a_n=-12-4(n-1)[/tex]

Step-by-step explanation:

The given sequence is

-12,-16,-20...

The first term of this sequence is [tex]a_1=-12[/tex].

The common difference is

[tex]d=-16--12[/tex]

[tex]d=-16+12=-4[/tex]

The nth term of this arithmetic sequence is;

[tex]a_n=a_1+d(n-1)[/tex]

We substitute the values for the first term and the common difference to obtain;

[tex]a_n=-12-4(n-1)[/tex]

Answer:

[tex]a_{n} = -12-4(n-1)[/tex]

Step-by-step explanation:

We have given a arithmetic sequence.

-12,-16,-20,...

We have to find formula for a given sequence.

The general formula for nth term of sequence is :

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

In given sequence,

[tex]a_{1} = -12[/tex]

d is the common difference between consecutive terms.

d = -16-(-12)  = -16+12

d = -4

Putting given values in formula, we have

[tex]a_{n} = -12-4(n-1)[/tex] which is the answer.

Answer:

So the answer is (3, 0) on a graph.

Step-by-step explanation:

What I use is a graphing calculator. It helps tremendously. It is called desmos.com. Hope this helps.

Answer:

The given function is

[tex]y=log(x-2)[/tex]

To graph any function, we just have to five arbitrary values to x-variable, and see the y-values that gives to formed coordinate pairs to graph them then.

In this case, the table that shows these values is gonna be

X       Y

2.5   -0.3

3        0

3.5    0.2

4       0.3

4.5 0.4

Notice that we started from [tex]x=2.5[/tex]. The reason is beacuse the given logarithmic function is not defined for values equal or lower than 2, when taht happens, the function becoms undetermined. This means the domain of the given function has to be restricted to [tex]D= (x>2)[/tex], otherwise the function won't be defined.

Next, we evalute the function for each case.

[tex]x=2.5[/tex]

[tex]y=log(x-2)=log(2.5-2)=log(0.5) \approx -0.3[/tex]

[tex]x=3\\y=log(x-2)=log(3-2)=log(1)=0[/tex]

[tex]x=3.5\\y=log(x-2)=log(3.5-2)=log(1.5) \approx 0.2[/tex]

[tex]x=4\\y=log(x-2)=log(4-2)=log(2) \approx 0.3[/tex]

[tex]x=4.5\\y=log(x-2)=log(4.5-2)=log(2.5) \approx 0.4[/tex]

When you have the table completed, you proceed to graph each coordinate.

In this case, the graph of the logarithmic function would be as the image attached. That's all the process to graph the function.

(In the graph, you can observe how fast the function drops when is near to x=2, that's the undetermination behaviour we talk about lines above).

The system has two solutions: [tex]\( (2, -2) \) and \( (-1, 4) \),[/tex] found by solving their simultaneous equations.

let's solve the system step by step:

Given the system of equations:

[tex]1. \( y = -2x + 2 \)\\2. \( y = x^2 - 3x \)[/tex]

We want to find the values of [tex]\( x \) and \( y \)[/tex] that satisfy both equations simultaneously.

First, we set the equations equal to each other since they both equal [tex]\( y \):[/tex]

[tex]\[ -2x + 2 = x^2 - 3x \][/tex]

Now, let's rearrange this equation to get a quadratic equation:

[tex]\[ x^2 - 3x + 2x - 2 = 0 \]\[ x^2 - x - 2 = 0 \][/tex]

Now, we can use the quadratic formula to solve for [tex]\( x \):[/tex]

[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \][/tex]

where [tex]\( a = 1 \), \( b = -1 \), and \( c = -2 \).[/tex]

[tex]\[ x = \frac{{-(-1) \pm \sqrt{{(-1)^2 - 4(1)(-2)}}}}{{2(1)}} \]\[ x = \frac{{1 \pm \sqrt{{1 + 8}}}}{2} \]\[ x = \frac{{1 \pm \sqrt{9}}}{2} \]\[ x = \frac{{1 \pm 3}}{2} \][/tex]

So, we have two potential values for [tex]\( x \): \( x = 2 \) and \( x = -1 \).[/tex]

Now, let's find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex] value using either of the original equations:

For [tex]\( x = 2 \):[/tex]

[tex]\[ y = -2(2) + 2 = -4 + 2 = -2 \][/tex]

For [tex]\( x = -1 \):[/tex]

[tex]\[ y = -2(-1) + 2 = 2 + 2 = 4 \][/tex]

So, we have two solutions for the system:[tex]\( (2, -2) \) and \( (-1, 4) \).[/tex] Therefore, the system has 2 solutions.

Answer:

=55m

Ahh. Correct me if I'm wrong.

Answer:

60%

Step-by-step explanation:

It went up 12 dollars, and you need to find how much 12 is of 20 as a percent. You do 12/20 times 5/5 and get 60/100, so 60% markup

Answer:

Step-by-step explanation:

The Answer Well -12$

20 - 32 = -12

length of rectangle =9cm

width of rectangle = b

perimeter of rectangle =24cm

perimeter of rectangle =2(l+b)

24=2(9+b)

9+b=12

b=3cm

Using the perimeter formula for a rectangle, the width of the rectangle is calculated to be 3 centimeters.

To find the width of the rectangle when the perimeter is 24 centimeters and the length is 9 centimeters, we can use the formula for the perimeter of a rectangle: P = 2l + 2w, where P is the perimeter, l is the length, and w is the width. From the given information:

P = 24 cm

l = 9 cm

We can set up the equation as follows:

24 = 2(9) + 2w

24 = 18 + 2w

2w = 24 - 18

2w = 6

We then divide both sides by 2 to solve for w:

w = 6 / 2

w = 3 cm

Thus, the width of the rectangle is 3 centimeters.

After calculating the perimeters, both the square and the rectangle have the same perimeter of 32 inches when the square has each side measuring 8 inches and the rectangle has dimensions of 6 inches by 10 inches.

To determine which shape has the greater perimeter, calculate the perimeter of both the square and the rectangle. The formula for the perimeter of a square is 4 × side length, and the formula for the perimeter of a rectangle is 2 × (length + width).

For the square with each side measuring 8 inches, its perimeter is 4 × 8 = 32 inches.

For the rectangle with a width of 6 inches and a length of 10 inches, its perimeter is 2 × (10 + 6) = 2 × 16 = 32 inches.

Therefore, both the square and the rectangle have the same perimeter of 32 inches.

Answer: x=12

Explanation:

In a parallelogram, adjacent angles equals 180 (consecutive angles). Therefore, (132-x)+(6x-12)=180

Simplify:

-x+6x+132-12=180

5x+120=180

5x=60

x=12

Answer:

The formula to calculate standard deviation from probability is \sqrt(n*p*(1-p)). n is the sample size, and 200 in this case (number of putts for practice). p is 80% or 0.8, the probability that he can make it. So the standard deviation is \sqrt(200*0.8*(1-0.8)=\sqrt(200*0.8*0.2)=\sqrt(16)=4.

1952 + 384 = 2336 years