Why do you think students have difficulty understanding the concept of a mathematical function?

Answer:Explained Below

Step-by-step explanation:

The given equation is similar to an ellipse which is in the form of

[tex]\frac{x^2}{a^2}[/tex]+[tex]\frac{y^2}{b^2}[/tex]=1

where

2a=length of major axis

2b=length of minor axis

Here after rearranging the given equation we get

[tex]\frac{x^2}{\frac{144}{9}}[/tex]+[tex]\frac{y^2}{\frac{144}{16}}[/tex]=1

[tex]\frac{x^2}{16}[/tex]+[tex]\frac{y^2}{9}[/tex]=1

[tex]\frac{x^2}{4^2}[/tex]+[tex]\frac{y^2}{3^2}[/tex]=1

therefore its origin is (0,0)

and vertices are[tex]\left ( \pm4,0\right )[/tex]&[tex]\left ( 0,\pm3\right )[/tex]

We can find origin by checking what is with x in the term [tex]\left ( x-something\right )^{2}[/tex]

same goes for y

for [tex]\left ( x-2\right )^{2}[/tex] here 2 is the x  coordinate of ellipse

and for vertices Each endpoint of the major axis is vertices and each endpoint of minor axis is co-vertices

Answer: 0.1880

Step-by-step explanation:

The binomial distribution formula is given by :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]

where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.

Given : The probability of new vehicles require warranty service within the first year : p =0.13.

Number of trials  : n= 12

Now, the required probability will be :

[tex]P(x=0)=^{12}C_0(0.13)^{0}(1-0.13)^{12-0}\\\\=(1)(0.13)^{0}(0.87)^{12}=0.188031682201\approx0.1880[/tex]

Thus,  the probability that none of these vehicles requires warranty service = 0.1880

The inside angle is half the outside angle.

2x +2 = 76 /2

2x +2 = 38

Subtract 2 from each side:

2x = 36

Divide both sides by 2:

x = 36 /2

x = 18

The answer is B.

Answer:

A participant in a cognitive psychology study is given 50 words to remember and she recalls 17 words.  

So, here variable will be the number of words the participant can remember, out of 50.

The possible values can be the whole numbers 0, 1, 2, 3, 4, upto... 50.

And the score is 17. That is the score she remembers out of 50.

Answer:  [tex]\hat{p}=0.5625[/tex]

Step-by-step explanation:

Given : Sample size : [tex]n=4000[/tex]

The number of people who are in favor of gun control legislation =2250

The proportion of people who are in favor of gun control legislation will be :-

[tex]p_0=\dfrac{2250}{4000}=0.5625[/tex]

We assume that the the given situation is normally distributed.

Then , the point estimate for the proportion [tex]\hat{p}[/tex] of citizens who are in favor of gun control legislation is equals to the sample proportion.

i.e.  [tex]\hat{p}=0.5625[/tex]

Answer:

Production cost is $20 per item.

Step-by-step explanation:

Fixed cost is $100 and 20 items cost $500 to produce.

[tex]C=100+x*production cost[/tex]

[tex]500=100+20*production cost[/tex]

[tex]400=20*production cost[/tex]

Production cost = $20.

So, [tex]C(x)=20x+100[/tex], where C is total cost and x is the number of items produced.

The linear cost function, based on a given fixed cost and the cost to produce a certain number of items, is found by identifying and adding the fixed and variable costs. In this scenario, the mathematical expression for the total cost function is C(x) = $100 + $20(x).

To determine the linear cost function for a production scenario with fixed and variable costs, we use the information provided: the fixed cost is $100, and the cost to produce 20 items is $500. Knowing that the cost function is linear, we can express it as C(x) = Fixed Cost + Variable Cost per Item (x), where C(x) is the total cost function and x is the number of items produced.

Since the fixed cost is given as $100, we have C(x) = $100 + Variable Cost per Item (x). To find the variable cost per item, we calculate the difference in total costs when producing 20 items. This is $500 (total cost to produce 20 items) minus the fixed cost of $100, which equals $400. Since this cost is associated with the production of 20 items, we divide $400 by 20 to find the variable cost per item, which is $20. Thus, our variable cost per item is $20.

Now, we combine the fixed cost with the variable cost per item to get the complete linear cost function: C(x) = $100 + $20(x).

Answer:

20 percent

Step-by-step explanation:

Total number of pieces in a puzzle = 500

No. of pieces in first container = 220

No. of pieces in second container = 180

Let no. of pieces in the third container  be x.

We get,

[tex]220+180+x=500[/tex]

On adding 220 and 180, we get

[tex]400+x=500[/tex]

On transposing 400 to RHS, we get

[tex]x=500-400=100[/tex]

Percentage of pieces in the third container = (no. of pieces in third container/total no. of pieces in a puzzle) [tex]\times 100[/tex]

[tex]=\frac{100}{500}\times 100=\frac{10000}{500}=20[/tex]

Therefore, percentage of pieces in the third container = 20 percent

Find the number of miles he drove by subtracting the odometer readings:

38735.5 - 38320.8 = 414.7 miles.

Now divide the number of miles driven by the number of gallons:

414.7 / 14.5 = 28.6 miles per gallon.

Answer:

8 Joule

Step-by-step explanation:

Mass of block = 2 kg

Displacement = x = 800 mm = 0.8 m

Spring constant = k = 25 N/m

Potential Energy of a spring

Work done = Difference in Potential Energy

Work Done = Δ P.E.

[tex]\Rightarrow \Delta\ P.E.=\frac{1}{2}kx^2[/tex]

⇒P.E. = 0.5×25×0.8²

⇒P.E. = 8 Nm = 8 Joule

Here already the spring constant and displacement is given so the mass will not be used while calculating the potential energy.

The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.

Differentiating the second equation with respect to t, we get: d^2y/dt^2 = -5 dx/dt, Substituting dx/dt from the first equation, we get: d^2y/dt^2 = -5(-5y) = 25y.

This is a second order differential equation in y. The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.

To find x as a function of t, we can substitute y(t) into the first equation and solve for x: dx/dt = -5y = -5(c1 cos(5t) + c2 sin(5t)) , Integrating both sides with respect to t, we get: x(t) = -c1 sin(5t) + c2 cos(5t) + k

where k is a constant of integration. Using the initial conditions x(0) = 4 and y(0) = 1, we can solve for the constants c1, c2, and k: x(0) = -c1 sin(0) + c2 cos(0) + k = c2 + k = 4, y(0) = c1 cos(0) + c2 sin(0) = c1 = 1

Substituting c1 = 1 and c2 + k = 4 into the equation for x, we get:

x(t) = -sin(5t) + 4

So the solution to the system of differential equations with initial conditions x(0) = 4 and y(0) = 1 is x(t) = -sin(5t) + 4 and y(t) = cos(5t).

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The probability that the sum of the two dice rolls is 5 given that the sum is not 6, is calculated by finding the ratio of favorable outcomes to total outcomes, in this case, 4/31.

The subject of this question is probability which comes under Mathematics. This is a high school-level problem. To answer the question, we first need to understand the rules of a die. A die is a cube, and each of its six faces shows a different number of dots from 1 to 6. When the die is thrown, any number from 1 to 6 can turn up. In this case, two dice are being rolled.

When two dice are rolled, the total possible outcomes are 36 (as each die has 6 faces & we have 2 dice, so 6*6=36 possible outcomes). The combinations that yield a sum of 5 are (1,4), (2,3), (3,2), (4,1), so there are 4 such combinations. Now, the outcome is given to be not 6, which means we exclude combinations where the sum is 6. The combinations of 6 are (1,5), (2,4), (3,3), (4,2), and (5,1) -- 5 combinations.

Excluding these combinations, we have 36 - 5 = 31 possible outcomes. So probability that the sum of the dice is 5 given that the outcome is not 6, is favorable outcomes/total outcomes = 4/31.

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Answer: Odds against rolling a sum of 3 = 17:1

Step-by-step explanation:

On rolling a pair of dice,

Total number of outcomes = 6 × 6 = 36

Outcomes with a sum of 3:

there is only 2 outcomes whose sum is 3, that is, (1,2) and (2,1)

∴ Favorable outcome = 2

Unfavorable outcome = 34

Odds against refers to the ratio of unfavorable outcomes to the favorable outcomes

so,

odds against rolling a sum of 3 = [tex]\frac{unfavorable\ outcomes}{favorable\ outcomes}[/tex]

= [tex]\frac{34}{2}[/tex]

= 17:1

Answer: 0.0031

Step-by-step explanation:

Binomial distribution formula :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of x successes in the n independent trials of the experiment and p is the probability of success.

Given : A binomial probability experiment is conducted with the given parameters.

[tex]n=9,\ p=0.8,\ x\leq3[/tex]

Now, [tex]P(x\leq3)=P(3)+P(2)+P(1)+P(0)[/tex]

[tex]=^9C_3(0.8)^3(1-0.8)^{9-3}+^9C_2(0.8)^2(1-0.8)^{9-2}+^9C_1(0.8)^1(1-0.8)^{9-1}+^9C_0(0.8)^0(1-0.8)^9\\\\=\dfrac{9!}{3!6!}(0.8)^3(0.2)^6+\dfrac{9!}{2!7!}(0.8)^2(0.2)^7+\dfrac{9!}{1!8!}(0.8)(0.2)^8+\dfrac{9!}{0!9!}(0.2)^9=0.003066368\approx0.0031[/tex]

Hence,  [tex]P(x\leq3)=0.0031[/tex]

The generating function is [tex]f(x)[/tex] where

[tex]f(x)=\displaystyle\sum_{k=0}^\infty a_kx^k[/tex]

with [tex]a_k=k^3[/tex] for [tex]k\ge0[/tex].

Recall that for [tex]|x|<1[/tex], we have

[tex]g(x)=\dfrac1{1-x}=\displaystyle\sum_{k=0}^\infty x^k[/tex]

Taking the derivative gives

[tex]g'(x)=\dfrac1{(1-x)^2}=\displaystyle\sum_{k=1}^\infty kx^{k-1}=\sum_{k=0}^\infty(k+1)x^k[/tex]

[tex]\implies g'(x)-g(x)=\dfrac x{(1-x)^2}=\displaystyle\sum_{k=0}^\infty kx^k[/tex]

Taking the derivative again, we get

[tex]g''(x)=\dfrac2{(1-x)^3}=\displaystyle\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k^2+3k+2)x^k[/tex]

[tex]\implies g''(x)-3g'(x)+g(x)=\dfrac{x^2+x}{(1-x)^3}=\displaystyle\sum_{k=0}^\infty k^2x^k[/tex]

Take the derivative one last time to get

[tex]g'''(x)=\dfrac6{(1-x)^4}=\displaystyle\sum_{k=3}^\infty k(k-1)(k-2)x^{k-3}=\sum_{k=0}^\infty(k^3+6k^2+11k+6)x^k[/tex]

[tex]\implies g'''(x)-6g''(x)+7g'(x)-g(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}=\displaystyle\sum_{k=0}^\infty k^3x^k[/tex]

So the generating function is

[tex]\boxed{f(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}}[/tex]

Find the volume of the silo.

The formula is: Volume =  PI x r^2 x h

Replace volume with the volume of grain and solve for h:

38000 = 3.14 x 24^2 x h

38000 = 3.14 x 576 x h

38000 = 1808.64 x h

Divide both sides by 1808.64

h = 38000 / 1808.64

h = 21.01

The grain would be 21.01 feet ( round to 21 feet.)

Answer:

Gold - $33, Silver - $5, Copper - $0.02

Step-by-step explanation:

Let $x be the price of one gram of gold, $y - price of 1 g of silver and $z - price of 1 g of copper.

1. The first alloy is​ 75% gold,​ 5% silver, and​ 20% copper, so in 100 g there are 75 g of gold, 5 g of silver and 20 g of copper.  If 100 g of the first alloy costs ​$2500.40​, then

75x+5y+20z=2500.40

2. The second alloy is​ 75% gold,​ 12.5% silver, and​ 12.5% copper, so in 100 g there are 75 g of gold, 12.5 g of silver and 12.5 g of copper.  If 100 g of the first alloy costs ​$2537.75​, then

75x+12.5y+12.5z=2537.75

3. The third alloy is​ 37.5% gold and​ 62.5% silver, so in 100 g there are 37.5 g of gold and 62.5 g of silver .  If 100 g of the first alloy costs ​$1550.00​, then

37.5x+62.5y=1550.00

Solve the system of three equations:

[tex]\left\{\begin{array}{l}75x+5y+20z=2500.40\\75x+12.5y+12.5z=2537.75\\37.5x+62.5y=1550.00\end{array}\right.[/tex]

Find all determinants

[tex]\Delta=\|\left[\begin{array}{ccc}75&5&20\\75&12.5&12.5\\37.5&62.5&0\end{array}\right] \|=28125\\ \\

\Delta_x=\|\left[\begin{array}{ccc}2500.40&5&20\\2537.75&12.5&12.5\\1550.00&62.5&0\end{array}\right] \|=928125\\ \\

\Delta_y=\|\left[\begin{array}{ccc}75&2500.40&20\\75&2537.75&12.5\\37.5&1550&0\end{array}\right] \|=140625\\ \\

\Delta_z=\|\left[\begin{array}{ccc}75&5&2500.40\\75&12.5&2537.75\\37.5&62.5&1550\end{array}\right] \|=562.5\\ \\[/tex]

So,

[tex]x=\dfrac{\Delta_x}{\Delta}=\dfrac{928125}{28125}=33\\ \\\\y=\dfrac{\Delta_y}{\Delta}=\dfrac{140625}{28125}=5\\ \\\\z=\dfrac{\Delta_z}{\Delta}=\dfrac{562.5}{28125}=0.02\\ \\[/tex]

Answer:

  ∠ABC = 84°

  ∠BEF = 64°

Step-by-step explanation:

∠ABC is supplementary to the 96° angle shown, so is 180° -96° = 84°.

__

∠ABD, marked as (x+y)°, is a vertical angle with ∠EBC, so has the same measure, 96°. ∠BEF, marked as y°, is a vertical angle with the one marked 2x°.

These relationships can be expressed as two equations:

Using the second of these equations to substitute for y in the first equation, we have ...

  x + 2x = 96

  x = 96/3 = 32

  y = 2x = 2·32 = 64 . . . . . . substitute the value of x into the second equation

Then ∠BEF = 64°.

Answer:

Given,

The jewelry business orders $ 1320 in supplies each month.

That is, the invested amount = $ 1320,

Also, the selling price of each earring/ring set = $ 55,

Part 1 : Let x be the number of set that have been sold each month to break even ( in which revenue and invested amount are equal )

So, Total revenue ( the cost of x sets) = 55x,

⇒ 55x = 1320

Divide both sides by 55,

We get,

x = 24

Hence, 24 sets are needed to sell each month to break even.

Part 2 : Let y be the number of sets in which the profit is $ 715,

Total revenue = 55x

Profit = Total revenue - invested amount

⇒ 55x - 1320 = 715,

⇒ 55x = 715 + 1320

⇒ 55x = 2035

⇒ x = 37

Hence, 37 sets are sold for the total profit of $715.

Part 3 :

Revenue in selling 7 sets = 55 × 7 = $ 385

Profit = $ 385 - $ 1320 = - $ 935

Hence, the profit is - $ 935 after selling 7 sets.

Answer:

slope -2/5

y-intercept -3/5

Step-by-step explanation:

Slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

Our goal is to write 2x+5y=-3 into y=mx+b to determine the slope and y-intercept.

So we need to isolate y.

2x+5y=-3

Subtract 2x on both sides:

   5y=-2x-3

Divide both side by 5:

   [tex]y=\frac{-2}{5}x-\frac{3}{5}[/tex]

Compare this to y=mx+b.

You should see m is -2/5 and b is -3/5

so the slope is -2/5 and the y-intercept is -3/5

Answer:

The slope is: [tex]-\frac{2}{5}[/tex] or [tex]-0.4[/tex]

The y-intercept is: [tex]-\frac{3}{5}[/tex] or [tex]-0.6[/tex]

Step-by-step explanation:

The equation of the line in Slope-Intercept form is:

[tex]y=mx+b[/tex]

Where "m" is the slope of the line and "b" is the y-intercept.

To write the given equation in this form, we need to solve for "y":

[tex]2x + 5y = -3\\\\5y=-2x-3\\\\y=-\frac{2}{5}x-\frac{3}{5}[/tex]

Therefore, you can identify that the slope of this line is:

[tex]m=-\frac{2}{5}=0.4[/tex]

And the y-intercept is:

[tex]b=-\frac{3}{5}=-0.6[/tex]

Answer:

The mathematical expression is false

Step-by-step explanation:

* Lets use the figure to answer the question

- There are four triangles in the figure

- Δ ROB and Δ PTA appear congruent because:

# The side RO appears equal the side PT

∴ RO ≅ PT

# The side OB appears equal the side TA

∴ OB ≅ TA

# The side RB appears equal the side PA

∴ RB ≅ PA ⇒ SSS

∴ Δ ROB ≅ Δ PTA

- Δ DEF and Δ YXW appear congruent because:

# The side DE appears equal the side YX

∴ DE ≅ YX

# The side EF appears equal the side XW

∴ EF ≅ XW

# The side DF appears equal the side YW

∴ DF ≅ YW

∴ Δ DEF ≅ Δ YXW ⇒ SSS

- Δ ROB and Δ DEF have different shapes and sizes

∵ Δ ROB not appear congruent to Δ DEF

∴ Δ ROB ≠ Δ DEF

∴ The mathematical expression is false

Answer : The mathematical expression is false.

Step-by-step explanation :

As we are given 4 triangles in which ΔROB & ΔPTA and ΔDEF & ΔYXW are appears congruent.

First we have to show that ΔROB and ΔPTA appear congruent.

Side RO appears equal to Side PT

Side OB appears equal to Side TA

Side RB appears equal to Side PA

∴ ΔROB ≅ ΔPTA   (by SSS)

Now we have to show that ΔDEF and ΔYXW appear congruent.

Side DE appears equal to Side YX

Side EF appears equal to Side XW

Side DF appears equal to Side YW

∴ ΔDEF ≅ ΔYXW   (by SSS)

According to given expression, ΔROB and ΔDEF have different shapes and sizes.

So, ΔROB not appear congruent to ΔDEF

Therefore, the mathematical expression is false.

Looks like the integral is

[tex]\displaystyle\iint_R2(x+1)y^2\,\mathrm dA[/tex]

where [tex]R=[0,1]\times[0,3][/tex]. (The inclusion of [tex]y=3[/tex] will have no effect on the value of the integral.)

Let's split up [tex]R[/tex] into [tex]mn[/tex] equally-sized rectangular subintervals, and use the bottom-left vertices of each rectangle to approximate the integral. The intervals will be partitioned as

[tex][0,1]=\left[0,\dfrac1m\right]\cup\left[\dfrac1m,\dfrac2m\right]\cup\cdots\cup\left[\dfrac{m-1}m,1\right][/tex]

and

[tex][0,3]=\left[0,\dfrac3n\right]\cup\left[\dfrac3n,\dfrac6n\right]\cup\cdots\cup\left[\dfrac{3(n-1)}n,3\right][/tex]

where the bottom-left vertices of each rectangle are given by the sequence

[tex]v_{i,j}=\left(\dfrac{i-1}n,\dfrac{3(j-1)}n\right)[/tex]

with [tex]1\le i\le m[/tex] and [tex]1\le j\le n[/tex]. Then the Riemann sum is

[tex]\displaystyle\lim_{m\to\infty,n\to\infty}\sum_{i=1}^m\sum_{j=1}^nf(v_{i,j})\frac{1-0}m\frac{3-0}n[/tex]

[tex]\displaystyle=\lim_{m\to\infty,n\to\infty}\frac3{mn}\sum_{i=1}^m\sum_{j=1}^n\frac{18}{mn^2}(j-1)^2(i-1+m)[/tex]

[tex]\displaystyle=\lim_{m\to\infty,n\to\infty}\frac{54}{m^2n^3}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}j^2(i+m)[/tex]

[tex]\displaystyle=\frac92\lim_{m\to\infty,n\to\infty}\frac{(3m-1)(2n^3-3n^2+n)}{mn^3}[/tex]

[tex]\displaystyle=\frac92\left(\lim_{m\to\infty}\frac{3m-1}m\right)\left(\lim_{n\to\infty}\frac{2n^3-3n^2+n}{n^3}\right)=\boxed{27}[/tex]

Answer: 297 miles

Step-by-step explanation:

The drive from Waterton to Middleton is 76 miles, from Middleton to Oak Hill is 87 miles, and from Oak Hill directly to Waterton it is 134 miles.

Then: [tex]76+87+134=297[/tex]

So Juan drives 297 miles.

Answer:

28

Step-by-step explanation:

substitute x for negative four -4(-4)+12=

solve -4 * -4= 16

add 16 and 12 equals 28

Answer:

28

Step-by-step explanation:

-4x + 12

Let x = -4

-4 (-4) +12

16+12

28

Answer:   0.0660

Step-by-step explanation:

Given : A particular fruit's weights are normally distributed with

Mean : [tex]\mu=353\text{ grams}[/tex]

Standard deviation : [tex]\sigma=6\text{ grams}[/tex]

The formula to calculate the z-score is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

Let x be the weight of randomly selected fruit.

Then for x = 334 , we have

[tex]z=\dfrac{334-353}{6}=-3.17[/tex]

for x = 344 , we have

[tex]z=\dfrac{344-353}{6}=-1.5[/tex]

The p-value : [tex]P(334<x<353)=P(-3.17<z<-1.5)[/tex]

[tex]P(-1.5)-P(-3.17)=0.0668072-0.000771=0.0660362\approx0.0660[/tex]

Thus, the probability that it will weigh between 334 grams and 344 grams = 0.0660.

Final answer:

To find the number of adults in the family, we need to solve the system of equations. By multiplying the first equation by 2 and subtracting it from the second equation, we can eliminate x and solve for y. Substituting the value of y back into the first equation, we can solve for x. The number of adults in the family is 3.

Explanation:

To find the number of adults in the family, we need to solve the system of equations:

Equation 1: 8x + 4y = 32

Equation 2: 16x + 12y = 72

We can solve this system by first multiplying Equation 1 by 2 to make the coefficients of x in both equations the same. This gives us:

Equation 1 (multiplied by 2): 16x + 8y = 64

Next, we can subtract Equation 1 (multiplied by 2) from Equation 2 to eliminate x:

Equation 2 - Equation 1 (multiplied by 2): (16x + 12y) - (16x + 8y) = 72 - 64

Simplifying the equation, we get:

4y = 8

Dividing both sides by 4, we find:

y = 2

So, there are 2 children in the family. Substituting this value back into Equation 1, we can solve for x:

8x + 4(2) = 32

8x + 8 = 32

8x = 24

Dividing both sides by 8, we find:

x = 3

Therefore, there are 3 adults in the family.

By Stokes' theorem, the integral of the curl of [tex]\vec F[/tex] across [tex]S[/tex] is equal to the integral of [tex]\vec F[/tex] along the boundary of [tex]S[/tex], call it [tex]C[/tex]. Parameterize [tex]C[/tex] by

[tex]\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath[/tex]

with [tex]0\le t\le2\pi[/tex]. So we have

[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_C\vec F\cdot\mathrm d\vec r[/tex]

[tex]=\displaystyle\int_0^{2\pi}(10\sin t\cos 0\,\vec\imath+e^{2\cos t}\sin0\,\vec\jmath+2\cos t\,e^{2\sin t}\,\vec k)\cdot(-2\sin t\,\vec\imath+2\cos t\,\vec\jmath)\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^{2\pi}-20\sin^2t\,\mathrm dt[/tex]

[tex]=\displaystyle-10\int_0^{2\pi}(1-\cos2t)\,\mathrm dt=\boxed{-20\pi}[/tex]

The problem makes use of Stokes' theorem to evaluate a given field over a hemisphere. We established the boundary curve of the surface and described it using a vector equation.

This problem can be solved using Stokes' theorem which asserts that the magnetic field flux through a surface is related to the circulation of the field encircling that surface. Stokes' theorem can be written in this form ∫ S curl F · dS = ∫ C F · dr. Given the field F(x, y, z) = 5y cos(z) i + ex sin(z) j + xey k and the hemisphere S defined by x² + y² + z² = 4, z ≥ 0, we need to look for its boundary curve C. C here is the circle in the xy-plane defined by x² + y² = 4, z = 0. We can describe this boundary using a vector equation r(t) = 2 cos(t) i + 2 sin(t) j + 0k with 0 ≤ t ≤ 2π.

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Answer with explanation:

Given two sets

          A={2, 4, 6, 8} and B={2, 3, 5, 7, 9}

⇒n(S)=Cardinality of a set

Means the number of distinct elements in a Set is called it's cardinal number.

→In Set A,total number of distinct elements is 4.

n(A)=4

Answer:

211 people

Step-by-step explanation:

1.  This is a least common multiple question, otherwise known as an LCM question. We know this because if the one extra person had not shown up to the party, all groups would have been formed evenly. This means the amount of people at the party is 1 more than a number 2, 3, 5, and 7 can go into.

2. We also know this is an LCM question because we are being asked for the smallest amount of people who could have possibly attended the party.

3.  From this we know the amount of people at the party must be an odd number. If the number were even, there would have been no left over when groups of 2 were formed.

4. The amount of people at the party must end in a 1. This is because all multiples of 5 always end in a 0 or 5. Because there is on extra person, we must add 1 to all multiples of 5 we check. However, 5 + 1 is 6. This is a problem because 6 is an even number, and as we already established, the amount of people at the party must end in an odd number. So, we now know the smallest amount of people at the party will end in a 1.

5. Because we know the largest teams attempted to be formed with 1 left over is teams of 7 people, we only need to check multiples of 7. It is the largest number, so doing this will save us time.

6. Since we know the amount of people at the party must end in a 1, and we are only checking multiples of 7, we only need to check multiples of 7 that end in a 0. This is because any multiple of 7, 7 will go into evenly without a remainder. So we must add 1 to every multiple we check in order to make a remainder of 1. The only number we can add 1 to in order to get 1 is 0, so we only need to check multiples of 7 that end in 0. The only multiples of 7 that end in a 0 are when 7 is multiplied by a ten, ex: 10, 20, 30, 40, 50, ect.

7. Only searching for odd numbers, numbers that end in 1, and multiples of 7 means we only have to check if any possible answer when divided by 3 has a remainder of 1. We only have to check by the number 3 because any number ending in 1 will automatically have a remainder of one for 2 and 5, and because we are using multiple of 7 we don't need to check through 7.

8. Now that we know all our rules, all we need to do is list multiples of 7 that end in 0. Then we will add 1 to them and check to see if they have a remainder of 1 when divided by 3.

Using these rules will narrow our search drastically.

Applicable multiples of 7

(7 × 10) = 70 → 70 + 1 = 71 → 71/3 = 23 R2 → not possible

(7 × 20) = 140 → 140 + 1 = 141 → 141/3 = 47 → not possible

(7 × 30) = 210 → 210 +1 = 211 → 211/3 = 70 R1 → possible

The smallest possible amount of people at the party is 211.

Final answer:

The smallest amount of people who could have been at the party is 210.

Explanation:

To find the smallest amount of people who could have been at the party, we need to find the least common multiple (LCM) of the numbers 2, 3, 5, and 7. The LCM is the smallest number that is divisible by all of the given numbers without leaving a remainder.

Therefore, the smallest amount of people who could have been at the party is 210.

Answer:

i). [tex]\$ 7500[/tex]

ii).[tex]\$ 67500[/tex]

Step-by-step explanation:

Given in the question-

Saving rate is s(t)= 3000(t+2)

We know that savings in the 1st year can be calculated as

    [tex]\int_{0}^{1}3000(t+2).dt[/tex]

    [tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^1[/tex]

    [tex]3000\left [ \frac{1}{2}+2 \right ][/tex]

 = [tex]\$ 7500[/tex]

So savings in the first 5 years can be calculated as

     [tex]\int_{0}^{5}3000(t+2).dt[/tex]

    [tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^5[/tex]

    [tex]3000\left [ \frac{25}{2}+5 \right ][/tex]

 = [tex]\$ 67500[/tex]

Answer:

840.02 square inches ( approx )

Step-by-step explanation:

Suppose x represents the side of each square, cut from the corners of the sheet,

Since, the dimension of the sheet are,

31 in × 17 in,

Thus, the dimension of the rectangular box must are,

(31-2x) in × (17-2x) in × x in

Hence, the volume of the box would be,

V = (31-2x) × (17-2x) × x

[tex]=(31\times 17 +31\times -2x -2x\times 17 -2x\times -2x)x[/tex]

[tex]=(527 -62x-34x+4x^2)x[/tex]

[tex]\implies V=4x^3-96x^2 +527x[/tex]

Differentiating with respect to x,

[tex]\frac{dV}{dx}=12x^2-192x+527[/tex]

Again differentiating with respect to x,

[tex]\frac{d^2V}{dx^2}=24x-192[/tex]

For maxima or minima,

[tex]\frac{dV}{dx}=0[/tex]

[tex]\implies 12x^2-192x+527=0[/tex]

By the quadratic formula,

[tex]x=\frac{192 \pm \sqrt{192^2 -4\times 12\times 527}}{24}[/tex]

[tex]x\approx 8\pm 4.4814[/tex]

[tex]\implies x\approx 12.48\text{ or }x\approx 3.52[/tex]

Since, at x = 12.48, [tex]\frac{d^2V}{dx^2}[/tex] = Positive,

While at x = 3.52, [tex]\frac{d^2V}{dx^2}[/tex] = Negative,

Hence, for x = 3.52 the volume of the rectangle is maximum,

Therefore, the maximum volume would be,

V(3.5) = (31-7.04) × (17-7.04) × 3.52 = 840.018432 ≈ 840.02 square inches