There are five nickels, five dimes and five quarters in your pocket. You randomly pick three coins and place them on the counter the first coin is a nickel, the second is a dime and the third is a quarter

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

[tex]Time=\frac{Distance}{Speed}[/tex]

Time took by James for climbing = [tex]\frac{2}{3}\ hours[/tex]

Time took for running back = [tex]\frac{2}{4}\ hours[/tex]

Total time = [tex]\frac{2}{3}+\frac{2}{4}=\frac{8+6}{12}=\frac{14}{12}[/tex]

Total time taken by James = [tex]\frac{7}{6}\ hours[/tex]

1 hour = 60

Total time taken by James = [tex]\frac{7}{6}*60=70\ minutes[/tex]

Time took by Lucas for climbing = [tex]\frac{2}{2}=\ 1\ hour[/tex]

Time took by Lucas for climbing = 60 minutes

Time took on return = [tex]\frac{2}{5} of\ an\ hour=\frac{2}{5}*60=24\ minutes[/tex]

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Keywords: distance, speed

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Answer:

Step-by-step explanation:

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

Time took by James for climbing =

Time took for running back =

Total time =

Total time taken by James =

1 hour = 60

Total time taken by James =

Time took by Lucas for climbing =

Time took by Lucas for climbing = 60 minutes

Time took on return =

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.®

Answer:

The least possible value is -2, obtained for a = -1 and b = 1.

Step-by-step explanation:

We want the minimum of [tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) [/tex] , where

First, lets simplify the expression given, we use common denominator on the second part, using ab as the common denominator. We obtain

[tex] \frac{1}{b} - \frac{1}{a} = \frac{a-b}{ab} [/tex]

As a result

[tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) = \frac{a-b}{(ab)^2} [/tex]

we need the minimum of the function

[tex] f(a,b) = \frac{a-b}{(ab)^2} [/tex]

with the restrictions [tex] -5 \leq a \ -1, 1 \leq b \leq 3 [/tex]

First, we calculate the gradient of f and find where it takes the zero value.

[tex]\nabla{f} = (f_a,f_b) [/tex]

with

[tex]f_a = \frac{(ab)^2 - (a-b) 2ab^2}{(ab)^2} = -1 + 2 \frac{b}{a}[/tex]

Since it has the reversed sign, we get

[tex]f_b = - (-1 + 2 \frac{a}{b}) =1 - 2 \frac{a}{b}[/tex]

In order for [tex] \nabla{f} [/tex] to be zero, we need both [tex] f_a [/tex] and [tex] f_b [/tex] to be zero, observe that

[tex]f_b = 0 \, \rightarrow 1 -2\frac{a}{b} = 0 \, \rightarrow 1 = 2 \frac{a}{b} \, \rightarrow b = 2a [/tex]

Which is impossible with the given restrictions. Hence, the minimum is realized in the border.

If we fix a value a₀ for a, with a₀ between -5 and -1 the function g(b) = f(a₀,b) wont have a minimum for b in [1,3] because the partial derivate of f over b didnt reach the value 0 in the restrictions given. On the other hand. by making a similar computation that before, we can obtain that the partial derivate of f over the variable a doesnt reach the value 0 either. This means that f doesnt reach the minimum on the sides. As a consecuence, it reach a minimum on the corners.

The 4 possible corner values are (-5,1), (-5,3), (-1,1) and (-1,3)

[tex] f(-5,1) = \frac{-6}{25} =  -0.24 [/tex]

[tex] f(-5,3) = \frac{-8}{225} = -0.0355 [/tex]

[tex] f(-1,1) = \frac{-2}{1} = -2 [/tex]

[tex] f(-1,3) = \frac{-4}{9} = -0.444 [/tex]

Clearly the least possible value between the four corners is -2.

Answer: he will need 28 cups of dough for 3 cups of chocolate.

Step-by-step explanation:

The baker uses 2 1/3 cups of cookie dough and 1/4 cup of chocolate chips to make 10 cookies. Converting 2 1/3 cups of cookie dough into improper fraction, it becomes 7/3 cups of cookie dough.

It means that for every 1/4 cup of chocolate, 7/3 cups of cookie dough is needed.

Let x represent the amount of cookie dough needed for 3 cups of chocolate chips.

If 7/3 dough = 1/4 cup of chocolate

x dough = 3 cups of chocolate

x × 1/4 =7/3 × 3

x/4 = 7

x = 7×4 = 28 cups of dough

The probability of selecting a black card or a jack is 15/26.

Given that, one card is selected from a deck of cards.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes

Total number of outcomes =52

The number of black cards in a deck =26

The number of jack cards in a deck =4

Probability of an event = 26/52 +4/52

= 30/52

= 15/26

Therefore, the probability of selecting a black card or a jack is 15/26.

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Final answer:

The probability of selecting a black card or a jack from a standard deck of 52 cards is 7/13.

Explanation:

To find the probability of selecting a black card or a jack from a standard deck of 52 cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.

In a deck, there are 26 black cards (13 clubs and 13 spades) and a total of 4 jacks.

However, since two of the jacks are black, we must avoid counting them twice.

The probability becomes:

P(Black or Jack) = P(Black) + P(Jack) - P(Black and Jack)

P(Black) = 26/52, P(Jack) = 4/52, and P(Black and Jack) = 2/52

Thus, the probability is:

P(Black or Jack) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13

Therefore, the probability of selecting a black card or a jack from a standard deck is 7/13.

Answer:

216 sq. units

Step-by-step explanation:

From the figure, we can see that only one pair of opposite angles are equal. So, the quadrilateral is a kite.

Formula to find the area of a kite:

Area, A = [tex]$ \frac{1}{2} \times d_1 \times d_2 $[/tex]

where, [tex]$ d_1 $[/tex] and [tex]$ d_2 $[/tex] are the lengths of the diagonals.

Here, [tex]$d_1 = 18 $[/tex] units.

And, [tex]$ d_2 = 24 $[/tex] units.

Therefore, the area A = [tex]$ \frac{1}{2} \times 18 \times 24 $[/tex]

= [tex]$ \frac{432}{2} $[/tex]

= 216 sq. units which is the required answer.

Answer:

2 hours

Step-by-step explanation:

45x + 40x = 170

85x = 170

x = 2

Answer:

40 mph: 4 hours 15 minutes

45 mph: 3 hours 46 minutes 40 seconds

Step-by-step explanation:

The rate of separation = 45 + 40 = 85 mph

speed = distance / time

85 = 170 / t

t= 170/85 hour

Answer:

Price of  hamburgers = $2

Price of white-dogs = $3

Step-by-step explanation:

Let

The number price of hamburgers be x

The number price of white-dogs  be  y

Then the first family buys 3 hamburgers and 2 white -dogs for $13

3x + 2y = 13-----------------------------(1)

Another first family buys 2 hamburgers and 5 white -dogs for $16

2x + 5y = 16----------------------------(2)

To solve let us multiply eq(1) by 2 and eq(2) by 3, we get

6x + 4y = 26-----------------------------(3)

6x + 15y = 48----------------------------(4)

subracting (3) from (4)

6x + 15y = 48

6x + 4y = 26

(-)

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

        11y  = 22

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

[tex]y =\frac{22}{11}[/tex]

y =2

Now substituting the value of y in eq(1),

3x + 2(2) = 13

3x + 4 = 13

3x = 13-4

3x = 9

x =[tex]\frac{9}{3}[/tex]

x =3

Answer:

The common difference of given AP is Option 2) -10.

Step-by-step explanation:

We are given the following information in the question:

First term of AP, a  = 100

The sum of its first 6 terms = 5(the sum of its next six terms)

We have to find the common difference of AP.

The sum of n terms of AP is given by:

[tex]S_n = \dfrac{n}{2}\big(2a + (n-1)d\big)[/tex]

where a is the first term and d is the common difference.

Thus, we can write:

[tex]S_6 = 5 (S_{12}-S_6)\\\dfrac{6}{2}\big(200 + (6-1)d\big) = 5\bigg(\dfrac{12}{2}\big(200 + (12-1)d\big)-\dfrac{6}{2}\big(200 + (6-1)d\big)\bigg)\\\\600 + 15d =5(1200+66d-600-15d)\\600+15d=3000+255d\\2400 = -240d\\d = -10[/tex]

Thus, the common difference of given AP is -10.

Answer:

The function

{\ displaystyle f (z) = {\ frac {z} {1- | z | ^ {2}}}} {\ displaystyle f (z) = {\ frac {z} {1- | z | 2}

It is an example of real and bijective analytical function from the open drive disk to the Euclidean plane, its inverse is also an analytical function. Considered as a real two-dimensional analytical variety, the open drive disk is therefore isomorphic to the complete plane. In particular, the open drive disk is homeomorphic to the complete plan.

However, there is no bijective compliant application between the drive disk and the plane. Considered as the Riemann surface, the drive disk is therefore different from the complex plane.

There are bijective conforming applications between the open disk drive and the upper semiplane and therefore determined as Riemann surfaces, are isomorphic (in fact "biholomorphic" or "conformingly equivalent"). Much more in general, Riemann's theorem on applications states that the entire open set and simply connection of the complex plane that is different from the whole complex plane admits a bijective compliant application with the open drive disk. A bijective compliant application between the drive disk and the upper half plane is the Möbius transformation:

{\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}} {\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}}

which is the inverse of the transformation of Cayley.

if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

To understand why a power series diverges on the unit circle when the analytic function it represents has a pole on the unit circle, we can use the concept of analytic continuation and the properties of poles and singularities.

Here's a step-by-step explanation:

1.Analytic function on the unit disk: Let's consider an analytic function defined on the open unit disk, denoted by[tex]\(D = \{z \in \mathbb{C} : |z| < 1\}\)[/tex]. This means the function is holomorphic (complex differentiable) at every point within this disk.

2.Power series representation: Since the function is analytic on[tex]\(D\)[/tex], it can be represented by a power series expansion around any point [tex]\(z_0\) in \(D\)[/tex]. Let's denote this function by[tex]\(f(z)\)[/tex], and its power series representation centered at[tex]\(z_0\) by \(\sum_{n=0}^{\infty} a_n (z - z_0)^n\)[/tex].

3.Pole on the unit circle: Suppose[tex]\(f(z)\)[/tex] has a pole (a point where the function becomes unbounded) on the unit circle[tex]\(|z| = 1\)[/tex], i.e., there exists a point [tex]\(z_1\)[/tex] on the unit circle such that [tex]\(f(z_1)\)[/tex] is infinite. Without loss of generality, let's assume [tex]\(z_1 = 1\)[/tex] (since the unit circle is symmetric about the origin).

4.Behavior near the pole: Near the pole at [tex]\(z = 1\)[/tex], the function[tex]\(f(z)\)[/tex]can be expanded in a Laurent series, which includes negative powers of [tex]\((z - 1)\)[/tex]. This expansion will have infinitely many terms with negative powers, indicating the singularity at [tex]\(z = 1\)[/tex].

5.Radius of convergence: The radius of convergence of the power series representation of [tex]\(f(z)\)[/tex]is at least the distance from the center of convergence to the nearest singularity. In this case, since the singularity (pole) is on the unit circle, the radius of convergence of the power series cannot exceed 1.

6.Divergence on the unit circle: Since the radius of convergence of the power series representation of [tex]\(f(z)\)[/tex] is at most 1, the power series diverges at every point on the unit circle (except possibly at the point of singularity itself, where it may converge by definition). This divergence occurs because the function has a singularity (pole) on the unit circle.

Therefore, if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

Answer:

There were 22 men in the sight-seeing tour

Step-by-step explanation:

If the number f women on the sight-seeing tour was less than 30, the closest number that can be divided by 5 is 25, so we suppose that there were 25 women, so with the ratio women(5): children(2) we can apply a rule of three, so we multiply 25*2 divided by 2, now we know that there were 10 children, now with the ratio of men(11): children(5) we apply another rule of three, and multiply 10*11 and then divide it by 5, and now we know that there were 22 men in the sight-seeing tour.

Answer:

256

Step-by-step explanation:

a₁ = 2

aₙ = (aₙ₋₁)²

a₂ = (a₂₋₁)²

a₂ = (a₁)²

a₂ = (2)²

a₂ = 4

a₃ = (a₃₋₁)²

a₃ = (a₂)²

a₃ = (4)²

a₃ = 16

a₄ = (a₄₋₁)²

a₄ = (a₃)²

a₄ = (16)²

a₄ = 256

Answer:

The data file is included in attached excel file.

Step-by-step explanation:

There are seven variables in the data file for four persons interviewed. Person type, marital status, ethnic group and gender are categorical variables while age, number of pets and GPA are quantitative variables. Person type is classified as A,B,C and D. Marital status consists of category married and single. Age of persons lies between 20-50. Ethic group has three categories that are white, Asian and Hispanic. Gender of persons has two categories male and female. Number of pets for the persons interviewed lies in the range of 2 to 5. Last GPA variable ranges from 2.2 to 3.8.

Answer:

Step-by-step explanation:

 102+202+302+402+502+602+702+802+902(4518)

+112+212+312+...+ 812+912(4608)

+122+222+322+...+822+922(4698)

+132+232+332+...+932(4788)

..........................................

+192+292+392+...+992(5328)

4518+4608+4698+...+5328

n=10

[tex]s=\frac{10}{2}(4518+5328)\\=5(9846)\\=49230[/tex]

Final answer:

To find the sum of all positive 3-digit numbers ending in 2, we calculate the total for each digit's place and sum them up, resulting in a total sum of 8280.

Explanation:

The problem requires finding the sum of all positive 3-digit numbers with a last digit of 2. To calculate this, we can identify that the first such number is 102 and the last is 992. There are 90 such numbers because they correspond to the tens digit going from 0 to 9 for each of the nine possible hundreds digits (1-9).

Since each number ends in 2, we can think of them as (100x + 10y + 2), where x is the hundreds digit (1 through 9) and y is the tens digit (0 through 9). To find the sum, we calculate the sum of the hundreds digits times their frequency, the sum of the tens digits times their frequency, and add 2 times the number of terms (90). The formula would be:

Sum = (Sum of hundreds values) * 10 * 9 + (Sum of tens values) * 1 * 90 + 2 * 90

The hundreds values are 1 through 9, whose sum is 45, and the tens values are 0 through 9, whose sum is 45 as well. Plugging these values into the formula, we get:

Sum = 45 * 10 * 9 + 45 * 1 * 90 + 2 * 90 = 4050 + 4050 + 180 = 8280.

Answer:

Step-by-step explanation:

Start with the unknown, which is the number of hours J worked and the number of hours A worked.  If A worked twice as many hours as J, then J worked x hours and A worked 2x hours.  If J can iron 25 shirts per hour, x, then the number of shirts he can iron in his shift is 25x.  If A can iron 35 shirts per hour, x, then the number of shirts he can iron in his shift is 35(2x).  The number of shirts they iron together in x hours is

25x + 35(2x) = 380 and

25x + 70x = 380 and

95x = 380 so

x = 4

This means that J worked 4 hours and A worked 8 hours.

Answer:

  b.  is $34 per ounce

Step-by-step explanation:

If the production cost were less, a competitor would drive the price down. If the production cost were more, the supplier would go out of business.

Since we're at equilibrium, the production cost must be equal to $34 per ounce.

Answer:

[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]

Step-by-step explanation:

Let's start with the original function.

[tex]y=a sin\frac{2\pi t}{T}[/tex]

We can immediately fill in the amplitude 'a' and period 'T' , as the question defines these for us, and provides values for 'a' and 'T', 4 and 4[tex]\pi[/tex] respectively.

[tex]y=4sin(\frac{2\pi t}{4\pi } )[/tex]

Now we only have phase shift and vertical shift to do. Vertical shift is very easy, you can just add it to the end of the right side of the expression. A positive value will shift the graph up, while a negative value will move shift the graph down. We have '-2' as our value for vertical shift, so we can add that on as so:

[tex]y=4sin(\frac{2\pit }{4\pi } )-2[/tex]

Now phase shift the most complicated of the transformations. Basically, it is just movement left or right. A negative phase shift moves the graph right, a positive phase shift moves the graph left (I know, confusing!). Phase shift applies directly to the x variable, or in this case the t variable. To achieve a -4/3 pi phase shift, we need to input +4/3 pi into the function, because of the aforementioned negative positive rule. Here is what the function looks like with the correct phase shift:

[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]

This function has vertical shift -2, phase shift -4/3 [tex]\pi[/tex], amplitude 4, and period 4[tex]\pi[/tex].

Desmos.com/calculator is a great tool for learning about how various parts of an equation affect the graph of the function, If you want you can input each step of this problem into desmos and watch the graph change to match the criteria.

Answer:

Step-by-step explanation:

Initial amount that was deposited into the savings account is $600 This means that the principal,

P = 600

The account earns 2.1 % interest compounded annually.. This means that it was compounded once in a year. So

n = 1

The rate at which the principal was compounded is 2.1%. So

r = 2.1/100 = 0.021

It was compounded for t years. So

t = t

a) The formula for compound interest is

A = P(1+r/n)^nt

A = total amount in the account at the end of t years. Therefore

A = 600 (1+0.021/1)^1×t

A = 600(1.021)^t

b)when A =$800, it becomes

800 = 600(1.021)^t

Dividing both sides by 600, it becomes

1.33 = (1.021)^t

Taking the tth root of both sides

t = 14 years

It will take 14 years

Why is this the answer?

The x value x = 4 shows up in the point (4, 3), which is in the given function set. Adding (4, -3) to this set will have x = 4 show up twice. We cannot have one x value pair up with more than one y value. In other words, any input cannot map to more than one output. Visually, the two points (4,3) and (4,-3) will fail the vertical line test, which means we wouldnt have a function.

Answer:

It's D; degree- 5; leading coefficient-10 :)

Step-by-step explanation:

I just took the test and got it right

The given polynomial function f(x) = 10x⁵ + 7x⁴ + 5 has Degree: 5; leading coefficient: 10 which is the correct answer would be option (D).

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.

The degree of a polynomial is the highest exponent of the variables in the polynomial. In this case, the highest exponent is 5, so the degree of the polynomial is 5.

The leading coefficient is the coefficient of the term with the highest degree.

In this case, the coefficient of the term with the highest degree (x⁵) is 10, so the leading coefficient is 10. Therefore, the correct answer is D).

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Answer:

B) Mean

Step-by-step explanation:

Standard Deviation is the measure of the amount of variation or dispersion of a set of values. Standard Deviation is represented by lower case Greek alphabet sigma σ. Standard Deviation is the square root of variance. Variance is the average of the squared differences or variation or dispersion from the Mean. Therefore, to compute standard deviation, the mean of the given data must be known.

    Standard Deviation, σ =  [tex]\sqrt{Variance}[/tex]

                     σ = [tex]\sqrt({\frac{1}{N-1}}[/tex] ∑[tex]_{i = 1}^{N}(x_{i} - X)^{2})[/tex]

where

[tex]x_{1},x_{2},x_{3}, ...x_{N},[/tex] are the values of the sample observed,

X is the mean value of these observations, and

N is the number of observations in the sample.

The correct equilibrium at one atmosphere pressure associated with its Kelvin temperature is the steam-water equilibrium at 373 K.

The equilibrium at one atmosphere pressure that is correctly associated with the Kelvin temperature at which it occurs is option d. steam-water equilibrium at 373 K. To explain, in the Kelvin temperature scale, the freezing point of water is 273.15 K and the boiling point is 373.15 K, both under standard atmospheric conditions (1 atmosphere pressure). So, at 373 K, the situation would be a steam-water equilibrium, not an ice-water equilibrium as in options a and b. The Kelvin temperature for ice-water equilibrium is 273.15 K and not 0 K and 32 K as stated in options a or b. Similarly, steam-water equilibrium does not occur at 212 K as suggested in option c.

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Option d. steam-water equilibrium at 373 K. The steam-water equilibrium at one atmosphere pressure occurs at 373 K.

The correct answer is:

Let's break down the reasons:

Thus, the correct association is the steam-water equilibrium occurring at 373 K at 1 atmosphere pressure.

Answer:

Option C.

Step-by-step explanation:

A dashed straight line has a negative slope and goes through (0, 2) and (4, 0).

The given inequality is

[tex]y<-\dfrac{1}{2}x+2[/tex]

We need find the point which is a solution to the given linear inequality.

Check the given inequality for point (2, 3).

[tex]3<-\dfrac{1}{2}(2)+2[/tex]

[tex]3<1[/tex]    

This statement is false. Option 1 is incorrect.

Check the given inequality for point (2, 1).

[tex]1<-\dfrac{1}{2}(2)+2[/tex]

[tex]1<1[/tex]

This statement is false. Option 2 is incorrect.

Check the given inequality for point (3, -2).

[tex]-2<-\dfrac{1}{2}(3)+2[/tex]

[tex]-2<0.5[/tex]

This statement is false. Option 3 is correct.

Check the given inequality for point (-1,3).

[tex]3<-\dfrac{1}{2}(1)+2[/tex]

[tex]3<1.5[/tex]

This statement is false. Option 4 is incorrect.

Therefore, the correct option is C.

Answer:

C

Step-by-step explanation:

(2,1)

Answer:

1. B.

2. B.

Step-by-step explanation:

Trigonometry

1) Coterminal angles can be found by adding or subtracting 360° (or 2\pi radians) to a given angle. If we have 120°, adding 360° gives 480°, adding again 360° gives 840°. There is no way to get -180°, so this option is not a coterminal angle to 120°

2)

A. [tex]Cos (-90^o)=0[/tex], and not undefined

B. [tex]Sin (-90^o)=-1[/tex]. This is correct

C.  [tex]Tan (-90^o)[/tex] is undefined, not zero

Thus the only correct option is B.

Answer: $180,000

Step-by-step explanation:

Gross Domestic Product (GDP) is the total monetary value of all finished goods and services made within a country during a specific period. It can be used to estimate the size and growth rate of the country's economy.

In the case above Company X sell leather which is not a finished good to Company Y, so it will not contribute to the gross domestic product (GDP). Company Y sells leather shoes which is a finished good to the consumers, which will contribute to the GDP.

Therefore the total contribution to GDP is $180,000

Answer:

  3x² +3xh +h²

Step-by-step explanation:

  [tex]\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^3-x^3}{h}=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}{h}\\\\=\dfrac{3x^2h+3xh^2+h^3}{h}=3x^2+3xh+h^2[/tex]

Answer: the measure of the indicated angle is 100 degrees

Step-by-step explanation:

The sum of angles in a triangle is 180 degrees. Let x represent the unknown angle in the bigger triangle. Therefore,

x + 80 + 25 = 180 degrees

x + 105 = 180

x = 180 - 105 = 75 degrees.

Let z represent the other unknown angle in the smaller triangle. Since the sum of the angles on a straight line is 180 degrees, therefore

75 + 55 + z = 180

130 +z = 180

z = 180 - 130 = 50 degrees

Let y represent the unknown angle that we are looking for. Therefore,

50 + y + 30 = 180

80 + y = 180

y = 180 - 80 = 100 degrees

Answer:

55

Step-by-step explanation:

Answer:

Step-by-step explanation:

The formula for the nth term of a geometric sequence is expressed as follows

Tn = ar^(n - 1)

Where

Tn represents the value of the nth term of the sequence

a represents the first term of the sequence.

n represents the number of terms.

From the information given,

r = 1/3

T3 = - 81

n = 3

Therefore,

- 81 = a× 1/3^(3 - 1)

-81 = a × (1/3)^2

-81 = a/9

a = -81 × 9 = - 729

The exponential equation for this sequence is written as

Tn = - 729 * (1/3)^(n-1)

Therefore, to find the second term,T2, n = 2. It becomes

T2 = - 729 * (1/3)^(2-1)

T2 = - 729 * (1/3)^1

T2 = - 729 * (1/3)

T2 = - 243

Answer:

a) [tex]5xz + xy + 4yz[/tex]

b) 10

Step-by-step explanation:

a) Here [tex]F(x,y,z)=(5z+y)i+(4z+x)j+(4y+5x)k[/tex]

Since the case [tex]F[/tex]  =  ∇[tex]f[/tex] holds, then

∇[tex]f = f_xi+f_yj+f_zk[/tex] = [tex](5z+y)i+(4z+x)j+(4y+5x)k[/tex]

So, [tex]f_x = 5z + y[/tex]

If we integrate [tex]f_x[/tex] with respect to x, we will get an integration constant C which is also a function that depends to y and z.

Hence,

[tex]f = \int f_xdx = 5xz + xy + g(y,z)[/tex]

Now we need to find g(y,z).

So first let's take the derivative of g(y,z) with respect to y.

[tex]f_y = x + g_y(y,z) = 4z + x[/tex]

Hence, [tex]g_y(y,z) = 4z[/tex]

So now, if we integrate [tex]g_y[/tex] with respect to y to find g(y,z)

[tex]g = \int g_ydy = 4yz + C[/tex]

Thus,

[tex]f = 5xz + xy + g(y,z) = 5xz + xy + 4yz + C[/tex]

And since [tex]f(0,0,0)=0[/tex], then [tex]C=0[/tex]

Thus,

[tex]f = f(x,y,z) = 5xz + xy + 4yz[/tex]

b) By the Fundamental Theorem of Line Integrals, we know that

[tex]\int\limits^a_b F. dr = F[r(b)]-F[r(a)][/tex]

Hence,

[tex]\int\limits^a_b F. dr = F(1,1,1)-F(0,0,0) =[(5+1+4)-(0+0+0)]=10[/tex]

To find ????, solve the system of partial differential equations. Use the function ???? from part a to compute the line integral.

To find a function ???? such that ????=∇???? and ????(0,0,0)=0, we can solve the system of partial differential equations. Let ????=????????+????????+????????, then compute the partial derivatives of ???? with respect to each variable. Equating these partial derivatives to the given vector field components, we can solve for the unknown function ???? and find its value at the point (0,0,0).

To compute the line integral ∫????????⋅????????, we can use the fundamental theorem of calculus for line integrals. Since ????(x,y,z) is the gradient of ????, the line integral is equal to the change in ???? along the curve C from (0,0,0) to (1,1,1). We can use the function ???? found in part a to evaluate this change in ????.

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Answer:

Step-by-step explanation:

The attached photo shows the diagram of quadrilateral QRST with more illustrations.

Line RT divides the quadrilateral into 2 congruent triangles QRT and SRT. The sum of the angles in each triangle is 180 degrees(98 + 50 + 32)

The area of the quadrilateral = 2 × area of triangle QRT = 2 × area of triangle SRT

Using sine rule,

q/SinQ = t/SinT = r/SinR

24/sin98 = QT/sin50

QT = r = sin50 × 24.24 = 18.57

Also

24/sin98 = QR/sin32

QR = t = sin32 × 24.24 = 12.84

Let us find area of triangle QRT

Area of a triangle

= 1/2 abSinC = 1/2 rtSinQ

Area of triangle QRT

= 1/2 × 18.57 × 12.84Sin98

= 118.06

Therefore, area of quadrilateral QRST = 2 × 118.06 = 236.12

Answer:

  216 square units

Step-by-step explanation:

Apparently, we're supposed to ignore the fact that the given geometry cannot exist. The short diagonal is too short to reach between the angles marked 98°. If Q and S are 98°, then R needs to be 110.13° or more for the diagonals to connect as described.

__

The equal opposite angles of 98° suggests that the figure is symmetrical about the diagonal RT. That being the case, diagonal RT will meet diagonal QS at right angles. Then the area is half the product of the lengths of the diagonals:

  (1/2)×18×24 = 216 . . . . square units

_____

In a quadrilateral, the area can be computed as half the product of the diagonals and the sine of the angle between them. Here, we have assumed the angle to be 90°, so the area is simply half the product of diagonal measures.

Average speed of slower car = 60 mph

Average speed of faster car = 70 mph

Solution:

Given that Two cars enter the Delaware turnpike at shoreline drive at 8 am, each heading for ocean city

Given that One cars average speed is 10 mph faster than the other

Let "x" be the average speed of slower car

Then x + 10 is the average speed of faster car

The faster car arrives at ocean city at 11 am, a half hour before the slower car

Time taken by faster car:

The faster car arrives at ocean city at 11 am but given that they start at 8 a.m

So time taken by faster car = 11 am - 8am = 3 hours

Time taken by slower car:

The faster car arrives at ocean city at 11 am, a half hour before the slower car

So slower car takes half an hour more than faster car

Time taken by slower car = 3 hour + half an hour = [tex]3\frac{1}{2} \text{ hour}[/tex]

Now the distance between Delaware turnpike at shoreline drive and ocean city will be same for both cars

Let us equate the distance and find value of "x"

The distance is given by formula:

[tex]distance = speed \times time[/tex]

Distance covered by faster car:

[tex]distance = (x + 10) \times 3 = 3x + 30[/tex]

Distance covered by slower car:

[tex]distance = x \times 3\frac{1}{2} = x \times \frac{7}{2} = 3.5x[/tex]

Equating both the distance,

3x + 30 = 3.5x

3x - 3.5x = -30

-0.5x = -30

x = 60

Thus average speed of slower car = 60 mph

Average speed of faster car = x + 10 = 60 + 10 = 70 mph