What is the farthest point on the sphere x² + y² + z² = 16 from the point (2, 2, 1) ?
a. (-8/3, -8/3, -4/3)
b. (-8/3, 8/3, 4/3)
c. (-8/3, -8/3, 4/3)
d. (8/3, -8/3, -4/3)
e. (8/3, 8/3, 4/3)

Answer:

see explanation

Step-by-step explanation:

the equation of a line in gradient- intercept form is

y = mx + c ( m is the gradient and c the y- intercept )

given

2x - 3y = 5 ( subtract 2x from both sides )

- 3y = - 2x + 5 ( divide through by - 3 )

y = \(\frac{2}{3}\) x - \(\frac{5}{3}\) ← in gradient- intercept form

with gradient m = \(\frac{2}{3}\)

Answer:

I'm not 100% sure, but I think it is A. 36 inches!

\(x + 30 + 2x + 90 = 180\)

\(3x + 120 = 180 \\ 3x = 180 - 120 \\ \: \: \: 3x = 60 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ x = \frac{60}{3} \: \: \: \: \: \: \: \: \: \: \: \\ x = 20 \: \: \: \: \: \: \: \: \: \: \: \: \)

Answer:

x+30 + 2x + 90 = 180

3x + 120 = 180

3x = 60

x = 20

Step-by-step explanation:

there are 180 degrees in a triangle

Answer:

\(\log_{9}4\approx 0.631\)

Step-by-step explanation:

The change of base formula we can use is

\(\log_{b}a=\frac{\log{a}}{\log{b}}\)

Not all calculators will allow the user to directly enter the logarithms with a given base, so the change of base can be used for all calculators, which use the common log.

Common logs have a base of 10.

Let's convert your expression:

\(\log_{9}4=\frac{\log4}{\log9}\)

Using our calculator, we will compute log 4 divided by log 9:

\(\frac{\log4}{\log9}\approx0.631\).

Based on the given circle with center Y, each of the measure include the following:

In Mathematics and Geometry, a circle simply refers to a closed, two-dimensional (2D) curved geometric shape with no edges or corners.

Based on the given circle with center Y and line AD is perpendicular to line DB, we can reasonably infer and logically deduce that arc AC is equal to arc BC;

mAC = mBC = 71 degrees.

mAB = mAC + mBC

mAB = 71 + 71

mAB = 142 degrees.

Since line DC is the perpendicular bisector of line AB, the measure of line AB and line BD can calculated as follows;

AD = BD = AB/2

AD = BD = 60/2

AD = BD = 30 units.

Note: YB is the radius of this circle and the hypotenuse of right-angle triangle YDB, which is equal to 34 units.

In order to determine the length of YD, we would have to apply Pythagorean's theorem as follows;

x² + y² = z²

BD² + YD² = YB²

30² + YD² = 34²

YD² = 1156 - 900

YD = √256

YD = 16 units.

DC = YB - YD

DC = 34 - 16

DC = 18 units.

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Answer:

the period of the graph of 4cos(1/2x)-3 is twice as long as the period of 4cos(x)-3

Using equations,

a. We can find that the required equation is p = 32n.

b. Also, Frankie reads 256 pages in 8 nights.

An equation can be defined in numerous ways. Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14. Algebraic equations in mathematics frequently have one or more variables.

As per the question,

a.

Frankie reads 32 pages of a book each night.

Number of pages = p

Number of nights = n

So, our equation will be:

p = 32 × n

⇒ p = 32n

b.

Now, we need to find how many pages Frankie reads in 8 nights:

n = 8

p = 32n

= 32 × 8

= 256

Therefore, Frankie reads 256 pages in 8 nights.

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Answer:

2 and 5

Step-by-step explanation:

Answer:

yes

Step-by-step explanation:

you can divide the whole ratio by 4 to get the 2nd as a result. Therefore, they are equivalent.

Step-by-step explanation:

I wrote down the steps Step-by-step for you

I hope you understand....

Answer:

Step-by-step explanation:

18.95+.16 = 19.11

answer is 2 miles is the maximum

The employee's net take-home pay is $2597.

The gross salary is the total salary before any deductions are made.

In this case, the employee's bi-weekly gross salary is $3000.

Deductions are made from the gross salary to arrive at the net take-home pay.

The deductions include income tax, CPP, El, and union dues.

The total deductions can be calculated by adding the individual deductions:

Total deductions = Income tax + CPP + El + union dues

Total deductions = $218 + $99 + $36 + $50Total deductions = $403

The net take-home pay is the amount that the employee receives after all the deductions have been made.

It can be calculated by subtracting the total deductions from the gross salary:

Net take-home pay = Gross salary - Total deductions

Net take-home pay = $3000 - $403Net take-home pay = $2597

Therefore, the employee's net take-home pay is $2597.

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The set H of polynomials of the form P(t) = a + bt², where a and b are real numbers with b > a, is not a vector space. It fails to satisfy property C: it is not closed under vector addition.

In order for a set to be a vector space, it must satisfy several properties: containing a zero vector, being closed under vector addition, and being closed under multiplication by scalars. Let's examine each property for the set H:

A) Contains zero vector: The zero vector in this case would be the polynomial P(t) = 0 + 0t² = 0. However, this polynomial does not have the form a + bt² with b > a, as required by H. Therefore, H does not contain a zero vector.

B) Closed under vector addition: To check this property, we take two arbitrary polynomials P(t) = a + bt² and Q(t) = c + dt² from H and try to add them. The sum of these polynomials is (a + c) + (b + d)t². However, it is possible to choose values of a, b, c, and d such that (b + d) is less than (a + c), violating the condition b > a. Hence, H is not closed under vector addition.

C) Closed under multiplication by scalars: Multiplying a polynomial P(t) = a + bt² from H by a scalar k results in (ka) + (kb)t². Since a and b can be any real numbers, there are no restrictions on their values that would prevent the resulting polynomial from being in H. Therefore, H is closed under multiplication by scalars.

In conclusion, the set H fails to satisfy property C: it is not closed under vector addition. Therefore, H is not a vector space.

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To calculate the covariance between x1 (the number of customers in the express checkout) and x2 (the number of customers in the superexpress checkout), we need the joint probability distribution or the joint probability mass function of x1 and x2. Without specific information about this distribution, it is not possible to directly calculate the covariance.

Covariance is a measure of how two random variables vary together. It quantifies the degree to which changes in one variable are associated with changes in the other variable. In order to calculate the covariance, we need to have a sample or probability distribution that provides the necessary information about the relationship between x1 and x2.

If we have a sample of observations for both x1 and x2, we can calculate the sample covariance using the following formula:

Cov(x1, x2) = Σ[(x1 - μ1)(x2 - μ2)] / (n - 1)

where Σ represents the summation over all observations, x1 and x2 are the individual observations, μ1 and μ2 are the sample means of x1 and x2, respectively, and n is the sample size.

If we have the joint probability distribution or the joint probability mass function, we can use the following formula to calculate the covariance:

Cov(x1, x2) = ΣΣ(x1 - μ1)(x2 - μ2) * P(x1, x2)

where ΣΣ represents the double summation over all possible values of x1 and x2, P(x1, x2) is the joint probability or probability mass function of x1 and x2, and μ1 and μ2 are the means of x1 and x2, respectively.

Without the specific probability distribution or a sample of observations, it is not possible to calculate the covariance between x1 and x2.

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Here are the vectors **t**, **n**, and **b** at the given point:

* **t** = (-3 sin t, 3 cos t, 0)

* **n** = (-3 cos t, -3 sin t, 3 / cos^2 t)

* **b** = (3 cos^2 t, -3 sin^2 t, -3)

The vector **t** is the unit tangent vector, which points in the direction of the curve at the given point. The vector **n** is the unit normal vector, which points in the direction perpendicular to the curve at the given point. The vector **b** is the binormal vector, which points in the direction that is perpendicular to both **t** and **n**.

To find the vectors **t**, **n**, and **b**, we can use the following formulas:

```

t(t) = r'(t) / |r'(t)|

n(t) = (t(t) x r(t)) / |t(t) x r(t)|

b(t) = t(t) x n(t)

```

In this case, we have:

```

r(t) = (3 cos t, 3 sin t, 3 ln cos t)

r'(t) = (-3 sin t, 3 cos t, 3 / cos^2 t)

```

Substituting these into the formulas above, we can find the vectors **t**, **n**, and **b** as shown.

The vectors **t**, **n**, and **b** are all orthogonal to each other at the given point. This is because the curve is a smooth curve, and the vectors are defined in such a way that they are always orthogonal to each other.

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The binormal vector (b) is perpendicular to both the tangent and normal vectors and completes the orthogonal coordinate system.

To find the vectors t, n, and b at the given point, we need to calculate the first derivative, second derivative, and third derivative of the position vector r(t).
Given r(t) = (3 cos t, 3 sin t, 3 ln cos t), we can calculate the derivatives as follows:
First derivative:
r'(t) = (-3 sin t, 3 cos t, -3 sin t / cos t)
Second derivative:
r''(t) = (-3 cos t, -3 sin t, -3 cos t / cos^2 t + 3 sin^2 t / cos t)
       = (-3 cos t, -3 sin t, -3 cos t / cos^2 t + 3 tan^2 t)
Third derivative:
r'''(t) = (3 sin t, -3 cos t, 6 cos t / cos^3 t - 6 sin t / cos t)
        = (3 sin t, -3 cos t, 6 sec^3 t - 6 tan t sec t)
At the given point (3, 0, 0), substitute t = 0 into the derivatives to find the vectors:
r'(0) = (0, 3, 0)
r''(0) = (-3, 0, 3)
r'''(0) = (0, -3, 6)
Therefore, at the given point, the vectors t, n, and b are:
t = r'(0) = (0, 3, 0)
n = r''(0) = (-3, 0, 3)
b = r'''(0) = (0, -3, 6)
These vectors represent the tangent, normal, and binormal vectors, respectively, at the given point.
The tangent vector (t) represents the direction of motion of the curve at that point. The normal vector (n) is perpendicular to the tangent vector and points towards the center of curvature.

The binormal vector (b) is perpendicular to both the tangent and normal vectors and completes the orthogonal coordinate system.
Remember to check your calculations and units when applying this method to different functions.

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The volume of a stack of 9 books is 1368.75 cubic inches.

To find the volume of a stack of 9 books, we first need to find the height of the stack. Since each book is 3/4 inch thick, the height of the stack is 9 times 3/4 inch, which is 6 3/4 inches.

Now we need to find the area of the base of the rectangular prism formed by the stack of books. Since each book has an area of 22 1/2 square inches, the total area of the base of the stack is 9 times 22 1/2 square inches, which is 202 1/2 square inches.

Therefore, the volume of the stack of 9 books is:

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Based on the monthly profits that accrued to the architecture firm of Stuart, Kennedy, Maxwell, Ltd., the average profit of those months was $215

To find the average profit of the 5 months of profit for the architecture firm of Stuart, Kennedy, Maxwell, Ltd., you need to sum the profits from that period and then divide by the number of months.

The average profit for those months is:

= ( 1,200 + 755 + (-450) + 210 + (-640) ) / 5 months

= 1,075 / 5 months

= $215

In conclusion, the average profit in those 5 months was $215.

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Step-by-step explanation:

The quotient of -256/6 represents the average rate of descent of the submarine, in feet per minute. To find this rate, we divide the total change in depth (-256 feet) by the time taken to descend (6 minutes):

-256 / 6 = -42.67...

The quotient is negative because the submarine is descending (i.e., decreasing in depth). The magnitude of the quotient is approximately 42.67 feet per minute, rounded to two decimal places.

Therefore, the correct answer is:

C. The depth of the submarine DECREASED by 42 2/3 feet per minute.

The quotient of the obtained from the long division of the volume expression by the width, (x - 3) is; w² - 6·w - 16

The long division of a polynomial is a method of dividing one polynomial by another polynomial of a lower or the same degree.

The function for the volume of water needed to fill the pond can be presented as follows;

V = w³ - 9·w² + 2·w + 48

The possible question obtained from a similar question on the internet requires the quotient of the fraction, (w³ - 9·w² + 2·w + 48)/(x - 3)

Therefore, using the long division for dividing polynomials, we get;

\({}\)            w² - 6·w - 16              

(w - 3) | (w³ - 9·w² + 2·w + 48)

\({}\)            w³ - 3·w²

\({}\)                   -6·w² + 2·w + 48

\({}\)                   -6·w² + 18·w + 48

\({}\)                 \({}\)             -16·w

\({}\)                               -16·w + 48

The correct options are; w², 6·w, 16

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Answer:

15 liters

Step-by-step explanation:

[] 9 / 24 = 0.375

-> Vito used 0.375 liters of water on the first 24 flower pots

[] 40 *  0.375 = 15

-> Vito will need  liters of water to water the 40 flower pots if he uses the same amount of water

Have a nice day!

     I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly.

- Heather

If 15 carpenters can build 3 houses in 3 hours, it means that each house takes 1 hour to complete with their combined effort. Therefore, it would take 5 carpenters the same amount of time, 1 hour, to build 1 house.

Given that 15 carpenters can build 3 houses in 3 hours, we can determine their collective rate of work. Since 3 houses are built in 3 hours, it means each house takes 1 hour to complete with the combined effort of the 15 carpenters. Now, if we reduce the number of carpenters to 5 while keeping the rate of work constant, it stands to reason that 5 carpenters would take the same amount of time, 1 hour, to build 1 house. The number of carpenters is inversely proportional to the time taken to complete a task, assuming the rate of work remains constant. Therefore, by reducing the number of carpenters from 15 to 5, we can still maintain the same rate of work and complete 1 house in 1 hour.

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Answer:

.66 cu. meters

Step-by-step explanation:

volume for cone is V=1/3 x TT x r^2 x height

TT (pi) is 3.14

radius is 1/2 of diameter (1/2 x 1.5 =  .75 )

height is 1.5 x radius (1.5 x .75 = 1.125)

V= 1/3 x 3.14 x .75^2 x 1.125

V= .66  cu. meters

Answer: (-1, 3)

Step-by-step explanation:

Midpoint formula: ((x1 + x2) / 2, (y1+y2) / 2)

((-5+7)/2, (4+2) / 2))

((-2/2), (6/2))

(-1, 3)

The probability of drawing 2 yellow markers and 2 red markers is approximately 0.2697.

To find the probability of drawing 2 yellow markers and 2 red markers, we need to calculate the total number of ways to draw 4 markers from the bag, as well as the number of ways to draw 2 yellow and 2 red markers.

The total number of ways to draw 4 markers from the bag is:

¹⁴C₄ = (14!)/(4!*(14-4)!) = 1001

This is because there are 14 markers in total, and we are choosing 4 of them.

To find the number of ways to draw 2 yellow and 2 red markers, we can use the combination formula again. The number of ways to choose 2 yellow markers from 4 yellow markers is:

⁴C₂ = (4!)/(2!*(4-2)!) = 6

Similarly, the number of ways to choose 2 red markers from 10 red markers is:

¹⁰C₂ = (10!)/(2!*(10-2)!) = 45

Therefore, the number of ways to draw 2 yellow and 2 red markers is:

6 * 45 = 270

Now we can find the probability of drawing 2 yellow and 2 red markers by dividing the number of ways to draw 2 yellow and 2 red markers by the total number of ways to draw 4 markers:

P(2 yellow and 2 red) = 270/1001 = 0.2697 (rounded to four decimal places)

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The probability of drawing 2 yellow marker and 2 red markers 15/1078

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty an event will occur is 1 which is equivalent to 100%.

Probability = sample space / total outcome

for the first draw ;

probability of picking yellow = 2/14 = 1/7

second draw;

probability of picking yellow = 1/13

third draw;

probability of picking red = 10/12 = 5/6

fourth draw;

probability of picking red = 9/11

Probability of picking two yellow = 2/14 × 1/7 = 2/98 = 1/49

probability of picking two reds = 5/6 × 9/11 = 45/66 = 15/22

probability of picking 2 yellow and 2 red = 1/49 × 15/22 = 15/1078

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Answer:

-x²+3

Step-by-step explanation:

(3x² + x - 2) - (4x² + x - 5).

3x²+x-2-4x²-x+

3x²-2-4x²+5

-x²-2+5

-x²+3

The area of the shaded region in this problem is given as follows:

995.44 cm².

The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:

A = πr²

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, hence it's measure is given as follows:

r = 21 cm.

Then the area of the entire circle is given as follows:

A = π x 21²

A = 1385.44 cm².

The right triangle has two sides of length 39 cm and 20 cm, hence it's area is given as follows:

A = 0.5 x 39 x 10

A = 390 cm².

Then the area of the shaded region is given as follows:

1385.44 - 390 = 995.44 cm².

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Therefore ,the heart beat the time per beat in units of seconds is 0.5s/beat.

A well-formed equation in algebra unites two variables with the equals sign to indicate the equality of the two terms. An equation, for instance, is any well-formed formula that consists of two expressions joined by the equals sign in English. An équation, on the other hand, is described as consisting of one or more variables in French.

Here,

Given: your heart rate is 120 beats per minute during strenuous exercise

Thus to calculate time per beat in units of seconds

We,

time  per second per beat =  total second in a minute /heart rate

time  per second per beat =60/120

time  per second per beat =1/2

time  per second per beat = 0.5 s/beat

Therefore ,the heart beat the time per beat in units of seconds is 0.5s/beat.

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Hence, the system does not have a consistent solution.

To determine whether the nonhomogeneous system Ax = b is consistent, we can perform row reduction on the augmented matrix [A | b] and check for any inconsistencies.

The given system of equations is:

2x - 4y + 5z = -7

-7x + 14y + 4z = -35

3x - 10y + z = 15

Rewriting the system as an augmented matrix [A | b]:

[ 2 -4 5 | -7 ]

[ -7 14 4 | -35 ]

[ 3 -10 1 | 15 ]

Now, let's perform row reduction to determine the consistency of the system:

R2 = R2 + 7R1

R3 = R3 - (3/2)R1

[ 2 -4 5 | -7 ]

[ 0 0 39 | -14 ]

[ 0 1 -17/2| 31/2]

The third row indicates a contradiction, as it implies 0 = -14, which is not possible. Therefore, the system is inconsistent.

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The distance a swimmer will travel if they swim from one corner to the opposite corner of an Olympic-size swimming pool is approximately 62.2 meters

The distance a swimmer will travel if they swim from one corner to the opposite corner of an Olympic-size swimming pool is approximately 62.2 meters. This can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the length of the pool is one side of the right triangle, the width of the pool is the other side of the right triangle, and the distance the swimmer travels is the hypotenuse.

Using the Pythagorean theorem, we can calculate the distance the swimmer travels as follows:

a² + b² = c²

where a is the length of the pool (50 meters), b is the width of the pool (25 meters), and c is the distance the swimmer travels.

To solve for c, we can plug in the values for a and b and simplify as follows:

50² + 25² = c²

500 + 625 = c²

125 = c²√3

125 = c

Approximately 62.2 meters (rounded to one decimal place)

Therefore, the distance a swimmer will travel if they swim from one corner to the opposite corner of an Olympic-size swimming pool is approximately 62.2 meters.

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The sequence of transformations that maps HIJK to ABCD is sequence B: reflection over line HÌ, then a translation 6 units to the right and 3 units down.

What is quadrilateral ?

A quadrilateral is a polygon with four sides and four vertices. The term "quadrilateral" comes from the Latin words "quadri" meaning "four" and "latus" meaning "side". Quadrilaterals can have different shapes and properties, including rectangles, squares, parallelograms, trapezoids, and kites.

Since quadrilaterals HIJK and ABCD are congruent, we can map one onto the other using a sequence of transformations. We need to determine which of the given sequences of transformations maps HIJK to ABCD.

Sequence A involves a 270° rotation about point J, followed by a translation 9 units to the right and 8 units down.

Sequence B involves a reflection over the line HÌ, followed by a translation 6 units to the right and 3 units down.

To determine which sequence maps HIJK to ABCD, we can apply each sequence to the vertices of HIJK and see if the resulting image matches the vertices of ABCD.

Starting with sequence A, we apply the rotation and translation to the vertices of HIJK:

Vertex H(-2, 1) is rotated 270° about J(-2, 2) to get H'(0, 0). Then we translate 9 units to the right and 8 units down to get H''(9, -8).

Vertex I(-3, -1) is rotated 270° about J(-2, 2) to get I'(1, -3). Then we translate 9 units to the right and 8 units down to get I''(10, -11).

Vertex J(-2, 2) is fixed by the rotation and then translated 9 units to the right and 8 units down to get J''(7, -6).

Vertex K(0, 0) is rotated 270° about J(-2, 2) to get K'(-2, -2). Then we translate 9 units to the right and 8 units down to get K''(7, -10).

The resulting image of HIJK under sequence A has vertices H''(9, -8), I''(10, -11), J''(7, -6), and K''(7, -10). We can see that these vertices do not match the vertices of ABCD, so sequence A does not map HIJK to ABCD.

Next, we apply sequence B to the vertices of HIJK:

Vertex H(-2, 1) is reflected over line HÌ to get H'(-4, 3). Then we translate 6 units to the right and 3 units down to get H''(2, 0).

Vertex I(-3, -1) is reflected over line HÌ to get I'(-1, 1). Then we translate 6 units to the right and 3 units down to get I''(5, -2).

Vertex J(-2, 2) is reflected over line HÌ to get J'(-4, 4). Then we translate 6 units to the right and 3 units down to get J''(2, 1).

Vertex K(0, 0) is reflected over line HÌ to get K'(-2, 2). Then we translate 6 units to the right and 3 units down to get K''(4, -1).

The resulting image of HIJK under sequence B has vertices H''(2, 0), I''(5, -2), J''(2, 1), and K''(4, -1). We can see that these vertices match the vertices of ABCD, so sequence B maps HIJK to ABCD.

Therefore, the sequence of transformations that maps HIJK to ABCD is sequence B: reflection over line HÌ, then a translation 6 units to the right and 3 units down.

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