1. Tom thought of a natural number, multiplied all its digits and after that he multiplied the result by the initial number. Is it possible to get 1716 as a result?

2. What is the largest prime factor of the factorial 49! ?

3. The GCD(a, b) = 18, LCM(a, b) = 108. If a=36, findb.

Each consecutive term in the sum is separated by a difference of 6, so the [tex]n[/tex]-th term is [tex]4+6(n-1)=6n-2[/tex] for [tex]n\ge1[/tex]. The last term is 70, so there are [tex]6n-2=70\implies n=12[/tex] terms in the sum.

Now,

[tex]S=4+10+\cdots+64+70[/tex]

but also

[tex]S=70+64+\cdots+10+4[/tex]

Doubling the sum and grouping terms in the same position gives

[tex]2S=(4+70)+(10+64)+\cdots+(64+10)+(70+4)=12\cdot74[/tex]

[tex]\implies\boxed{S=444}[/tex]

Answer:

a < -12

Step-by-step explanation:

Isolate the variable, a. Treat the < as a equal sign, what you do to one side, you do to the other. Subtract 2 from both sides:

a + 2 < -10

a + 2 (-2) < -10 (-2)

Simplify:

a < -10 - 2

a < -12

a < -12 is your answer.

~

Answer:

[tex]\huge \boxed{a<-12}\checkmark[/tex]

Step-by-step explanation:

Subtract by 2 both sides of equation.

[tex]\displaystyle a+2-2<-10-2[/tex]

Simplify, to find the answer.

[tex]-10-2=-12[/tex]

[tex]\displaystyle a<-12[/tex], which is our answer.

Answer:

The correct answer is 206550 pigs.

Step-by-step explanation:

First of all you need to calculate the constant growth rate, which is given by the formula [tex]p=((\frac{f}{s} )^{\frac{1}{y} } -1)*100[/tex] where f is the value at the end of the year, s is the start value of that year, and y is the number of years.

From the excercise facts, we know that for 1 year (y=1), the final value is 421 (f=421), and the start value is 16 (s=16). Replacing them in the formula we get : [tex]((\frac{421}{16}) ^{\frac{1}{1} } -1)*100 = 2531.25[/tex] So, the constant growth rate equals 2531.25.

Next, we have to multiply the starting amount of pigs times the constant growth rate times the amount of time that passed to get the final quantity of pigs. This would be [tex]16*2531.25*5.1[/tex] and this gives us a total amount of 206550 pigs.

Have a nice day.

Answer:

Since living expenses are 11% higher in the new city. Marla would need an 11% increase in her current income to maintain her current buying power when moving. We can solve this by simply multiplying her current income by 11%.

Step-by-step explanation:

[tex]95,000*1.11 = 105,450[/tex]

If we multiply her current income of $95,000 by 11% we see that Marla would need an income of $105,450 in order to maintain her buying power in the new city. Since her new job offer is paying $118,000 she will have more buying power in the new city.

Answer: Marla must earn an Income of $105,450 to maintain her buying power.

Marla would need to earn $105,450 in the new city to maintain her current buying power after taking into account an 11 percent increase in living expenses.

The student has asked how much Marla Opper must earn at the new job in another city to maintain her current buying power, considering that the new city has living expenses that are 11 percent higher. To calculate this, we first need to figure out what amount of income in the new city would be equivalent to Marla's current income of $95,000, given the increase in living expenses.

Firstly, we find 11 percent of Marla's current income:

This means Marla needs an additional $10,450 on top of her current income to maintain the same buying power in the new city. So, the amount Marla needs to earn in the new city is:

Thus, Marla would need to earn $105,450 in the new city to keep her current buying power.

Answer:

(a) $13811.68

(b) $13914.30

(c) $14004.55

(d) $14022.54

(e) $14023.15

Step-by-step explanation:

Since, the amount formula in compound interest,

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]

Where,

P = Principal amount,

r = annual rate,

n = number of periods,

t = number of years,

Here, P = $ 9,400, r = 8% = 0.08, t = 5 years,

If the amount is compounded annually,

n = 1,

Hence, the amount of investment would be,

[tex]A=9400(1+\frac{0.08}{1})^5=9400(1.08)^5=\$ 13811.6839219\approx \$ 13811.68[/tex]

(a) If the amount is compounded annually,

n = 1,

The amount of investment would be,

[tex]A=9400(1+\frac{0.08}{1})^5=9400(1.08)^5=\$ 13811.6839219\approx \$ 13811.68[/tex]

(b) If the amount is compounded semiannually,

n = 2,

The amount of investment would be,

[tex]A=9400(1+\frac{0.08}{2})^{10}=9400(1.04)^{10}=\$13914.2962782\approx \$ 13914.30[/tex]

(c) If the amount is compounded Monthly,

n = 12,

The amount of investment would be,

[tex]A=9400(1+\frac{0.08}{12})^{60}=9400(1+\frac{1}{150})^{60}=\$ 14004.549658\approx \$ 14004.55[/tex]

(d) If the amount is compounded Daily,

n = 365,

The amount of investment would be,

[tex]A=9400(1+\frac{0.08}{365})^{365\times 5}=9400(1+\frac{2}{9125})^{1825}=\$ 14022.5375476\approx \$ 14022.54[/tex]

(e) Now, the amount in compound continuously,

[tex]A=Pe^{rt}[/tex]

Where, P = principal amount,

r = annual rate,

t = number of years,

So, the investment would be,

[tex]A=9400 e^{0.08\times 5}=9400 e^{0.4}=\$14023.1521578\approx \$14023.15[/tex]

Answer:

R is not reflexive,not irreflexive ,symmetric,not asymmetric,not antisymmetric and transitive.

Step-by-step explanation:

1.Reflexive : Relation R is not reflexive because it does not contain identity relation on A.

Identity relation on A:{(1,1),(2,2),(3,3),(4,4)}

If we take a relation R on A :{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}

The relation R is reflexive on set A because it contain identity relation.

2.Irreflexive: Given relation R is not reflexive because it contain some elements of identity relation .If a relation is irreflexive then it does not contain any element of identity relation. We can say the intersection of R with identity relation is empty.

[tex]R\cap I=\left\{(1,1),(2,2),(4,4)\right\} \neq \phi[/tex]

If we take a relation on A R:{(1,2),(2,1)}

Then relation is irreflexive because it does not contain any element of identity relation

[tex]R\cap I=\phi[/tex]

3.Symmetric : The given relation R is symmetric because it satisfied the property of symmetric relation.

(1,2)belongs to relation (2,1) also belongs to relation ,(1,4) belongs to relation and (4,1) also belongs to given relation Hence we can say it is symmetric relation.

4.Asymmetric: The given relation is not asymmetric relation.Because it does not satisfied the property of asymmetric relation

Asymmetric relation: If (a,b) belongs to relation then (b,a) does not belongs to given relation.

Here (1,2) belongs to given relation and (2,1) also belongs to given relation Therefore, it is not asymmetric.

If we take a relation R on A

R:{(1,2),(1,3)}

It is asymmetric relation because it does not contain (2,1) and (3,1).

5.Antisymmetric: The given relation is not antisymmetric because it does not satisfied the property of antisymmetric relation.

Antisymmetric relation: If (a,b) and (b,a) belongs to relation then a=b

If we take a relation R

R;{(1,2) (1,1)}

It is antisymmetric because it contain (1,1) where 1=1 .Hence , it is antisymmetric.

6.Transitive: The given relation is transitive because it satisfied the property of transitive relation

Transitive relation: If (a,b) and (b,c) both belongs to relation R then (a,c) belongs to R.

Here, we have (1,2) and (2,1)belongs to relation then (1,1)  and (2,2) are also belongs to relation.

(1,4) and (4,1)  then (1,1) and (4,4) are also belongs to the relation.

Answer:

33

Step-by-step explanation:

The mode is the number that sets up the most in an answer.

Remember, the stem is the number that is in the "tens" place value, while the leaves is the number in the "ones" place value.

Expand the stem-leaves:

10, 13, 16, 20, 21, 23, 27, 27, 28, 29, 30, 32, 33, 33, 33, 33, 38, 39, 46, 46, 46, 48

The number that shows up the most is 33, and is your answer.

~

Answer:

  y = 2(x + 2)^2 – 5

Step-by-step explanation:

When y = ax^2 +bx +c is written in vertex form, it becomes ...

  y = a(x +b/(2a))^2 +(c -b^2/(4a))

The constant term in the squared binomial is b/(2a) = 8/(2(2)) = +2. Only one answer choice matches:

  y = 2(x +2)^2 -5

There are 9,072 different routes possible for the delivery driver.

To find the number of different routes possible, we can use the concept of permutations. Since the driver has to deliver to 7 out of the 9 remaining locations, we can calculate the number of ways to choose 7 out of 9 and then multiply it by the number of ways to arrange those 7 locations. The formula for permutations is P(n, r) = n! / (n - r)!. In this case, n = 9 and r = 7.

P(9, 7) = 9! / (9 - 7)! = 9! / 2! = 9 × 8 × 7 × 6 × 5 × 4 × 3 = 9,072

Therefore, there are 9,072 different routes possible for the delivery driver.

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To find the number of different routes possible, we can use the combination formula: C(n, k) = n! / (k!(n-k)!). Plugging in the values, we get C(9, 7) = 36. Therefore, there are 36 different routes possible.

To find the number of different routes possible, we can use the combination formula:

C(n, k) = n! / (k!(n-k)!)

Where n is the total number of locations remaining (9) and k is the number of locations the driver can make deliveries to (7).

Plugging in the values, we get:

C(9, 7) = 9! / (7!(9-7)!)

= 9! / (7!2!)

= (9 * 8 * 7!)/(7! * 2!)

= (9 * 8)/(2 * 1)

= 36

Therefore, there are 36 different routes possible.

The number of different possible 7-number selections from a pool of 48, where order does not matter, can be calculated using the mathematical concept of combinations. The formula to calculate the total number of combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of options and k is the number of options selected.

In the LOTTO game you described, you must select 7 numbers from a pool of 48, and the order of the numbers does not matter. This is a problem of combinations in mathematics. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of options, k is the number of options selected, and '!' denotes a factorial. In this case, n=48 (the numbers 1 through 48) and k=7 (the seven numbers you select).

By plugging these values into the formula, we can calculate the total number of different selections possible: C(48, 7) = 48! / [7!(48-7)!]. This calculation would give us the total number of combinations of 7 numbers that can be selected from a pool of 48, which represents all the different possible LOTTO selections. It should be remembered that factorials such as 48! or 7! represent a product of an integer and all the integers below it, down to 1.

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There are 85,900,584 different selections possible in the LOTTO game.

To calculate the number of different selections possible in the LOTTO game, we need to focus on the concept of permutations.

With 48 numbers to choose from and 7 numbers to be selected, the number of permutations can be calculated using the formula P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items to be selected.

In this case, n = 48 and r = 7.

Using the formula, we have P(48, 7) = 48! / (48-7)! = (48 * 47 * 46 * 45 * 44 * 43 * 42) / (7 * 6 * 5 * 4 * 3 * 2 * 1)

= 85,900,584 different selections possible.

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Step-by-step explanation:

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

Kyd's expected value for this game is -$0.38.

North's expected value for this game is  $0.38.Kyd is expected to lose money, and North is expected to gain money.  

North has the advantage.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Given

In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings).  The remaining 40 cards are non-face cards.

The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.

a) Kyd's expected value is:

K = (12/52)(5) + (40/52)(-2)

K = -5/13

K ≈ -$0.38

b) North's expected value is:

N = (12/52)(-5) + (40/52)(2)

N = 5/13

N ≈ $0.38

c) Kyd is expected to lose money, and North is expected to gain money.  North has the advantage.

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Answer: [tex]\dfrac{2}{1001}[/tex]

Step-by-step explanation:

Given : The number of female applicants = 9

The number of male applicants = 6

Total applicants = 15

The number of ways to select 5 applicants from 15 applicants :-

[tex]^{15}C_5=\dfrac{15!}{5!(15-5)!}=3003[/tex]

The number to select 5 applicants from 15 applicants such that no female applicant is selected:-

[tex]^{9}C_0\times^6C_5=1\times\dfrac{6!}{5!(6-5)!}=6[/tex]

Now, the required probability :-

[tex]\dfrac{6}{3003}=\dfrac{2}{1001}[/tex]

Answer:

B. 3√2/2

Step-by-step explanation:

The value of x can be found with the tan rule.

Step 1: Identify the sides as opposite, adjacent and hypotenuse to apply the formula

Opposite = (opposite from angle 45 degrees)

Adjacent = x (between angle 45 degrees and 90 degrees)

Hypotenuse = 3 (opposite from 90 degrees)

Step 2: Apply the tan formula

Cos (angle) = adjacent/hypotenuse

Cos (45) = x/3

√2/2 = x/3

x = √2/2 x 3

x = 3√2/2

Therefore, the correct answer is B; x = 3√2/2

!!

With 22 packed cheese sandwiches made, bread is the 'limiting' ingredient, and we have three slices of cheese in excess.

The number of sandwiches that can be made depends on the availability of the critical ingredients needed to complete a sandwich: 2 slices of bread and three slices of cheese. To calculate the number of sandwiches, we can determine how many sandwiches each ingredient batch could provide, then find the minimum of those two, as the 'limiting' factor will be the ingredient that finishes first.

For bread, we have 44 pieces/ 2 pieces per sandwich = 22 sandwiches. With cheese, we have 69 pieces/ 3 pieces per sandwich = 23 sandwiches. Therefore, we can fully make a maximum of 22 sandwiches because we would run out of bread first. This means the bread is the 'limiting' ingredient, while cheese is in excess. With 22 sandwiches made, we would use three slices of cheese per sandwich for 66 slices used, leaving three slices of cheese remaining. The quantity in excess is three slices of cheese.

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There are 0 slices of bread in excess, and there are 3 slices of cheese in excess.

Let's start by calculating the maximum number of sandwiches that can be made using 44 slices of bread:

Each sandwich requires 2 slices of bread.

So, the maximum number of sandwiches that can be made using 44 slices of bread is 44/2 = 22 sandwiches.

Now, let's calculate the maximum number of sandwiches that can be made using 69 slices of cheese:

Each sandwich requires 3 slices of cheese.

So, the maximum number of sandwiches that can be made using 69 slices of cheese is 69/3 = 23 sandwiches.

Now, we compare the results:

With 44 slices of bread, we can make 22 sandwiches.

With 69 slices of cheese, we can make 23 sandwiches.

Since we can only make 22 sandwiches due to the limitation of the bread, the bread is the limiting ingredient.

The quantity of the ingredient in excess can be found by subtracting the number of sandwiches that can be made using the limiting ingredient from the total number of slices of that ingredient.

For the bread:

Quantity of bread in excess = Total slices of bread - (Number of sandwiches × Slices of bread per sandwich)

Quantity of bread in excess = 44 - (22 *2)

= 44 - 44

= 0

For the cheese:

Quantity of cheese in excess = Total slices of cheese - (Number of sandwiches × Slices of cheese per sandwich)

Quantity of cheese in excess = 69 - (22 *s 3)

= 69 - 66

= 3

So, there are 0 slices of bread in excess, and there are 3 slices of cheese in excess.

Answer:

Step-by-step explanation:Find the slope of the line that passes through the points shown in the table.

The slope of the line that passes through the points in the table is

.

Answer:

0.1971 ( approx )

Step-by-step explanation:

Let X represents the event of weighing more than 20 pounds,

Since, the binomial distribution formula is,

[tex]P(x)=^nC_r p^r q^{n-r}[/tex]

Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

Given,

The probability of weighing more than 20 pounds, p = 25% = 0.25,

⇒ The probability of not weighing more than 20 pounds, q = 1-p = 0.75

Total number of samples, n = 16,

Hence, the probability that fewer than 3 weigh more than 20 pounds,

[tex]P(X<3) = P(X=0)+P(X=1)+P(X=2)[/tex]

[tex]=^{16}C_0 (0.25)^0 (0.75)^{16-0}+^{16}C_1 (0.25)^1 (0.75)^{16-1}+^{16}C_2 (0.25)^2 (0.75)^{16-2}[/tex]

[tex]=(0.75)^{16}+16(0.25)(0.75)^{15}+120(0.25)^2(0.75)^{14}[/tex]

[tex]=0.1971110499[/tex]

[tex]\approx 0.1971[/tex]

Answer with explanation:

To test the Significance of the population which is Normally Distributed we will use the following Formula Called Z test

   [tex]z=\frac{\Bar X - \mu}{\frac{\sigma}{n}}[/tex]

[tex]\Bar X =31.1\\\\ \sigma=6.3\\\\ \mu=25\\\\n=15\\\\z=\frac{31.1-25}{\frac{6.3}{\sqrt{15}}}\\\\z=\frac{6.1\times\sqrt{15}}{6.3}\\\\z=\frac{6.1 \times 3.88}{6.3}\\\\z=3.756[/tex]

→p(Probability) Value when ,z=3.756 is equal to= 0.99992=0.9999

⇒Significance Level (α)=0.01

We will do Hypothesis testing to check whether population mean is different from 25 at the alpha equals 0.01 level of significance.

→0.9999 > 0.01

→p value > α

With a z value of 3.75, it is only 3.75% chance that ,mean will be different from 25.

So,we conclude that results are not significant.So,at 0.01 level of significance population mean will not be different from 25.

Answer:

Step-by-step explanation:

Each Calculus text returns $10/2 = $5 per unit of shelf space. For History and Marketing texts, the respective numbers are $4/1 = $4 per unit, and $8/4 = $2 per unit. Using 1200 units of shelf space for 600 Calculus texts returns ...

  $5/unit × 1200 units = $6000 . . . profit

Any other use of units of shelf space will reduce profit.

The College should purchase only 600 Calculus textbooks to maximize College profits.

Data and Calculations:

Budget limit on the number of textbooks per semester = 700

Shelf space for the purchased textbooks = 1,200 units

                                                       Calculus      History      Marketing

Shelf-space occupied

by each textbook                              2                   1                  4

Profit per textbook                        $10                 $4               $8

Profit per shelf-space                    $5 ($10/2)    $4 ($4/1)     $2 ($8/4)

The highest profit per shelf space is $5 generated by Calculus.

The highest profit that the College can make over the purchase of used Calculus textbooks = $6,000 ($5 x 1,200) or ($10 x 600).

Thus, for the College to maximize its profits, it should purchase 600 Calculus textbooks, which will not exceed the textbook limit for the semester, and at the same time maximally utilize the shelf space available.

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Answer:

[tex]x+\sqrt{3}y=8[/tex]

Step-by-step explanation:

Given equation of curve,

[tex]y^2+x^2=16[/tex]

[tex]\implies y^2=16-x^2[/tex]

Differentiating with respect to x,

[tex]2y\frac{dy}{dx}=-2x[/tex]

[tex]\implies \frac{dy}{dx}=-\frac{x}{y}[/tex]

Since, the tangent line of the curve contains the point (2, 2√3),

Thus, the slope of the tangent line,

[tex]m=\left [ \frac{dy}{dx} \right ]_{(2, 2\sqrt{3})}=-\frac{1}{\sqrt{3}}[/tex]

Hence, the equation of tangent line would be,

[tex]y-2\sqrt{3}=-\frac{1}{\sqrt{3}}(x-2)[/tex]

[tex]\sqrt{3}y-6=-x+2[/tex]

[tex]\implies x+\sqrt{3}y=8[/tex]

Answer:

Simple interest=4500

maturity value= 10500

Step-by-step explanation:

Given: Principal P = 6000

Rate % R = 5%

time T = 15 years

we know that simple interest [tex]SI=\frac{PRT}{100}[/tex]

⇒[tex]SI=\frac{6000\times5\times15}{100}[/tex]

on calculating we get SI = 4500

and maturity value = Principal amount + Simple interest

maturity value = 6000+4500 = 10500

hence the final answers are

Simple interest= 4500

and maturity value= 10500

hope this helps!!

Answer:

The center is the point (3,1) and the radius is 3 units

Step-by-step explanation:

we know that

The equation of a circle in standard form is equal to

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

we have

[tex]x^{2}+y^{2}-6x-2y+1=0[/tex]

Convert to standard form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](x^{2}-6x)+(y^{2}-2y)=-1[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex](x^{2}-6x+9)+(y^{2}-2y+1)=-1+9+1[/tex]

[tex](x^{2}-6x+9)+(y^{2}-2y+1)=9[/tex]

Rewrite as perfect squares

[tex](x-3)^{2}+(y-1)^{2}=9[/tex]

The center is the point (3,1) and the radius is 3 units

Answer:

  6 cm by 17 cm

Step-by-step explanation:

The area is the product of the dimensions; the perimeter is double the sum of the dimensions.

So, we want to find two numbers whose product is 102 and whose sum is 23.

  102 = 1·102 = 2·51 = 3·34 = 6·17

The last of these factor pairs has a sum of 23.

The dimensions are 6 cm by 17 cm.

Answer:

A (9, 3)

Step-by-step explanation:

First the point is rotated 90° counterclockwise about the origin.  To do that transformation: (x, y) → (-y, x).

So S(-3, -5) becomes S'(5, -3).

Next, the point is translated +4 units in the x direction and +6 units in the y direction.

So S'(5, -3) becomes S"(9, 3).

Answer:

C. 62 degrees

Step-by-step explanation:

Alright I see a circle and a half a rotation where the diameter is at. A half of 360 degrees is 180 degrees.

So the arc measure in degrees for EG is 180 degrees (both the left piece and right piece have this measure).

Since EG is 180 then FG=EG-EF=180-56=124.

To find x we have to half 124 since it is the arc measure where x is but x is the inscribed angle.

x=124/2=62

C.

62 degrees ! All the rest wouldn’t make sense (:

Answer:

112.77 gallons of water

Step-by-step explanation:

We will calculate the area by Trapezoid formula :

[tex]A=\frac{a+b}{2}\times h\times l[/tex]

Given Base a = 12 inches  = 1 feet

          Base b = 3 inches  =  0.25 feet

          height  = 8 inches  = 0.67 feet

          length  = 36 feet    = 36 feet

[tex]Area=\frac{1+2.5}{2}\times 0.67\times 36[/tex]

= 0.625 × 0.67 × 36

= 15.075 cubic feet.

As we know 1 cubic feet = 7.48052 per liquid gallon

Therefore, 15.075 cubic feet = 15.075 × 7.48052

                                               = 112.768839 ≈ 112.77 liquid gallon

When full it will hold 112.77 gallons of water

Answer:

5377

Step-by-step explanation:

C(x) = x^2 - 400x + 45,377

To find the location of the minimum, we take the derivative of the function

We know that is a minimum since the parabola opens upward

dC/dx = 2x - 400

We set that equal to zero

2x-400 =0

Solving for x

2x-400+400=400

2x=400

Dividing by2

2x/2=400/2

x=200

The location of the minimum is at x=200

The value is found by substituting x back into the equation

C(200) = (200)^2 - 400(200) + 45,377

           =40000 - 80000+45377

            =5377

Answer:

The minimum unit cost is 5377

Step-by-step explanation:

Note that we have a cudratic function of negative principal coefficient.

The minimum value reached by this function is found in its vertex.

For a quadratic function of the form

[tex]ax ^ 2 + bx + c[/tex]

the x coordinate of the vertex is given by the following expression

[tex]x=-\frac{b}{2a}[/tex]

In this case the function is:

[tex]C(x) = x^2 - 400x + 45,377[/tex]

So:

[tex]a=1\\b=-400\\c=45,377[/tex]

Then the x coordinate of the vertex is:

[tex]x=-\frac{-400}{2(1)}[/tex]

[tex]x=200\ cars[/tex]

So the minimum unit cost is:

[tex]C(200) = (200)^2 - 400(200) + 45,377[/tex]

[tex]C(200) = 5377[/tex]

I assume the cone has equation [tex]z=\sqrt{x^2+y^2}[/tex] (i.e. the upper half of the infinite cone given by [tex]z^2=x^2+y^2[/tex]). Take

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

The volume of the described region (call it [tex]R[/tex]) is

[tex]\displaystyle\iiint_R\mathrm dx\,\mathrm dy\,\mathrm dz=\int_0^{2\pi}\int_0^{\sqrt2}\int_{\pi/4}^\pi\rho^2\sin\varphi\,\mathrm d\varphi\,\mathrm d\rho\,\mathrm d\theta[/tex]

The limits on [tex]\theta[/tex] and [tex]\rho[/tex] should be obvious. The lower limit on [tex]\varphi[/tex] is obtained by first determining the intersection of the cone and sphere lies in the cylinder [tex]x^2+y^2=1[/tex]. The distance between the central axis of the cone and this intersection is 1. The sphere has radius [tex]\sqrt2[/tex]. Then [tex]\varphi[/tex] satisfies

[tex]\sin\varphi=\dfrac1{\sqrt2}\implies\varphi=\dfrac\pi4[/tex]

(I've added a picture to better demonstrate this)

Computing the integral is trivial. We have

[tex]\displaystyle2\pi\left(\int_0^{\sqrt2}\rho^2\,\mathrm d\rho\right)\left(\int_{\pi/4}^\pi\sin\varphi\,\mathrm d\varphi\right)=\boxed{\frac43(1+\sqrt2)\pi}[/tex]

Answer:

Step-by-step explanation:

In March, Your Co. will collect 20% of January's sales, 30% of February's sales, and 50% of March's sales:

  .20×50 +.30×40 +.50×60 = 10 +12 +30 = 52

Similarly, in April, collections will be ...

  .20×40 + .30×60 + .50×30 = 8 +18 +15 = 41

Answer:Am nevoie de Puncte

Step-by-step explanation:

Answer:

The value of r(s(3)) = -21

Step-by-step explanation:

It is given that,

r(x) = -2x + 1

s(x) = -x^2 + 2

To find the value of r(s(3))

s(x) = -x^2 + 2

s(3) = (-3)^2 + 2    [Substitute 3 instead of x]

 = 9 + 2

 = 11

Therefore s(3) = 11

r(x) = -2x + 1

r(s(3)) = r(11)   [Substitute 11 instead of x]

 =   -2(11) + 1

 = -22 + 1

  = -21

Therefore the value of r(s(3)) = -21

The answer is:

[tex]r(s(3))=15[/tex]

To solve the problem, first, we need to compose the functions, and then evaluate the obtained function. Composing function means evaluating a function into another function.

We have that:

[tex]f(g(x))=f(x)\circ g(x)[/tex]

From the statement we know the functions:

[tex]r(x)=-2x+1\\s(x)=-x^{2}+2[/tex]

We need to evaluate the function "s" into the function "r", so:

[tex]r(s(x))=-2(-x^2+2)+1\\\\r(s(x))=2x^{2}-4+1=2x^{2}-3[/tex]

Now, evaluating the function, we have:

[tex]r(s(3))=2(3)^{2}-3=2*9-2=18-3=15[/tex]

Have a nice day!

Answer: 0.4772

Step-by-step explanation:

Given : The distribution is bell shaped , then the distribution must be normal distribution.

Mean : [tex]\mu=\$280[/tex]

Standard deviation :[tex]\sigma= \$40[/tex]

The formula to calculate the z-score :-

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

For x = $200

[tex]z=\dfrac{200-280}{40}=-2[/tex]

For x = $280

[tex]z=\dfrac{280-280}{40}=0[/tex]

The p-value = [tex]P(-2<z<0)=P(z<0)-P(z<-2)[/tex]

[tex]0.5-0.0227501=0.4772499\approx0.4772[/tex]

Hence, the proportion of students will spend between $200 and $280 this semester = 0.4772

Approximately 34% of the college students will spend between $200 and $280 on book purchases for a semester according to the empirical rule for a bell-shaped and symmetric distribution.

The distribution of the amount of money spent on book purchases for a semester by college students is described as bell-shaped and symmetric with a mean of $280 and a standard deviation of $40. To find the proportion of students that will spend between $200 and $280, we can use the empirical rule (68-95-99.7 rule) which indicates that approximately 68% of the data fall within one standard deviation from the mean. Since we are concerned with the range from $200 (which is 2 standard deviations below the mean) to $280 (the mean), we are effectively looking at half of this 68% range. Therefore, approximately 34% of the students are expected to spend between $200 and $280 on books for a semester.

To calculate the proportion, we take 68% of the range (which covers -1 to +1 standard deviation from the mean) and divide it by 2:

Thus, the proportion of students spending between $200 and $280 is 0.34, which when rounded to four decimal places, gives us 0.3400.