On January 9, 2008, an earthquake in northern Algeria measuring 4.81 on the Richter scale killed 1 person, and on May 12, 2008, an earthquake in China measuring 7.9 on the Richter scale killed 67,180 people. How many times more intense was the earthquake in China than the one in northern Algeria? How are logarithms used in this type of problem?

The solution to this expression (√5 - 3√2)(√5 + 3√2) is equal to -13.

In Mathematics and Geometry, the standard form for a difference of two (2) squares is modeled or represented by this mathematical expression:

a² - b² = (a + b)(a - b).

Where:

a and b represent numerical values (numbers or numerals).

Based on the information provided, we have the following mathematical expression:

(√5 - 3√2)(√5 + 3√2);

By comparison, we have the following:

(√5 - 3√2)(√5 + 3√2) = (a + b)(a - b)

a² - b² = (√5)² - (3√2)²

a² - b² = 5 - 18

a² - b² = -13.

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He borrowed $24,000.

We can see here that determining whether quadrilateral ABCD is a parallelogram, we have: B. Yes; I used the Distance Formula to find that both pairs of opposite sides are congruent.

A parallelogram is a geometric shape that is defined as a quadrilateral with two pairs of parallel sides. This means that the opposite sides of a parallelogram are parallel and equal in length.

In addition, a parallelogram has several other properties. For example:

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The bottom of the ladder will be about 13.3 feet from the bottom of the building.

What is Pythagorean theorem?

The Pythagorean theorem is a fundamental concept in mathematics that relates to the lengths of the sides of a right triangle.

We can use the Pythagorean theorem to solve this problem. If we let x be the distance from the bottom of the building to the bottom of the ladder, then we can create a right triangle where the ladder is the hypotenuse, the building is one leg, and x is the other leg.

Using the Pythagorean theorem, we have:

ladder² = building² + x²

Substituting the given values, we have:

24² = 20² + x²

Simplifying, we get:

576 = 400 + x²

Subtracting 400 from both sides, we get:

176 = x²

Taking the square root of both sides, we get:

x = 13.266

Rounding to the nearest tenth, we have:

x ≈ 13.3 feet

Therefore, the bottom of the ladder will be about 13.3 feet from the bottom of the building.

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Using the properties of the adjacent angles of a parallelogram, the value of x is calculated as: x = 9.

Recall that the adjacent angles in a parallelogram are always supplementary, that is, they have a sum of 180 degrees.

Using the above fact, we can create the equation below to find the value of x:

13x + 11 + 6x - 2 = 180

13x + 6x + 11 - 2 = 180

19x + 9 = 180

19x = 180 - 9

19x = 171

Divide both sides by 19

19x/19 = 171/19

x = 9

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The probability of selecting a blue marble from the bag is 10/40 or 1/4. This is because there are 10 blue marbles out of a total of 40 marbles in the bag.

Probability is a measure of how likely an event is to occur as a fraction or percentage between 0 and 1, or 0% and 100%. It is used to describe the likelihood of an event happening, and can be used to make predictions and decisions about the future.

To calculate the probability, divide the number of blue marbles by the total number of marbles.

Therefore, the probability of selecting a blue marble from this bag is 10/40 or 1/4.

This means that out of every 4 marbles taken from the bag, one of them will be blue.

The probability of any event can be expressed as a fraction, decimal, or percentage. In this case, the probability of selecting a blue marble is 25%, or 0.25.

This means that there is a 25% chance that the marble taken from the bag will be blue.

In conclusion, the probability of selecting a blue marble from a bag containing 8 red, 10 blue, and 22 green marbles is 10/40 or 1/4, which is equivalent to 25% or 0.25.

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The type of scale represented by the question, "What is your primary mode of transportation to commute to work?" is nominal. This is because the options provided (subway, bus, car, train, other) are categorical and cannot be meaningfully ranked or assigned numerical values.

The scale represented by the question "What is your primary mode of transportation to commute to work?" is a categorical scale. Specifically, it is a nominal scale since it categorizes the modes of transportation into distinct categories with no inherent order or ranking. However, it is worth noting that the question does not represent a ratio scale since it does not measure a numerical quantity with a true zero point. It also does not represent an ordinal scale since there is no inherent order or ranking to the modes of transportation listed. Lastly, it is not an interval scale since there is no uniform or consistent measure of the differences between the modes of transportation.

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for the sake of readability, let's change "a" to "z", so for what values of "z" there's only one rational root?

well, we can look at the discriminant of a quadratic, and if the discriminant spits out a 0, or equals 0, then we have only one rational root, so let's reword that.

what values of "z", make the equation 0?

[tex](2z-5)x^2-2(z-1)x+3=0\implies (2z-5)x^2-(2z-2)x+3=0 \\\\\\ (2a-5)x^2+(2-2z)x+3=0 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ y=\stackrel{\stackrel{a}{\downarrow }}{(2z-5)}x^2\stackrel{\stackrel{b}{\downarrow }}{+(2-2z)}x\stackrel{\stackrel{c}{\downarrow }}{+3} ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex](2-2z)^2~~ - ~~4(2z-5)(3)~~ = ~~0\implies (4-8z+4z^2)-(8z-20)(3)=0 \\\\\\ (4-8z+4z^2)-(24z-60)=0\implies 4z^2-32z+64=0 \\\\\\ 4(z^2-8z+16)=0\implies z^2-8z+16=0 \\\\\\ (z-4)(z-4)=0\implies \boxed{z=4}[/tex]

Factorizing the given expressions completely, we have:

a. (x - 3y)(x - 3y - 2); b. (1 - a)(1 + a)(1 + b²).

(a) To factorize the expression x² +9y² - 6xy - 2x + 6y, we can group the terms as follows:

(x² - 2xy + 9y²) - 2x + 6y

Now, we can factor the quadratic expression inside the parentheses using the formula for the difference of squares:

(x - 3y)² - 2x + 6y

Next, we can factor out the common factor of -2 from the last two terms:

(x - 3y)² - 2(x - 3y)

Finally, we can factor out the common factor of (x - 3y) to obtain the fully factorized expression:

(x - 3y)(x - 3y - 2)

Therefore, the expression x² +9y² - 6xy - 2x + 6y is equal to (x - 3y)(x - 3y - 2).

(b) To factorize the expression 1-a²b² + b²-a², we can rearrange the terms as follows:

(1 - a²) (1 + b²)

Now, we can factor the difference of squares expression (1 - a²) using the formula:

(1 - a)(1 + a) (1 + b²)

Therefore, the expression 1-a²b² + b²-a² is equal to (1 - a)(1 + a)(1 + b²).

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Answer:

18%

Step-by-step explanation:

If you ever wondered how to write 0.18 as a percent, you are not alone. Many people struggle with this simple math problem, but don't worry, I'm here to help. The trick is to multiply 0.18 by 100 and add the percent sign. Sounds easy, right? Well, let's see how it works. First, we need to move the decimal point two places to the right to get 18. Then, we need to add the percent sign (%) to show that it is a percentage. So, 0.18 = 0.18 x 100% = 18%. Congratulations, you have just learned how to write 0.18 as a percent! Now you can impress your friends and teachers with your math skills. Just don't ask me how to write 0.18 as a fraction, because that's a whole different story.

To use a model to represent the number, we can use a grid of 100 squares and shade 18 of them. For example, the model below shows that 18% of the grid is shaded. It also shows that I have a lot of free time and a very artistic sense of humor. I could have just written 0.18 or 18/100, but where's the fun in that? This way, you can see how much 18% really is and also enjoy my beautiful masterpiece.

✧☆*: .｡. That's all folks, have fun with math! (✧ω✧) .｡.:*☆✧

The integral ∫ ∫ 3sin(25x² + 100y²) dA is evaluated as u = 5x, v = 2y: (3/10) ∫0¹ ∫[tex]0^{(1-25u)[/tex]/100 sin(u²/25 + v²/16) dv du.

This substitution transforms the given region in the first quadrant bounded by the ellipse 25x + 100y = 1 into the region R in the uv-plane bounded by the ellipse

u²/5² + v²/2² = 1

We can express the double integral in terms of u and v as:

∫∫ 3sin(25x² + 100y²) dA = ∫∫ 3sin(25(u²/25² + v²/4²)) (1/5)(1/2) dudv

= (3/10) ∫∫ sin(u²/25 + v²/16) dudv

The limits of integration for the region R in the uv-plane are:

0 ≤ u ≤ 5/5 = 1

0 ≤ v ≤ (1 - 25u)/100

Using these limits, we can evaluate the double integral as:

(3/10) ∫∫ sin(u²/25 + v²/16) dudv

= (3/10) ∫0¹ ∫[tex]0^{(1-25u)[/tex]/100 sin(u²/25 + v²/16) dv du

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To represent the final price (Px) of the items purchased, we first apply the 30% holiday sale discount and then the 15% off coupon. Let's find the correct function:

1. Apply the 30% holiday sale: Price becomes 0.7x (as 100% - 30% = 70%)
2. Apply the 15% off coupon: Final price becomes 0.85 * 0.7x (as 100% - 15% = 85%)

Now, multiply 0.85 by 0.7, which equals 0.595.

Therefore, the function representing the final price is Px(x) = 0.595x.

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Answer: B) as x→-∞, f(x) →∞ as x→∞, f(x) →-∞

Step-by-step explanation:

Answer:

x=20

Step-by-step explanation:

3x+20=80

3x=80-20

3x=60

x=60/3

x=20

To solve for 'x' to make 'A||B', understand the rules that make lines parallel which mostly depend on the equality of corresponding or interior angles. Applying these rules, find a solution for x that makes these angles equal.

In the context of this mathematics question, when you are asked to solve for 'x' to make 'A||B', it implies that there are lines A and B in a geometric scenario that are to be made parallel (A||B refers to A is parallel to B). Parallel lines are lines in the same plane that never intersect and always maintain an equivalent distance apart.

Normally, the unknown 'x' would be part of an angle or line segment's measurement in this geometric scenario. Two lines are parallel if their corresponding angles are equal or their alternate interior angles are equal.

For instance, consider lines A and B cut by a transversal. If angle a on line A equals angle b on line B (or a corresponding angle), then lines A and B are parallel; therefore, x in this case is the value that makes angle a equal to angle b.

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There is a 9.16% probability that exactly seven claims will be filed in the next week.

There is a 0.0123% probability that no claims will be filed in the next week.

There is a 5.73% probability that less than four claims will be filed in the next week.

In this scenario, the insurance company has determined that the average number of claims filed in their Atlanta branch each week is nine. To calculate the probability of specific outcomes for the next week, we can use the Poisson distribution formula, which is commonly used to model the probability of rare events occurring over time.

The formula is:

P(x) = [tex](e^-\lambda \times \lambda ^x)[/tex] / x!

Where:

P(x) = the probability of x events occurring

e = the mathematical constant approximately equal to 2.71828

λ = the average number of events that occur in a given time period (in this case, nine claims per week)

x = the number of events we are interested in

Using this formula, we can calculate the probability of the following outcomes for the next week:

Exactly seven claims will be filed:

P(7) = [tex](e^{-9} \times 9^7)[/tex] / 7! = 0.0916 or approximately 9.16%

No claims will be filed:

P(0) = [tex](e^{-9} \times 9^0)[/tex] / 0! = 0.000123 or approximately 0.0123%

Less than four claims will be filed:

P(0) + P(1) + P(2) + P(3) = [tex](e^{-9} \times 9^0) / 0! + (e^{-9} \times 9^1) / 1! + (e^{-9} \times 9^2) / 2! + (e^{-9} \times 9^3) / 3![/tex] = 0.0573 or approximately 5.73%

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The following can be answered by the concept of Trigonometry.

To convert degrees to radians, you need to use the formula:

radians = (degrees x pi)/180

1. 225 degrees = (225 x pi)/180 = (5/4)pi radians
2. 20 degrees = (20 x pi)/180 = (1/9)pi radians
3. -255 degrees = (-255 x pi)/180 = (-17/12)pi radians
4. -140 degrees = (-140 x pi)/180 = (-7/9)pi radians
5. 75 degrees = (75 x pi)/180 = (5/12)pi radians
6. -300 degrees = (-300 x pi)/180 = (-5/3)pi radians

To convert radians to degrees, you need to use the formula:

degrees = (radians x 180)/pi

7. 23x/128 radians = (23x/128 x 180)/pi degrees
8. -31x/369 radians = (-31x/369 x 180)/pi degrees
9. x/1210 radians = (x/1210 x 180)/pi degrees
10. 5x/911 radians = (5x/911 x 180)/pi degrees
11. -x/-212 radians = (-x/-212 x 180)/pi degrees
12. -7x/-613 radians = (-7x/-613 x 180)/pi degrees
13. 115014 radians = (115014 x 180)/pi degrees
14. 350015 radians = (350015 x 180)/pi degrees
15. -40016 radians = (-40016 x 180)/pi degrees
16. -4600 radians = (-4600 x 180)/pi degrees

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a. The number of ways to select a man or a girl is 17.

b. The number of ways to select one person is 25.

c. The number of ways to select a group of 2 men and 2 women is 1170.

d. The number of ways for the children to sit in a row, if the two boys are to sit together, is 239500800.

a. To select a man or a girl, we can add the number of ways to select a man and the number of ways to select a girl, and then subtract the cases where we select both a man and a girl at the same time (as we want to count only one person). The number of ways to select a man is 13 and the number of ways to select a girl is 4, so the total number of ways to select a man or a girl is

13 + 4 - 0 = 17

b. To select one person from the group, we can add the number of ways to select a man, a woman, a boy, or a girl. The number of ways to select a man is 13, the number of ways to select a woman is 6, the number of ways to select a boy is 2, and the number of ways to select a girl is 4, so the total number of ways to select one person is

13 + 6 + 2 + 4 = 25

c. To select a group of 2 men and 2 women, we can use the combination formula

C(13, 2) × C(6, 2) = 78 × 15 = 1170

Here, C(13, 2) is the number of ways to select 2 men from a group of 13 men, and C(6, 2) is the number of ways to select 2 women from a group of 6 women. We multiply these two values to get the total number of ways to select a group of 2 men and 2 women.

d. We can treat the two boys who sit together as a single entity. Then we have 11 people (13 men and 4 girls) and 1 entity (the two boys) to arrange in a row. The number of ways to arrange these 12 entities in a row is

12! = 479001600

However, we have overcounted the arrangements where the two boys switch positions. Since the two boys can be arranged in 2! = 2 ways, we need to divide the total number of arrangements by 2 to get the number of distinct arrangements

479001600 / 2 = 239500800

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So the equation of the normal line to the Surface z=6x^3y^4 at the point (-1,-2,-96) is:

x = -1 + 324t

y = -2 - 216t

z = -96 - t

To find the equation of the normal line to the surface z=6x^3y^4 at the point (-1,-2,-96), we need to find the gradient of the surface at that point, which is a vector perpendicular to the tangent plane at that point.

The gradient vector of the surface f(x,y,z)=6x^3y^4 is:

∇f = ( ∂f/∂x, ∂f/∂y, ∂f/∂z )

= ( 18x^2y^4, 24x^3y^3, -1 )

Substituting the coordinates of the given point (-1,-2,-96) into the gradient vector, we get:

∇f(-1,-2,-96) = ( 324, -216, -1 )

So the gradient vector at the given point is (324,-216,-1).

The equation of the normal line passing through the point (-1,-2,-96) can be written in vector form as:

r(t) = r0 + tN

where r0 is the position vector of the given point (-1,-2,-96), N is the normal vector to the surface at that point, and t is a scalar parameter.

Substituting the values, we get:

r(t) = <-1,-2,-96> + t<324,-216,-1>

= <-1+324t, -2-216t, -96-t>

So the equation of the normal line to the surface z=6x^3y^4 at the point (-1,-2,-96) is:

x = -1 + 324t

y = -2 - 216t

z = -96 - t

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The upper bound of the 99% confidence interval for the difference in population construction times between people listening to classical music as they work and people working in silence is 18.31 minutes.

To find the confidence interval, we first need to calculate the standard error of the difference between the means. This can be done using the formula:

standard error = sqrt((s₁²/n₁) + (s₂²/n₂))

where s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and sqrt represents the square root function.

Plugging in the given values, we get:

standard error = sqrt((9.5²/55) + (11²/45)) = 2.872

Next, we need to find the z-score corresponding to a 99% confidence level. Using a standard normal distribution table, we find that the z-score is 2.576.

Finally, we can calculate the confidence interval using the formula:

confidence interval = (X₁ - X₂) ± zc × standard error

where X₁ and X₂ are the sample means, and zc is the z-score corresponding to the desired confidence level.

Plugging in the given values, we get:

confidence interval = (45 - 55) ± 2.576 × 2.872 = -10 ± 7.39

Since we are only interested in the upper bound of the confidence interval, we take the positive value:

upper bound = -10 + 7.39 = 18.31

Therefore, we can be 99% confident that the true difference in population construction times between people listening to classical music and people working in silence is no more than 18.31 minutes.

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Probability that a customer ordered a hot drink given that he or she ordered a large is 29.33 %

Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are.

From the table, we have:

n(hot and large) = 22

n(hot and small) = 5

n(hot and medium) = 48

n(hot) = 22 + 5 + 48 = 75

So, the probability P(hot | large) is calculated as:

P(hot | large) = 22/75

P( hot | large) = 0.2933

Round to the nearest hundredth

P(hot | large) = 29.33 %

Probability that a customer ordered a hot drink given that he or she ordered a large is 29.33 %

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Based on the stated expenditure and sales tax information, the Mrs. R's meal cost around $14.8.

Let us assume Mr. Rodriguez's meal cost $x. So, the percentage of the sum of their meal cost will be the sales tax.

Thus, representing the equation -

(6.25 + x + 9.98 ) 7.25% = 2.25

Rewriting the equation with percentage

(6.25 + x + 9.98 ) = 2.25×100/7.25

Performing multiplication and division on Right Hand Side

(6.25 + x + 9.98 ) = 31.03

Adding the digits on Left Hand Side

16.23 + x = 31.03

Rewriting the equation in terms of x

x = 31.03 - 16.23

Performing subtraction

x = $14.8

Hence, the cost of Mrs. R's meal is $14.8.

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The vertex form of a quadratic function is given by:

�

(

�

)

=

�

(

�

−

ℎ

)

2

+

�

f(x)=a(x−h)

2

+k

where $(h, k)$ is the vertex of the parabola.

Substituting the given vertex $\left(-2,\ -6\right)$, we get:

�

(

�

)

=

�

(

�

−

(

−

2

)

)

2

−

6

f(x)=a(x−(−2))

2

−6

Simplifying:

�

(

�

)

=

�

(

�

+

2

)

2

−

6

f(x)=a(x+2)

2

−6

To find the value of $a$, we use the fact that the function passes through the point $\left(-1,\ 2\right)$:

2

=

�

(

−

1

+

2

)

2

−

6

2=a(−1+2)

2

−6

Solving for $a$:

\begin{align*}

2 + 6 &= a(1)^2 \

a &= 8

\end{align*}

Therefore, the quadratic function in vertex form that satisfies the given conditions is:

�

(

�

)

=

8

(

�

+

2

)

2

−

6

f(x)=8(x+2)

2

−6

The linear transformation t from R2 to R2 that rotates any vector through an angle of 135 degrees in the counterclockwise direction is: a = | -1/sqrt(2)  1/sqrt(2) | ,  | 1/sqrt(2)   -1/sqrt(2) |

To find the matrix of the linear transformation that rotates any vector through an angle of 135 degrees counterclockwise, we first need to know the standard matrix for a rotation of theta degrees. This matrix is given by:
cos(theta)  -sin(theta) ,sin(theta)   cos(theta)

For our specific transformation with theta = 135 degrees, we have:cos(135)  -sin(135),sin(135)   cos(135).Using trigonometric identities, we can simplify this to: (-sqrt(2)/2)  -(-sqrt(2)/2),(sqrt(2)/2)   (-sqrt(2)/2). Simplifying further, we get: -1/sqrt(2)  1/sqrt(2),1/sqrt(2)   -1/sqrt(2)

Therefore, the matrix a of the linear transformation t from R2 to R2 that rotates any vector through an angle of 135 degrees in the counterclockwise direction is: a = | -1/sqrt(2)  1/sqrt(2) | ,  | 1/sqrt(2)   -1/sqrt(2) |

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Answer:

x=2

Step-by-step explanation:

3x+4=10 (Carry the 4 to the other side)

3x=6 (Divide both sides by 3)

x=2

Hello !

Answer:

[tex]\Large \boxed{\sf x=2}[/tex]

Step-by-step explanation:

We want to find the value of x that verifies the following equation :

[tex]\sf 3x + 4 = 10[/tex]

Let's isolate x !

First, substract 4 from both sides :

[tex]\sf 3x+4-4=10-4\\3x=6[/tex]

Now divide both sides by 3 :

[tex]\sf\frac{3x}{3}=\frac{6}{3} \\ \boxed{\sf x=2}[/tex]

Have a nice day ;)

If a function f has an inverse and f(-3) = 6, then we can conclude that

[tex]f^-1(6) = -3[/tex]

which means that if we plug in 6 as the input value to the inverse function

[tex]f^-1[/tex]

we get -3 as the output value.

If a function f has an inverse, it means that for every output value of f, there exists a unique input value that produces that output value. The inverse function of f, denoted as

[tex]f^-1[/tex]

takes an output value of f and returns the input value that produced it.

Given that f(-3) = 6, we know that the input value -3 produced an output value of 6. We can say that

[tex]f^-1(6) = -3[/tex]

This means that if we plug in 6 as the input value to the inverse function

[tex]f^-1[/tex]

we get -3 as the output value.

To see why this is true, we can think of it in terms of reversing the process of the original function f. If we start with an input value of -3, the function f produces an output value of 6. If we then take this output value of 6 and plug it into the inverse function

[tex]f^-1[/tex]

we should get back our original input value of -3.

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The area of the region common to the interiors of the cardioids r = 1 + cosθ and r = 1 - cosθ is (π + 2)/4.

We can find the area of the region common to the interiors of the cardioids by setting the two equations equal to each other and solving for θ. This will give us the values of θ that define the boundaries of the region. We can then integrate the area enclosed between the two curves over those boundaries.

1 + cosθ = 1 - cosθ

2cosθ = 0

cosθ = 0

θ = π/2, 3π/2

So the boundaries of the region are θ = π/2 and θ = 3π/2.

To find the area of the region, we can integrate the area enclosed between the two curves over these boundaries:

A = 2 ∫[0, π/2] (1 + cosθ) / 2 × dθ

Note that we multiply by 2 to account for the area in the other half of the region, between θ = 3π/2 and θ = 2π, which is symmetric.

Simplifying the integrand:

A = ∫[0, π/2] (1 + cosθ) / 2 × dθ

= (1/2) ∫[0, π/2] (1 + cosθ) dθ

= (1/2) (∫[0, π/2] dθ + ∫[0, π/2] cosθ dθ)

Evaluating the integrals:

A = (1/2) (θ ∣[0, π/2] + sinθ ∣[0, π/2])

= (1/2) (π/2 + sin(π/2) - 0 - sin(0))

= (1/2) (π/2 + 1)

= (π + 2) / 4

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An overall loss of number of points is - [tex]11\frac{13}{35}[/tex] points.

Given that, last week, by Wednesday Hopt LLC's stock had decreased [tex]5\frac{4}{5}[/tex]   points. By Friday it was down an additional [tex]5\frac{2}{7}[/tex] points.

Here, [tex]5\frac{4}{5}[/tex] = 29/5 and [tex]5\frac{2}{7}[/tex] = 39/7

Since the points decreased on Wednesday and Friday, we get

-29/5 -  39/7

= -203/35 - 195/35

= -398/35

= - [tex]11\frac{13}{35}[/tex] points

Therefore, an overall loss of number of points is - [tex]11\frac{13}{35}[/tex] points.

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The linearization of the function [tex]f(x,y) = 2x^2 + y^2 - 5y[/tex] at the point (1,2) is L(x,y) = 4x - y - 5. This linearization is an approximation of the original function near the point (1,2) that is valid for small values of x and y.

To find the linearization of a function, we need to use the formula [tex]L(x, y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)[/tex], where [tex]f_x(a,b)[/tex] and[tex]f_y(a,b)[/tex] are the partial derivatives of the function at the point (a,b).

In this case, our function is [tex]z = f(x,y) = 2x^2 + y^2 - 5y,[/tex] and we want to find the linearization at the point (1,2). So we need to first find the partial derivatives of f with respect to x and y:  [tex]f_x(x,y) = 4x f_y(x,y) = 2y - 5[/tex]

Next, we can plug in our point (1,2) into these partial derivatives to get their values at that point:
[tex]f_x(1,2) = 4(1) = 4 f_y(1,2) = 2(2) - 5 = -1[/tex]. Now we can plug all of these values into the formula for the linearization:  [tex]L(x,y) = f(1,2) + f_x(1,2)(x-1) + f_y(1,2)(y-2) L(x,y) = (2)(1)^2 + (2)^2 - 5(2) + 4(x-1) - 1(y-2) L(x,y) = 4x - y - 5[/tex]

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The number of hundreds which is equal to the sum of the tens and ones is 661 according to the given information.

Let's break down the clues:

The number is greater than 600 and less than 1000, so it must be in the range of 601 to 999.

Since there are four 1's, it must end with four 1's. So the last digit is 1.

The number of 10s is 3 less than 1, so in case, the number of 1s is x, the number of 10s is x-3. 

The hundredth place is equal to the sum of 10 and 1, so the hundredth place is 2x-3. Putting all this data together, we are able to compose a number in terms of x.

number = [tex](2x-3) * 100 + (x-3) * 10 + 1[/tex]

Since x has as it were 10 digits (0 to 9), we know it must be between 9. Let's try some values ​​of x:

For x = 4, the number is[tex](2*4-3)*100 + (4-3)*10 + 1 = 841.[/tex] There are no four ones in this.

If x = 5, the number is [tex](2*5-3)*100 + (5-3)*10 + 1 = 951[/tex]. There are no four ones in this either. For x = 6, the number is (2*6-3)*100 + (6-3)10 + 1 = 661.

For greater than 600 and less than 1000, there are four 1's. The tens place is 6-3=3 and the hundreds place is 26-3=9, which is the sum of tens and ones.

therefore, the number is 661.  

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