The following illustrates an application of optimization using parabolas.
A study found that the cost C per pupil of operating a high school in a certain country depends on the number n of students enrolled. The cost is given by
C = 70,000 − 500n + n2 dollars per pupil.
What enrollment produces the minimum cost per pupil?
n = students

The correct factorization for the Difference of Squares or a²-b² is (a+b)(a-b). This is because when you multiply (a+b)(a-b), you get a²- ab + ab - b², which simplifies to a²-b².

Here is a step-by-step explanation of how to factor the Difference between Squares:

Step 1: Identify the two terms that are being squared. In this case, a and b are the terms being squared.
Step 2: Write the two terms with a plus sign in between them in one set of parentheses and with a minus sign in between them in another set of parentheses. This will give you (a+b)(a-b).
Step 3: Multiply the two sets of parentheses together to check your answer. You should get a²-b².
So, the correct factorization for the Difference of Squares or a²-b² is (a+b)(a-b).

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1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).

What do you mean by decimal?

In mathematics, a decimal is a number that represents a fraction or a part of a whole using a base-ten positional numeral system.

To multiply 1 and 1/7 by 3/5, we can first convert the mixed number to an improper fraction.

1 and 1/7 can be written as:

(7/7 * 1) + 1/7 = 7/7 + 1/7 = 8/7

So, we have:

1 and 1/7 = 8/7

Now, we can multiply 8/7 by 3/5 as follows:

(8/7) * (3/5) = (8 * 3) / (7 * 5) = 24/35

Therefore, 1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).

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Answer:

1369.1

Step-by-step explanation:

Answer:

1369.1 feet

Step-by-step explanation:

Answer:$72 Total

Step-by-step explanation:

20%=1/5

1/5 + 5/5(total) = 6/5

60 * 6/5 = $72

Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.

To answer mathematical issues involving proportional relationships between quantities, one uses the unitary method. This approach involves calculating the value of a single unit of a quantity and using that value to calculate the value of any number of additional units. For instance, if we know that five items cost $20 each, we can calculate the price of any other quantity of items by first determining the price of one item, then increasing that figure by the desired quantity of items.

given

The quantity of lotion needed for each application, assuming a 4-oz container of self-tanning lotion yields seven applications, is:

4 oz / 7 = 0.57 oz (rounded to two decimal places) (rounded to two decimal places)

The quantity needed for each application must be multiplied by the number of applications needed by each individual, which equals 1, to determine how many ounces of self-tanning lotion the ensemble of 15 needs to buy:

15. People at 0.57 ounce each equals 8.55 oz.

Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.

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Answer: The angles are classified as interior angles, where x=9

Step-by-step explanation:

The angle measure of a straight line is 180°, meaning that 7x-10 and 12x+19 must add up to 180.

It is clear from the visual that the former is an acute angle, and the latter is an obtuse angle. This means that when solving each, 7x-10 must be less than 90, and 12x+19 must be greater than 90, but less than 180.

If you set up the equation to solve for x; (7x-10) + (12x+19) = 180, you must combine like terms.

19x+9=180 is the new equation. Since x needs to be isolated, subtract 9 from each side, to get 19x=171. Dividing each side by 19 results in x=9.

The required value of x for the triangle is 9 units.

The theorem you are referring to is known as the "Parallel Line Theorem" or "Thales' Theorem".

If a line is drawn parallel to one side of a triangle, then the other two sides of the triangle are divided proportionally.

Consider a triangle ABC with a line parallel to one of its sides, say side AB. Let the parallel line intersect sides AC and BC at points D and E, respectively.

Then,

AB/BD = CB/BE

In triangle;

[tex]$\frac{4x}{x-1} =\frac{27}{6}[/tex]

24x = 27x - 27

= 3x = 27

= x = 9

Thus, required value of x is 9.

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Answer:

Step-by-step explanation:

0.16x + 0.20y = 11

we can write 0.16 = 16/100 = 4/25 and 0.20 = 20/100 = 1/5

[tex]\frac{4}{25}x + \frac{1}{5}y = 11[/tex] --------(1)

[tex]x + y = 60 \\[/tex] --------(2)

multiply both side by 5 in equation (1) we get

[tex]\frac{4}{5}x + y =55[/tex]   --------(3)

Subtract equation 3 from 2

[tex]x - \frac{4}{5}x + y - y = 60 - 55\\\frac{x}{5} = 5\\ x = 25[/tex]

put the value of x in equation 2

25 + y = 60

y = 35

The compound interest on N2,000 for 2 years at the rate of 10% per annum is N441.00.

Compound interest is the interest earned not only on the principal amount but also on the accumulated interest. To calculate the compound interest, we use the formula:

where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

In this case, P = N2,000, r = 0.1 (10%), n = 1 (compounded annually), and t = 2 years.

So, A = 2000[tex](1 + 0.1/1)^{(1*2)[/tex] = N2,441.00

Therefore, the compound interest is N2,441.00 - N2,000.00 = N441.00.

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The system of equations 4x + 2y = 6 and y = -2x + 6  have infinite many solutions

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

Given the system of equation:

4x + 2y = 6

Divide through by 2:

2x + y = 3

y = -2x + 6     (1)

The second equation is:

y = -2x + 6      (2)

Since both equations are the same hence the system of equations have infinite many solutions

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Answer:

Below

Step-by-step explanation:

Diameter = 20 inches     radius = 10 inches

Area = pi r^2 = 3.14 * 10^2 = 314 in^2

The answer of third integer is 15.

To solve this problem, we can use algebra. Let's call the first of the three consecutive odd integers "x." The next two consecutive odd integers would then be "x + 2" and "x + 4."

The problem tells us that three times the first integer is 3 more than twice the third, so we can write an equation:

3x = 2(x + 4) + 3

Simplifying the equation, we get:

3x = 2x + 8 + 3

x = 11

So the first integer is 11, and the third integer is 11 + 4 = 15.

Therefore, the correct answer is O 15.

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30 peοple are playing chess and 30 peοple are playing cribbage.

A system οf linear equatiοns is a set οf twο οr mοre linear equatiοns with multiple variables that are sοlved simultaneοusly. The gοal οf sοlving a system οf linear equatiοns is tο find the values οf the variables that satisfy all the equatiοns in the system.

A linear equatiοn is an equatiοn in which the variables appear οnly tο the first pοwer, and the cοefficients οf the variables are cοnstants. Fοr example, the equatiοn y = 3x + 2 is a linear equatiοn, where y and x are the variables, and 3 and 2 are the cοefficients.

A system οf linear equatiοns can have οne unique sοlutiοn, nο sοlutiοn, οr infinitely many sοlutiοns. The number οf sοlutiοns depends οn the number οf equatiοns and variables in the system, and the relatiοnships between them. There are variοus methοds tο sοlve a system οf linear equatiοns, such as substitutiοn, eliminatiοn, and matrix methοds.

Let's assume that there are x peοple playing chess and y peοple playing cribbage. Since everyοne is playing either chess οr cribbage, we knοw that:

x + y = 60

We alsο knοw that chess is a twο-player game, while cribbage is a fοur-player game. Therefοre, the tοtal number οf players in the games must be a multiple οf 2 and 4. This means that:

2x + 4y = 60

We can simplify this equatiοn by dividing bοth sides by 2

x + 2y = 30

Nοw we have twο equatiοns:

x + y = 60

x + 2y = 30

We can sοlve fοr x by subtracting the secοnd equatiοn frοm the first equatiοn:

x + y - (x + 2y) = 60 - 30

-y = -30

y = 30

Substituting this value οf y intο the first equatiοn, we get:

x + 30 = 60

x = 30

Therefοre, 30 peοple are playing chess and 30 peοple are playing cribbage.

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Complete question:

60 people attend a game night. Everyone chooses to play chess, a two-player game, or cribbage, a four-player game. All 60 people are playing either chess or cribbage. There are 3 more games of cribbage being played than games of chess being played. How many of each game are being played?

8 chess, 15 cribbage

30 chess, 30 cribbage

8 chess, 11 cribbage

30 chess, 15 cribbage

Ship B is 22.8 km away from ship A on a bearing of N50°W.

The topmost point of the town hall is 28 metres above ground level.
The distance travelled by a point on the roundabout in one second is 113 cm for point A and 63 cm for point B.
Ship B is 22.8 km away from ship A on a bearing of N50°W.

Explanation:
1) For the first question, we can use trigonometry to find the height of the town hall. Let h be the height of the topmost point above ground level, and d be the distance between point A and the base of the town hall. Then we have:
tan(35°) = h/d and tan(60°) = h/(d-30)
Solving for h, we get:
h = d*tan(35°) and h = (d-30)*tan(60°)
Equating the two expressions for h, we get:
d*tan(35°) = (d-30)*tan(60°)
d = 30*tan(60°)/(tan(60°)-tan(35°)) ≈ 48.5 metres
Substituting back into the first equation, we get:
h = 48.5*tan(35°) ≈ 28 metres
Therefore, the topmost point of the town hall is 28 metres above ground level.

2) For the second question, we can use the formula for the circumference of a circle to find the distance travelled by a point on the roundabout in one second. The circumference of a circle is given by C = 2πr, where r is the radius of the circle. The distance travelled by a point in one second is then given by C/5, since the roundabout makes one revolution every five seconds. For point A, which is 1.8 metres from the centre of rotation, we have:
C = 2π*1.8 = 11.31 metres
Distance travelled in one second = 11.31/5 = 2.26 metres ≈ 226 cm ≈ 113 cm to the nearest centimetre
For point B, which is 1 metre from the centre of rotation, we have:
C = 2π*1 = 6.28 metres
Distance travelled in one second = 6.28/5 = 1.26 metres ≈ 126 cm ≈ 63 cm to the nearest centimetre

3) For the third question, we can use the law of cosines to find the distance between ship A and ship B. The law of cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite side c. Let x be the distance between ship A and ship B, and let θ be the angle between the two ships. Then we have:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(θ)
To find the angle θ, we can use the fact that the sum of the angles in a triangle is 180°. We have:
θ = 180° - 60° - 80° = 40°
Substituting back into the equation, we get:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(40°) ≈ 520.3
x ≈ 22.8 km
To find the bearing of ship B from ship A, we can use the law of sines. The law of sines states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the angles opposite those sides. Let α be the bearing of ship B from ship A. Then we have:
14.1/sin(α) = 22.8/sin(40°)
Solving for α, we get:
α = sin^-1(14.1*sin(40°)/22.8) ≈ 30°
Since ship B is to the west of ship A, the bearing of ship B from ship A is N50°W.
Therefore, ship B is 22.8 km away from ship A on a bearing of N50°W.

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Answer:

The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.

Substituting r = 15, we get:

V = (4/3)π(15)^3

V = (4/3)π(3375)

V = 4π(1125)

V = 4500π

Therefore, the volume of the sphere with a radius of 15 units is 4500π cubic units. This can also be approximated as 14,137.17 cubic units by using a value of 3.14 for π and rounding to the nearest hundredth.

Step-by-step explanation:

To determine the intervals where a function is increasing or decreasing, we need to take the derivative of the function and find the critical points. The critical points are where the derivative is equal to zero or undefined.

First, let's take the derivative of the function:
g'(x) = -1/(zx)^2, x<-2 . 3/2, x>-2

Now, let's find the critical points:
-1/(zx)^2 = 0 -> There are no values of x that will make this equation true, so there are no critical points for x<-2.
3/2 = 0 -> There are no values of x that will make this equation true, so there are no critical points for x>-2.

Since there are no critical points, the function is either always increasing or always decreasing. To determine which one it is, we can pick a value of x in each interval and plug it into the derivative:

For x<-2, let's pick x=-3:
g'(-3) = -1/(-3z)^2 = 1/(9z^2) > 0

Since the derivative is positive, the function is increasing on the interval (-infinity, -2).
For x>-2, let's pick x=0:
g'(0) = 3/2 > 0

Since the derivative is positive, the function is also increasing on the interval (-2, infinity).

Therefore, the function is increasing on the entire domain of the function, which is (-infinity, infinity).

Answer: Increasing on: (-infinity, infinity) Decreasing on: None

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Synthetic Division of the polynomials (x^(4)-5x^(3)+4x-17) and (x-5) gives answer "x^(3)+4+3/(x-5)."

Step 1: Write the coefficients of the polynomial in descending order. In this case, the coefficients are 1, -5, 0, 4, -17.

Step 2: Write the value of x from the divisor (x-5) in the left column. In this case, the value of x is 5.

Step 3: Bring down the first coefficient to the bottom row.

Step 4: Multiply the value of x by the number in the bottom row and write the result in the next column.

Step 5: Add the numbers in the second column and write the result in the bottom row.

Step 6: Repeat steps 4 and 5 until you have completed all the columns.

Step 7: The numbers in the bottom row are the coefficients of the quotient, and the last number is the remainder.

The synthetic division will look like this:

5 | 1 -5 0 4 -17
 |   5 0 0 20
 ----------------
   1  0 0 4   3

So, the quotient is x^(3)+0x^(2)+0x+4, or simply x^(3)+4, and the remainder is 3.

Therefore, the answer is (x^(4)-5x^(3)+4x-17)÷(x-5) = x^(3)+4+3/(x-5).

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The minimum value of the objective function is -15, which occurs at the corner point (0, 5).

The corner points of a linear programming problem are the points where the constraints intersect. These points can be found by solving the system of inequalities for each pair of constraints.

For this problem, we can find the corner points by solving the system of inequalities for each pair of constraints:

3x – y >= 2 and x + y <= 5:
- Add y to both sides of the first inequality: 3x >= 2 + y
- Subtract 2 from both sides of the first inequality: 3x - 2 >= y
- Substitute 3x - 2 for y in the second inequality: x + (3x - 2) <= 5
- Simplify: 4x <= 7
- Divide by 4: x <= 7/4
- Substitute 7/4 for x in the first inequality: 3(7/4) - 2 >= y
- Simplify: 5/4 >= y

The first corner point is (7/4, 5/4).

3x – y >= 2 and x >= 0:
- Set x = 0 and solve for y: 3(0) - y >= 2, y <= -2
- Set y = 0 and solve for x: 3x - 0 >= 2, x >= 2/3

The second corner point is (2/3, 0).

x + y <= 5 and x >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5

The third corner point is (0, 5).

x + y <= 5 and y >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5

The fourth corner point is (5, 0).

Now we can plug each corner point into the objective function to find the minimum value:

C = 3x – 3y

C = 3(7/4) - 3(5/4) = 3
C = 3(2/3) - 3(0) = 2
C = 3(0) - 3(5) = -15
C = 3(5) - 3(0) = 15

Therefore, the solution to the linear programming problem is (0, 5).

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The required, graph of the function f(x) = √[x+3]-1 is a curve that starts at (-3,-1) and extends to the right, increasing in value but at a decreasing rate due to the square root function.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
The graph of the function f(x) = √[x+3]-1 is a curve that starts at the point (-3,-1) and extends to the right indefinitely. The square root function √x has a domain of x ≥ 0, so in this case, the domain of the function is x ≥ -3.

The graph is always above the x-axis, as the square root function can only output non-negative values. The graph also approaches but never touches the horizontal line y = 0 as x increases without bound, since the -1 term in the function only shifts the graph downward by one unit.

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Answer:

3 Marbles were left.

Step-by-step explanation:

To find the number of marbles Ronen gave to each teammate, we can divide the total number of marbles by the number of teammates:

1003 ÷ 8 = 125 with a remainder of 3

Therefore, each teammate received 125 marbles, and there were 3 marbles left over.

So, 125 marbles were shared with each teammate, and 3 marbles were left.

The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

a. A simple event is an event that has only one possible outcome. An example of a simple event is getting a head on the third coin. This is represented by option C: "Getting a head on the third coin and a head on the fourth coin."
b. A joint event is an event that involves two or more simple events happening at the same time. An example of a joint event is getting a head on the first coin and a tail on the third coin. This is represented by option C: "Getting a head on the first coin and a tail on the third coin."
c. The complement of getting a tail on the fourth coin is getting a head on the fourth coin. This is represented by option A: "Getting a head on the fourth coin."
d. The sample space consists of all possible outcomes of the four flipped coins. This is represented by option D: "Getting a head or a tail on any of the four coins." The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

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The answer to the question is that \( A_{1} \) is smaller than \( A_{2} \)

To begin, it is important to note that the terms "emalier" and "5PRECALC7" are not relevant to the question and can be ignored. Additionally, there are several typos and extraneous information that can also be ignored. The main focus of the question is to determine the relationship between \( A_{1} \) and \( A_{2} \).

From the information provided, it is clear that \( A_{1} \) is smaller than \( A_{2} \). This is because the value of \( A_{1} \) is given as 2, while the values of \( b \) and \( c \) are 27 and 33, respectively. Since \( A_{2} \) is the sum of \( b \) and \( c \), it is clear that \( A_{2} \) is larger than \( A_{1} \).

Therefore, the answer to the question is that \( A_{1} \) is smaller than \( A_{2} \).

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The value of k is ∛√199.

Consider the expansion of x^2(3x^2 + k/x)^8. The constant term is 16128. We need to find the value of k.

The expansion of (3x^2 + k/x)^8 will have terms of the form (3x^2)^a(k/x)^b, where a + b = 8. The constant term will be the term where the powers of x cancel out, so we need to find a and b such that 2a - b = 0.

Solving for a and b, we get a = 4 and b = 8. So the constant term will be (3x^2)^4(k/x)^8 = 81x^8(k^8/x^8) = 81k^8.

Setting this equal to 16128 and solving for k, we get:

81k^8 = 16128
k^8 = 16128/81
k^8 = 199
k = ∛√199

Therefore, the value of k is ∛√199.

k = ∛√199.

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The quotient and remainder using long division for [tex](2x^(3)-6x^(2)+7x-11)/(2x^(2)+5)[/tex] are x-11/2 and 55/2x-11, respectively.

The quotient and remainder using long division for the given expression can be found by following these steps:

Step 1: Set up the long division as follows:
[tex]```2x^(2)+5 | 2x^(3)-6x^(2)+7x-11         |----------------------         |```[/tex]

Step 2: Divide the first term of the dividend (2x^(3)) by the first term of the divisor (2x^(2)) to get the first term of the quotient (x):
[tex]```2x^(2)+5 | 2x^(3)-6x^(2)+7x-11         |----------------------         | x```[/tex]

Step 3: Multiply the first term of the quotient (x) by the divisor (2x^(2)+5) and subtract the result from the dividend:
[tex]```2x^(2)+5 | 2x^(3)-6x^(2)+7x-11         |-(2x^(3)+5x)         |----------------------         |        -11x^(2)+7x-11```[/tex]

Step 4: Repeat steps 2 and 3 until the degree of the remainder is less than the degree of the divisor:
[tex]```2x^(2)+5 | 2x^(3)-6x^(2)+7x-11         |-(2x^(3)+5x)         |----------------------         | x     -11x^(2)+7x-11         |      -(-11x^(2)-55/2x)         |----------------------         |         55/2x-11```[/tex]

Step 5: The final result is the quotient and remainder:
[tex]```Quotient = x-11/2Remainder = 55/2x-11```[/tex]

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As per the concept of integral, the series ∑ne⁻ⁿ converges.

Let's consider the series ∑ne⁻ⁿ. Since each term of the series is positive, the first condition of the Integral Test is satisfied. To check the second condition, we need to determine whether the function f(x) = xe⁻ˣ is decreasing for x ≥ 1.

To do this, we can take the derivative of f(x) with respect to x:

f'(x) = e⁻ˣ - xe⁻ˣ

Setting f'(x) = 0, we get:

e⁻ˣ - xe⁻ˣ = 0

x = 1

So f(x) has a maximum value at x = 1. Since f'(x) < 0 for x ≥ 1, f(x) is decreasing for x ≥ 1.

Now, we can set up the corresponding integral:

∫₁^∞ xe⁻ˣ dx

To evaluate this integral, we can use integration by parts:

u = x, dv = e⁻ˣ dx

du = dx, v = -e⁻ˣ

∫₁^∞ xe⁻ˣ dx = -xe⁻ˣ │₁^∞ + ∫₁^∞ e⁻ˣ dx

= 0 + e⁻ˣ │₁^∞

= 1

Since the integral converges, the series ∑ne⁻ⁿ also converges by the Integral Test.

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Complete Question:

How do you use the Integral Test to determine convergence or divergence of the series: ∑ne⁻ⁿ from n=1 to infinity?

Answer: X = 14

Step-by-step explanation: (:

Answer:88 miles

Step-by-step explanation:

1.1x80+88

The possible number of distinct complex roots for a third-degree polynomial function are 0, 1, 2, or 3, making option (B) the correct answer.

A polynomial function of degree n can have at most n distinct roots. For a third-degree polynomial, n=3, so it can have at most 3 roots.

Complex roots of a polynomial come in conjugate pairs. So if a third-degree polynomial has one real root, then the other two roots must be a conjugate pair of complex roots. If the polynomial has two real roots, then the third root must be a complex root that is not real.

Therefore, the possible number of distinct complex roots for a third-degree polynomial function is 0, 1, 2, or 3, making option (B) the correct answer.

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Answer:

Kayla and Rachel raised $8.39, Francesca raised $18.89.

Step-by-step explanation:

Knowing that $35.67 is the total amount raised, that Rachel and Kayla raised the same amount (x) each, and that Francesca raised $10.50 more than them (x + 10.50), then the equation would look like:

35.67 = x + x + (x + 10.50)

or

35.67 = 3x + 10.50

From here, you can solve algebraically.

Isolate the variable by subtracting 10.50 from both sides

25.17 = 3x

Then, find X by dividing both sides by 3:

x = 8.39

Knowing that Kayla and Rachel raised $8.39 each, to find Francesca's contribution you can either:

A) Subtract Kayla and Rachel's contributions from the total amount:

     35.67 - 8.39 - 8.39 = 18.89

or

B) Add $10.50 to the amount that Kayla and Rachel raised:

     8.39 + 10.50 = 18.89