Algebra Question

Jasmine played outside today. She asked her mother if she could play outside tomorrow. Her mother said she could play outside only if the difference between today’s temperature and tomorrow’s temperature is less than 10 degrees. If today’s temperature was 93, what is the range of temperatures that would allow Jasmine to play outside tomorrow?

Address each of these parts:

a. Define the variable for the situation.

b. Write an absolute value inequality to represent the situation (tolerance).

c. Solve the inequality.


Referring back to the last question, using words, write the interpretation of the solution in terms of the range of temperatures that will allow Jasmine to play outside. Then, graph the solution.

There is just one solution to this system. The answer is (460/7), (-13/7), and (-24/7). The correct answer is option A.

Given the system of equations as X+ y- z=63x- y+ z=2x- 4y+ 2z-34.We have to use row operations to solve this system of equations. Let us start by writing down the augmented matrix of the given system of equations.  [1, 1, -1 | 6][3, -1, 1 | 2][1, -4, 2 | -34]

The first step is to change the first element of the second row to zero. For that, we subtract three times the first row from the second row to get the following:  [1, 1, -1 | 6][0, -4, 4 | -16][1, -4, 2 | -34]

Now, we need to change the first element of the third row to zero. For that, we subtract the first row from the third-row to get the following:  [1, 1, -1 | 6][0, -4, 4 | -16][0, -5, 3 | -40]

The next step is to change the second element of the third row to zero. For that, we add 5/4 times the second row to the third row to get the following:  [1, 1, -1 | 6][0, -4, 4 | -16][0, 0, 7 | -24]

Now, we solve the system of equations using back-substitution. We have 7z = -24 ⇒ z = -24/7

Substituting this value of z in the second equation, we get -4y = 4 - z = 4 + 24/7 = 52/7⇒ y = -13/7

Substituting these values of y and z in the first equation, we get x = 63 - y + z = 63 + 13/7 - 24/7 = 460/7Thus, the solution of the given system of equations is (x, y, z) = (460/7, -13/7, -24/7). Therefore, the correct choice is A. This system has exactly one solution. The solution is (460/7, -13/7, -24/7).

The given system of equations is solved using row operations. It is found that this system has exactly one solution which is (460/7, -13/7, -24/7). Therefore, the correct choice is A. This system has exactly one solution. The solution is (460/7, -13/7, -24/7).

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a) Passed at least one subject: 100 students

b) Passed exactly two subjects: 90 students

c) Passed Geography and failed Mathematics: 20 students

d) Passed all three subjects: 15 students

e) Failed Mathematics given that they passed History: 30 students.

To solve this problem, let's draw a Venn diagram to visualize the information given:

In the Venn diagram above, the circles represent the three subjects: Mathematics (M), History (H), and Geography (G). The numbers outside the circles represent the students who did not pass that particular subject, and the numbers inside the circles represent the students who passed the subject. The numbers in the overlapping regions represent the students who passed multiple subjects.

Now, let's answer the questions:

a) Passed at least one subject:

To find the number of students who passed at least one subject, we add the number of students in each circle (M, H, and G), subtract the students who passed two subjects (since they are counted twice), and add the students who passed all three subjects.

Total = M + H + G - (M ∩ H) - (M ∩ G) - (H ∩ G) + (M ∩ H ∩ G)

Total = 70 + 60 + 45 - 30 - 25 - 35 + 15

Total = 100

Therefore, 100 students passed at least one subject.

b) Passed exactly two subjects:

To find the number of students who passed exactly two subjects, we sum the students in the overlapping regions (M ∩ H, M ∩ G, and H ∩ G).

Total = (M ∩ H) + (M ∩ G) + (H ∩ G)

Total = 30 + 25 + 35

Total = 90

Therefore, 90 students passed exactly two subjects.

c) Passed Geography and failed Mathematics:

To find the number of students who passed Geography and failed Mathematics, we subtract the number of students in the intersection of M and G from the number of students who passed Geography.

Total = G - (M ∩ G)

Total = 45 - 25

Total = 20

Therefore, 20 students passed Geography and failed Mathematics.

d) Passed all three subjects:

To find the number of students who passed all three subjects, we look at the overlapping region (M ∩ H ∩ G).

Total = (M ∩ H ∩ G)

Total = 15

Therefore, 15 students passed all three subjects.

e) Failed Mathematics given that they passed History:

To find the number of students who failed Mathematics given that they passed History, we subtract the number of students in the intersection of M and H from the number of students who passed History.

Total = H - (M ∩ H)

Total = 60 - 30

Total = 30

Therefore, 30 students failed Mathematics given that they passed History.

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The area of the triangle in this problem is given as follows:

34 ft².

The area of a rectangle of base b and height h is given by half the multiplication of dimensions, as follows:

A = 0.5bh.

The parameters for this problem are given as follows:

b = 10 ft, h = 6.8 ft.

Hence the area is given as follows:

A = 0.5 x 10 x 6.8

A = 34 ft².

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Answer:

The equation of a hyperbola with co-vertices at (0, 7) and (0, -7) and a transverse axis that is 12 units long is:

x²/36 - y²/49 = 1.

Therefore, the correct equation is: x²/36 - y²/49 = 1.

The statement that is correct option d: The spread of the number of activities for Class N is less than for Class M.

The term 'spread' in mathematics refers to the difference between the largest and smallest values in a dataset or the range of the data. It's the extent to which the dataset is spread out.The median is the center of a dataset. It's the number that lies in the middle of the sorted values. Half the values are greater than the median, while the other half are lesser than the median.

An outlier is a value that is very different from the other values in the dataset.In class M, there are no outliers. The distribution is skewed to the right since most students have only a few activities, and some have many. The median is between 2 and 3.

In class N, there are no outliers. Most students have a moderate number of activities, and the spread is less than in Class M. The median is between 5 and 6.Hence, the correct statement is The spread of the number of activities for Class N is less than for Class M.The correct answer is d

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The measure of the Intercepted arc having an inscribed angle of 40 degrees is 80 degrees.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the figure:

Inscribed angle = 40 degrees

Intercepted arc = ?

Plug the given value into the above formula and solve for the arc:

Inscribed angle = 1/2 × intercepted arc.

40  = 1/2 × intercepted arc

Multiply both sides by 2:

40 × 2 = 2 × 1/2 × intercepted arc

40 × 2 = intercepted arc

Intercepted arc = 80°

Therefoore, the Intercepted arc is 80 degrees.

Option B) 80° is the correct answer.

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The angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

To find the angle the slide forms with the tower, we can use trigonometric ratios. Let's consider the right triangle formed by the height of the tower (18 m), the length of the slide (20 m), and the angle we want to find.

Using the tangent function, we have:

tan(angle) = opposite/adjacent

In this case, the opposite side is the height of the tower (18 m) and the adjacent side is the length of the slide (20 m). Therefore:

tan(angle) = 18/20

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(18/20)

Using a calculator, we find that arctan(18/20) is approximately 40.56 degrees.

Therefore, the angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

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Answer:

60 cm³

Step-by-step explanation:

volume of a rectangular pyramid

v = ( l * w * h ) / 3

v = ( 4 * 5 * 9 ) / 3

v = 60

Let's assume the dimensions of the iPad's screen are represented by length (L) and width (W). We can use the given information to set up two equations.

Perimeter equation:

The perimeter of a rectangle is given by the formula: 2(L + W). In this case, the perimeter is given as 36 inches.

So, 2(L + W) = 36.

Area equation:

The area of a rectangle is given by the formula: L * W. In this case, the area is given as 80 square inches.

So, L * W = 80.

We now have a system of two equations:

2(L + W) = 36

L * W = 80

From equation 1, we can simplify it to L + W = 18 and rewrite it as L = 18 - W.

Now substitute this value of L into equation 2:

(18 - W) * W = 80

Expanding the equation:

18W - W^2 = 80

Rearranging the equation to quadratic form:

W^2 - 18W + 80 = 0

Factoring or using the quadratic formula, we find the two possible values for W.

The solutions are W = 8 and W = 10.

Now we can substitute these values back into the equation L = 18 - W to find the corresponding values of L.

For W = 8:

L = 18 - 8 = 10

For W = 10:

L = 18 - 10 = 8

Therefore, the dimensions of the iPad's screen can be either 8 inches by 10 inches or 10 inches by 8 inches.

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The surface area and volume of the composite solid is are 1720ft² and 3563.33 ft³ respectively.

The area occupied by a three-dimensional object by its outer surface is called the surface area.

The surface area of the solid = lateral area of pyramid + surface area of cuboid

lateral area of pyramid = 4 × 1/2 bh

= 4 × 1/2 × 10× 12

= 120×2 = 240 ft²

Surface area of the cuboid = 2( 100+ 320+ 320)

= 2( 740)

= 1480 ft²

Surface area of the composite solid = 240 + 1480

= 1720 ft²

Volume of the composite solid = volume of cuboid + volume of pyramid

volume of cuboid = 10×10×32 = 3200ft²

volume of pyramid = 1/3base area × height

height of the pyramid is calculated as;

diagonal of base = √ 10²+10²

= √200

= 14.14

h² = 13²-7.07²

h² = 169 - 49.98

h² = 119.02

h = 10.9 ft

Volume of pyramid = 1/3 × 100 × 10.9

= 363.33 ft³

Volume of the composite solid = 3200+363.33

= 3563.33 ft³

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Answer:

See below

Step-by-step explanation:

Domain would be all the x-values, so this is {3, 8, 8, 9}
Range would be all the y-values, so this is {0, 3, p, q}

Answer: the answer is 110.8

Step-by-step explanation: add um all up

Answer:

(a) The graph is an exponential growth curve.

(b) f(0) = 2,  f(1) = 10

(c) The graph is entirely above the x-axis and increases from left to right, more steeply than the graph of y=5ˣ.

Step-by-step explanation:

Given function:

[tex]k(x)=2(5)^x[/tex]

[tex]\hrulefill[/tex]

The graph is an exponential growth curve.

The curve begins in quadrant II, crosses the y-axis at (0, 2) into quadrant I, and increases rapidly as x increases.

In general, an exponential growth function is in the form:

[tex]\boxed{y = ab^x}[/tex]

where:

In this case, the y-intercept is 2 and the base is 5.

As the value of a is positive, and the value of b is greater than 1, it indicates exponential growth.

[tex]\hrulefill[/tex]

The term "second coordinates" refers to the y-values of the points on a graph.

To find the second coordinates of the points with first coordinates 0 and 1, substitute x = 0 and x = 1 into the function and solve.

[tex]\begin{aligned}k(0)&=2(5)^{0}\\&=2(1)\\&=2\end{aligned}[/tex]

[tex]\begin{aligned}k(1)&=2(5)^{1}\\&=2(5)\\&=10\end{aligned}[/tex]

[tex]\hrulefill[/tex]

The graph is entirely above the x-axis and increases from left to right, more steeply than the graph of y=5ˣ.

As 2(5)ˣ > 0, the graph is entirely above the x-axis.

The end behaviour of the function is:

Therefore, the graph increases from left to right.

In y = 2(5)ˣ, the base is 5, but it is multiplied by 2. This means that for every unit increase in x, the corresponding y-value is doubled compared to the graph of y = 5ˣ. Therefore, the multiplication by 2 makes the graph of y = 2(5)ˣ steeper than the graph of y = 5ˣ.

The graph is completely on the x-axis; It then increases from left to right. it is less steep that y = 5ˣ

The values are (0, 2) and (1, 10)

From the question, we have the following parameters that can be used in our computation:

f(x) = 2(5ˣ)

The above function is an exponential growth curve

This means that

Here, we have

f(x) = 2(5ˣ)

This gives

f(0) = 2(5⁰)

f(0) = 2

f(1) = 2(5¹)

f(1) = 10

So, the values are (0, 2) and (1, 10)

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The equivalent function that best shows the x-intercepts on the graph of f(x) = 4x² - 1 is option (a) or (b): Of(x) = (4x + 1)(4x - 1).

The correct answer to the given question is option a or b.

The given function is f(x) = 4x² - 1. We need to choose the equivalent function that best shows the x-intercepts on the graph.The x-intercepts are the points where the graph of a function intersects the x-axis. At the x-intercepts, the value of y is zero.

Therefore, to find the x-intercepts, we need to solve the equation f(x) = 0 for x. The function f(x) = 4x² - 1 can be factored as:(2x + 1)(2x - 1)

To find the x-intercepts, we set f(x) = 0:4x² - 1 = 0(2x + 1)(2x - 1) = 0So, either 2x + 1 = 0 or 2x - 1 = 0. Solving these equations, we get:

x = -1/2 or x = 1/2

These are the x-intercepts of the graph of f(x) = 4x² - 1.Now, let's look at the given options and determine which one shows the x-intercepts on the graph:

Option (a):

Of(x) = (4x + 1)(4x - 1)

This is the factored form of f(x) = 4x² - 1. It correctly shows the x-intercepts.

Option (b):

Of(x) = (2x + 1)(2x - 1)

This is the same as option (a) and correctly shows the x-intercepts.

Option (c): f(x) = 4(x² + 1)

This function does not have any x-intercepts. It has a minimum value of 4 at x = 0.

Option (d): f(x) = 2(x² - 1)

This function has x-intercepts at x = -1 and x = 1. It does not show the x-intercepts of the given function.

Therefore, the equivalent function that best shows the x-intercepts on the graph of f(x) = 4x² - 1 is option (a) or (b):

Of(x) = (4x + 1)(4x - 1).

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The true options are:

A. If p = a number is negative and q = the additive inverse is positive, the original statement is p → q.

B. If p = a number is negative and q = the additive inverse is positive, the inverse of the original statement is ~p → ~q.

E. If q = a number is negative and p = the additive inverse is positive, the converse of the original statement is q → p.

Option A represents the original statement accurately. It states that if a number is negative (p), then the additive inverse is positive (q). This corresponds to the implication p → q, where the antecedent is p and the consequent is q.

Option B represents the inverse of the original statement. It states that if a number is not negative (~p), then the additive inverse is not positive (~q). This is the negation of the original statement and can be written as ~p → ~q.

Option C represents the converse of the original statement. It states that if the additive inverse is not positive (~q), then the number is not negative (~p). The converse swaps the positions of the antecedent and consequent, resulting in ~q → ~p.

Options D and E are not true. Option D represents the contrapositive of the original statement, which would be if the additive inverse is not positive (~q), then the number is not negative (~p). However, the contrapositive should have the negation of both the antecedent and the consequent, so the correct contrapositive would be ~q → ~p.

Option E incorrectly represents the converse by stating that if the additive inverse is negative (q), then the number is positive (p), which is not an accurate representation of the converse.

In summary, the true options are A, B, and C, as they accurately represent the original statement, its inverse, and its converse, respectively.

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The complete question is :

Given the original statement "If a number is negative, the additive inverse is positive,” which are true? Select three options.

A. If p = a number is negative and q = the additive inverse is positive, the original statement is p → q.

B. If p = a number is negative and q = the additive inverse is positive, the inverse of the original statement is ~p → ~q.

C. If p = a number is negative and q = the additive inverse is positive, the converse of the original statement is ~q → ~p.

D. If q = a number is negative and p = the additive inverse is positive, the contrapositive of the original statement is ~p → ~q.

E. If q = a number is negative and p = the additive inverse is positive, the converse of the original statement is q → p.

Answer:

0.090909

Step-by-step explanation:

I used a calculator

Answer:

63

Step-by-step explanation:

Given the expression:

[tex]\displaystyle{x^2+3x-7}[/tex]

Substitute x = -7:

[tex]\displaystyle{7^2+3(7)-7}[/tex]

Evaluate:

[tex]\displaystyle{(7)(7)+3(7)-7}\\\\\displaystyle{=49+21-7}\\\\\displaystyle{=63}[/tex]

Answer:

200x - 40 (5x-1)

Step-by-step explanation:

2 x 10 = 20

20 x 10 = 200

20 x -2 = -40

200 - 40 = 100 - 20, 50 - 10, 20 - 4, 10 - 2, 5 -1

so the answer is either 5x - 1 or 200x - 40

Answer:

1/12 is the correct answer

(a) Chin had 93.1 marbles after the game.

(b) The three children had a total of 271.44 marbles altogether.

Let's break down the problem step by step to find the answers:

Initial marbles

Before the game:

Let's assume Afiq had x marbles.

Bala had 1/6 fewer marbles than Afiq, so Bala had (x - 1/6x) marbles.

Chin had 3/5 as many marbles as Bala, so Chin had (3/5)(x - 1/6x) marbles.

After the game

After the game, Bala lost 20% of his marbles to Chin, so he has 80% (or 0.8) of his initial marbles remaining.

Afiq lost 2/3 of his marbles to Chin, so he has 1/3 (or 0.33) of his initial marbles remaining.

Calculating the marbles

(a) How many marbles did Chin have after the game?

To find Chin's marbles after the game, we add the marbles gained from Bala to Chin's initial marbles and the marbles gained from Afiq to Chin's initial marbles.

Chin's marbles = Initial marbles + Marbles gained from Bala + Marbles gained from Afiq

Chin's marbles = (3/5)(x - 1/6x) + 0.8(x - 1/6x) + 0.33x

Chin's marbles = (3/5)(5x/6) + 0.8(5x/6) + 0.33x

Chin's marbles = (3/6)x + (4/6)x + 0.33x

Chin's marbles = (7/6)x + 0.33x

We are given that Chin gained 105 marbles, so we can equate the equation above to 105 and solve for x:

(7/6)x + 0.33x = 105

(7x + 2x) / 6 = 105

9x / 6 = 105

9x = 105 * 6

x = (105 * 6) / 9

x = 70

Substituting the value of x back into the equation for Chin's marbles:

Chin's marbles = (7/6)(70) + 0.33(70)

Chin's marbles = 10(7) + 0.33(70)

Chin's marbles = 70 + 23.1

Chin's marbles ≈ 93.1

Therefore, Chin had approximately 93.1 marbles after the game.

(b) After the game, the 3 children each bought another 40 marbles. To find the total number of marbles the 3 children have altogether, we need to sum up their marbles after the game and the additional 40 marbles for each.

Total marbles = Afiq's marbles + Bala's marbles + Chin's marbles + Additional marbles

Total marbles = 0.33x + 0.8(x - 1/6x) + (7/6)x + 40 + 40 + 40

Total marbles = 0.33(70) + 0.8(70 - 1/6(70)) + (7/6)(70) + 120

Total marbles = 23.1 + 0.8(70 - 11.7) + 81.7 + 120

Total marbles = 23.1 + 0.8 × 58.3 + 201.7

Total marbles = 23.1 + 46.64 + 201.7

Total marbles = 271.44

The three children had a total of 271.44 marbles altogether.

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Question

Afiq. Bala and Chin played a game of marbles. Before the game, Bala had 1/ 6 fewer marbles than Afig and Chinhad 3/5 as many marbles as Bala.

After the game, Balahad lost 20% of his marbles to Chinwhile Afig had lost

2/3 of his marbles to Chin. Chingained 105 marbles at the end of the

game.

(a)How many marbles didChinhave after the game?

(b)After the game, the 3 children each bought another 40 marbles. How manymarbles did the 3 children have altogether?

The number of possible integral values of 'a' is 4 and the possible value of a are 1, 2, 5, and 10.

here we have to  find the number of possible integral values of 'a' that satisfy the given conditions, we need to compare the two inequalities:

x² - 6ax + 5a² <= 0

x² - 14x + 40 <= 0

Let's analyze each inequality separately:

x² - 6ax + 5a² <= 0

x² - 5ax -xa + 5a²<=0

(x - a)(x - 5a) <= 0

Case 1: (x - a) <= 0 and (x - 5a) <= 0

This implies a <= x <= 5a.

Case 2: (x - a) >= 0 and (x - 5a) >= 0

This implies x >= a and x >= 5a.

x² - 14x + 40 <= 0

x² - 10x-4x + 40 <= 0

(x - 4)(x - 10) <= 0

Case 3: (x - 4) <= 0 and (x - 10) <= 0

This implies 4 <= x <= 10.

Case 4: (x - 4) >= 0 and (x - 10) >= 0

This implies x >= 4 and x >= 10, which simplifies to x >= 10.

Case 1 (a <= x <= 5a) and Case 4 (x >= 10).

Since x >= 10, the lower bound of the intersection should be 10. We can substitute this value into the first inequality:

a <= 10 <= 5a

Dividing both sides by an (assuming a is positive), we get:

1 <= 10/a <= 5

To satisfy this condition, 'a' must be an integer divisor of 10. The integral values of 'a' that satisfy this condition are 1, 2, 5, and 10.

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Complete question:

if all the solutions of the inequality x² -6ax + 5a²<=0 are also the solutions of inequality x²- 14x + 40<=0 then find the number of possible integral values of a.

The question pertains to a quadratic inequality. A solution process could be carried out given the correct quadratic formula, although the initial inequality seems to contain a typo due to the lack of a comparison operator.

The question you asked is about finding the solution to quadratic inequality x^2-6ax+5a^2. In general, the solutions or roots for any quadratic equation can be calculated using the formula: -b ± √b² - 4ac / 2a. Therefore, you can potentially apply this formula to your inequality.

However, it appears that there might be a typo in your question, as an inequality should have a comparison operator (like <, >, ≤, or ≥). If the full equation was x^2-6ax+5a^2 ≤ 0 or ≥ 0, we could carry out the solution process with the given formula.

I would recommend reviewing the question to ensure that it's written correctly. Once you have the correct inequality, you can apply the quadratic formula and solve for your variable 'x'.

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The value of x in ΔSTU is in the range (2, 32).

Given that ΔSTU is a triangle,

ST = 15,

US = x,

UT = 17.

We need to find the length of x in the triangle STU.

To solve this question, we need to apply the triangle inequality theorem.

According to the theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Therefore, using this theorem, we get:

ST + US > UT => 15 + x > 17

=> x > 17 - 15

=> x > 2US + UT > ST

=> x + 17 > 15 => x > 15 - 17

=> x > -2UT + ST > US

=> 17 + 15 > x

=> 32 > x

In this case ST = 15, US = x, UT = 17. Let's apply the triangle inequality to the sides of the triangle STU:

ST + US > UT

15 + x > 17

Simplify the inequality:

x > 17 - 15

x > 2

Triangle STU is valid The length of US(x) must be greater than 2 to be a perfect triangle.

However, without further information, we cannot determine the exact value of x.

You can specify any value greater than 2.

Therefore, from the above conditions, we conclude that 2 < x < 32.

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Answer:

Region A: 20,178 people/521 km²

= 38.7 people/km²

Region B: 1,200 people/451 km²

= 2.7 people/km²

Region C: 13,475 people/395 km²

= 34.1 people/km²

Region D: 6,980 people/426 km²

= 16.4 people/km²

The regions, in order from the most densely populated to the least densely populated: A, C, D, B

The type of medals is a categorical nominal variable, while the volume of water is a numerical continuous variable.

Note: This question is in Spanish, here is the question in English:

Write the type of variable and level of measurement for the following group of variables: A) type of medals at Olympic test. B) Volume of water in a tank

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Answer:

The total amount with annual compounding is 23,304.79 .

The interest with annual compounding is $18,304.79.

The total amount with semi annual compounding is 24,005.10.

The interest with semi annual compounding is $19,005.10.

The total amount with quarterly compounding is 24,377.20.

The interest with quarterly compounding is $19,377.20.

What are the total amount and interest?

The formula for determining the total amount is:

FV = PV(1 + r/m)^nm

Where:

FV =total amount

PV = amount deposited

r  = interest rate

n  = number of years

m = number of compounding

Interest = FV - amount deposited

FV = 5000 x (1.08)^20 = 23,304.79

Interest = 23,304.79 - 5000 = $18,304.79

FV =5000 x (1.08/2)^(2 x 20) =24,005.10

Interest = 24,005.10 - 5000 = $19,005.10

FV =5000 x (1.08/4)^(20 x 4) =24,377.20

Interest = 24,377.20 - 5000 = $19,377.20

Step-by-step explanation:

Answer:

[tex]_{13}C_5=1287[/tex]

Step-by-step explanation:

[tex]\displaystyle _nC_r=\frac{n!}{r!(n-r)!}\\\\_{13}C_5=\frac{13!}{5!(13-5)!}\\\\_{13}C_5=\frac{13!}{5!\cdot8!}\\\\_{13}C_5=\frac{13*12*11*10*9}{5*4*3*2*1}\\\\_{13}C_5=\frac{154440}{120}\\\\_{13}C_5=1287[/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5!(13 - 5)!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5!(8)!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13!}{5! \times 8!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8!}{5! \times 8!} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 \times \cancel{8!}}{5! \times \cancel{8!}} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{13 \times 12 \times 11 \times 10 \times 9 }{5 \times 4 \times 3 \times 2 \times 1} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{156 \times 110 \times 9 }{20 \times 6 \times 1} \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{156 \times 990 }{20 \times 6 } \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = \frac{154440 }{120 } \\ [/tex]

[tex] \sf \hookrightarrow \: \: {}^{13}{ C}_{5} \: = 1287 \\ [/tex]

The average rate of change of f(x) is less than average rate of change of g(x). Then the correct option is A.

A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.

The function f(x) is given in the table, then the rate of change of the function f(x) will be

[tex]\text{Rate of change of f(x)} = \dfrac{(-2 + 4.5)}{(-2 + 3)}[/tex]

[tex]\text{Rate of change of f(x)} = 2.5[/tex]

If function g is a quadratic function that contains the points (-3, 5) and (0, 14).

Then the rate of change of the function g(x) will be

[tex]\text{Rate of change of g(x)} = \dfrac{(14 - 5)}{(0 + 3)}[/tex]

[tex]\text{Rate of change of g(x)} = \dfrac{9}{3}[/tex]

[tex]\text{Rate of change of g(x)} = 3[/tex]

Thus, the average rate of change of f(x) is less than average rate of change of g(x).

Then the correct option is A.

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The minimum distance between the point (0, 4) on the y-axis and points on the graph is 4.

To find the minimum distance between the point (0, 4) on the y-axis and points on the graph of the function y = x², we can use the concept of perpendicular distance.

The distance between a point (x, y) on the graph and the point (0, 4) is given by the formula:

distance = √((x - 0)² + (y - 4)²) = √(x² + (y - 4)²)

Substituting the function y = x² into the distance formula, we get:

distance = √(x² + (x² - 4)²) = √(x² + (x⁴ - 8x² + 16))

Simplifying further, we have:

distance = √(x⁴ + x² - 8x² + 16) = √(x⁴ - 7x² + 16)

To find the minimum distance, we need to minimize the expression x⁴ - 7x² + 16. Since this is a quadratic-like expression, we can use calculus to find the minimum.

Taking the derivative of x⁴ - 7x² + 16 with respect to x, we get:

d/dx (x⁴ - 7x² + 16) = 4x³ - 14x

Setting the derivative equal to zero to find critical points:

4x³ - 14x = 0

Factorizing, we have:

2x(2x² - 7) = 0

This gives us two critical points: x = 0 and x = ±√(7/2).

Next, we evaluate the expression x⁴ - 7x² + 16 at these critical points and the endpoints of the interval:

f(0) = 0⁴ - 7(0)² + 16 = 16

f(±√(7/2)) = (√(7/2))⁴ - 7(√(7/2))² + 16 ≈ 4.157

Comparing these values, we find that the minimum distance occurs at x = 0, giving us a minimum distance of √(0⁴ - 7(0)² + 16) = √16 = 4.

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The cost of one share in the software company (x) is $32, and the cost of one share in the biotech company (y) is approximately $93.

Let's denote the cost of one share in the software company as 'x' dollars and the cost of one share in the biotech company as 'y' dollars.

According to the given information, Sally purchased 7 shares in the software company and 87 shares in the biotech company, which cost a total of $8,315. We can express this as the following equation:

7x + 87y = 8315 ----(Equation 1)

Similarly, Katle invested a total of $7,338 in 84 shares in the software company and 50 shares in the biotech company. This can be represented as the following equation:

84x + 50y = 7338 ----(Equation 2)

To solve this system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

Multiply Equation 1 by 50 and Equation 2 by 87 to eliminate the 'y' terms:

350x + 4350y = 415750 ----(Equation 3)

7308x + 4350y = 638526 ----(Equation 4)

Now, subtract Equation 3 from Equation 4:

(7308x + 4350y) - (350x + 4350y) = 638526 - 415750

Simplifying the equation:

7308x - 350x = 222776

6958x = 222776

Divide both sides of the equation by 6958:

x = 222776 / 6958

x = 32

Now that we have the value of 'x', we can substitute it back into either Equation 1 or Equation 2 to solve for 'y'. Let's substitute it into Equation 1:

7(32) + 87y = 8315

224 + 87y = 8315

87y = 8315 - 224

87y = 8091

Divide both sides of the equation by 87:

y = 8091 / 87

y ≈ 93

Therefore, the cost of one share in the software company (x) is $32, and the cost of one share in the biotech company (y) is approximately $93.

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