80 athletes are running a race. a gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. assume that there are no ties. in how many possible ways can the 3 medals be distributed?

Answer:

48

Step-by-step explanation:

So I attached an image explaining how I solved the problem, but essentially you use the Pythagorean Theorem to define the two other sides, which I named "a" and "b", and then defined them in terms of x, using the Pythagorean Theorem. I finally defined the hypotenuse of the entire triangle which is 36+64 in terms of these two equations. Once that entire thing was set up it was a matter of solving for x, to get 48

Answer:

X^2+(13-2)x -26

x^2+13x-2x-26

x(x+13) -2(x+13)

(x+13) (x-2)

Answer:

Step-by-step explanation

A) To factorize x^2 + 11x - 26, we need to find two numbers that multiply to give -26 and add to give 11. These numbers are 13 and -2. Therefore, we can write:

x^2 + 11x - 26 = (x + 13)(x - 2)

B) To factorize x^2 -5x -24, we need to find two numbers that multiply to give -24 and add to give -5. These numbers are -8 and 3. Therefore, we can write:

x^2 -5x -24 = (x - 8)(x + 3)

C) To factorize 9x^2 + 6x - 8, we first need to factor out the common factor of 3:

9x^2 + 6x - 8 = 3(3x^2 + 2x - 8)

Now we need to find two numbers that multiply to give -24 and add to give 2. These numbers are 6 and -4. Therefore, we can write:

9x^2 + 6x - 8 = 3(3x + 4)(x - 2)

Replace -0.5 with the fraction -1/2 if needed.

======================================================

Explanation:

The legs of any right triangle are always perpendicular. They meet up to form a right angle, aka 90 degree angle.

The line y = 2x+4 goes through (2,8). This can be confirmed with a graph. Or note how plugging x = 2 leads to y = 8.

Therefore, y = 2x+4 forms one of the legs of this triangle.

The slope here is 2 = 2/1; the negative reciprocal is -1/2 = -0.5, which is the slope of the perpendicular line. We want this line to go through (2,8)

So,

y = mx+b

8 = -0.5*2+b

8 = -1+b

8+1 = b

9 = b

b = 9

The equation of the perpendicular line is y = -0.5x+9

Replace -0.5 with the fraction -1/2 if needed.

Check out the graph below.

Answer:

(90 , 9)

(70 , 7)

Step-by-step explanation:

I'm not too sure of the answer, thanks

Answer:

32.9

Step-by-step explanation:

7*9.4/2=32.9

Answer:

Ok???????????????????

The equivalent expressions of \(x^3\) are\(\frac{(x ^ 6)}{(x ^ 3)}\)and \(\frac{(x^{-3}) }{(x^{-6}) }\), the expressions \(\frac{(x^{30}) }{(x^{10})}\) and \(\frac{(x ^ 4) ^ 2)}{x ^ 2}\) are not equivalent to \(x^3\).

What are equivalent expressions?

Equivalent expressions are different algebraic expressions that have the same value when evaluated for a particular set of variables.

The equivalent expressions  \(x^3\) are:

\(\frac{(x ^ 4) ^ 2)}{x ^ 2}\) : This expression can be simplified as follows:

=\(\frac{(x ^ 4) ^ 2)}{x ^ 2}\)

= \(\frac{(x ^ 8)}{(x ^ 2) }\)

=\(x^{8-2}\)

= \(x^6\)

\(\frac{(x ^ 6)}{(x ^ 3)}\) : This expression can be simplified as follows:

=\(\frac{(x ^ 6)}{(x ^ 3)}\)

= \(x^{6-3}\)

=\(x^3\)

\(\frac{(x^{30}) }{(x^{10})}\) : This expression can be simplified as follows:

=\(\frac{(x^{30}) }{(x^{10})}\)

=\(x^{30-10}\)

=\(x^{20}\)

\(\frac{(x^{-3}) }{(x^{-6}) }\) : This expression can be simplified as follows:

=\(\frac{(x^{-3}) }{(x^{-6}) }\)

=\(x^{-3-(-6)}\)

=\(x^{-3+6}\)

=\(x^{3}\)

Therefore, the equivalent expressions of \(x^3\) are\(\frac{(x ^ 6)}{(x ^ 3)}\)and \(\frac{(x^{-3}) }{(x^{-6}) }\) , the expressions \(\frac{(x^{30}) }{(x^{10})}\) and \(\frac{(x ^ 4) ^ 2)}{x ^ 2}\) are not equivalent to \(x^3\).

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Answer:

3/7

Step-by-step explanation:

12/28=3*4/4*7=3/7

Answer:

The answer would be 3

Step-by-step explanation:

Do the () first

3 to the third power would be 27

28-27 = 1

3 x 1 = 3

There are 13 elements of the sample space that will have a green ball.

The various ways in which objects from a set may be selected, generally without replacement, to form subsets.

Since, when a green ball is drawn, three coins are tossed. So, sample points are  OHHH, OHHT, OHTH, and OHTT. OTHH, 0THT, OTTH, OTTT

So, clearly, 9 sample points will have an orange ball.

When a blue ball is drawn, two coins are tossed. So, sample points are BHH, BHT, BTH, and BTT.

So, clearly, 4 sample points will be when a blue ball is drawn.

Total samples points=4+9=13

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Answer:

y = 18 + 12x

Step-by-step explanation:

To find the total cost Mr. Montenegro pays in dollars when x shirts are ordered, we can use the equation y = 18 + 12x, where y is the total cost in dollars and x is the number of shirts ordered. This equation takes into account the delivery fee of $18 and the cost of each shirt, which is $12. Therefore, we can use the equation y = 18 + 12x to find the total cost Mr. Montenegro pays when he orders x shirts.

Answer:

∠D, ∠C, ∠E

Step-by-step explanation:

"The angle opposite to the shortest side is the smallest angle and the side opposite to the longest side is the largest angle."

Using strong induction, we can prove that the square root of -2 is irrational by showing that it cannot be expressed as a fraction of coprime odd integers.

To prove that √-2 is irrational using strong induction, we need to show that for any natural number n, if the square root of -2 can be expressed as a fraction a/b, where a and b are coprime integers, then a and b must be odd.

We can start by using the base case, n = 1. Assume that √-2 can be expressed as a fraction a/b where a and b are coprime integers. Then, we have

√-2 = a/b

Squaring both sides gives

-2 = a^2/b^2

Multiplying both sides by b^2 gives

-2b^2 = a^2

This implies that a^2 is even, and therefore a is also even. We can express a as 2k for some integer k, which means

-2b^2 = (2k)^2

Simplifying, we get

-2b^2 = 4k^2

Dividing both sides by -2 gives

b^2 = -2k^2

This implies that b^2 is even, which means that b is also even. However, this contradicts our assumption that a and b are coprime integers. Therefore, √-2 cannot be expressed as a fraction a/b where a and b are coprime integers.

Now, let's assume that for all n ≤ k, the square root of -2 cannot be expressed as a fraction a/b where a and b are coprime integers with a and b odd. We want to prove that this also holds for n = k+1.

Assume that √-2 can be expressed as a fraction a/b where a and b are coprime integers with a and b odd. Then, we have

√-2 = a/b

Squaring both sides gives

-2 = a^2/b^2

Multiplying both sides by b^2 gives

-2b^2 = a^2

This implies that a^2 is even, and therefore a is also even. We can express a as 2k for some integer k, which means

-2b^2 = (2k)^2

Simplifying, we get

-2b^2 = 4k^2

Dividing both sides by -2 gives

b^2 = -2k^2

This implies that b^2 is even, which means that b is also even. However, this contradicts our assumption that a and b are coprime integers with a and b odd. Therefore, √-2 cannot be expressed as a fraction a/b where a and b are coprime integers with a and b odd.

By strong induction, we have proven that for any natural number n, the square root of -2 cannot be expressed as a fraction a/b where a and b are coprime integers with a and b odd. Therefore, √-2 is irrational.

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Answer:

The probability of picking 1 red candle followed by another red candle is 16.66%.

Step-by-step explanation:

Since Alfred draws candles randomly from a pack containing 4 colored candles of the same shape and size, and there are 2 red candles, 1 green candle, and 1 blue candle, and he draws 1 candle and then draws another candle without replacing the first one, to find the probability of picking 1 red candle followed by another red candle, the following calculation must be performed:

2/4 x 1/3 = X

0.5 x 0.333 = X

0.16666 = X

Therefore, the probability of picking 1 red candle followed by another red candle is 16.66%.

Answer:

1/6

Step-by-step explanation:

2/4 x 1/3 = 1/6

False. In the position coordinate system, P(r,θ), r represents the radial coordinate, while θ represents the angular coordinate, not the transverse coordinate.

The transverse coordinate is typically denoted by z and is used in three-dimensional Cartesian coordinates (x,y,z) to represent the position of a point in space.

In polar coordinates, such as P(r,θ), r represents the distance from the origin to the point, while θ represents the angle between the positive x-axis and the line connecting the origin to the point. Together, they determine the position of a point in a two-dimensional plane. The radial coordinate gives the distance from the origin, while the angular coordinate determines the direction or orientation of the point with respect to the reference axis.

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Average rate of growth over this period is 0.4%.

Average rate of growth over a period is given by:

A=P\(e^{rT}\) where,

A- Population at time T

P- Initial population

r-average rate of growth

T-Time period

Now, P=250,000, A=310,000 and T=21 (2000-1980+1)

therefore, putting values in the equation, we get:

310,000=250,000\(e^{21r}\)

⇒ \(e^{21r}\) = 1.24

⇒ 21r = log(1.24)

⇒ r = 0.0044 or 0.4%.

so, the average rate of growth over this time period is 0.4%.

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The expected number of days per week the captain can leave port is 3.45.

The expected number of days per week the captain can leave port is calculated by the formula

μ = Σ [x P(x)], where μ is the expected value, x is the variable, and P(x) is the probability.

The given probability distribution is given below:

X         0       1       2       3      4        5         6       7

P(x) 0.05  0.10  0.15  0.20  0.25  0.10   0.10   0.05

Expected value,

μ = Σ [x P(x)]

μ = 0 (0.05) + 1(0.10) + 2(0.15) + 3(0.20) + 4(0.25) + 5(0.10) + 6(0.10) + 7(0.05)

μ = 0 + 0.10 + 0.30 + 0.60 + 1.00 + 0.50 + 0.60 + 0.35

μ = 3.45

Therefore, the expected number of days per week the captain can leave port is 3.45.

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The answer of this question is √17 units

(a) a graph with the specified properties does exist because this is a non-increasing sequence where all degrees are positive.

(b) 1. A, B, C, D, E, F.

2. A, B, D, E, F, C.

(c) V₁, V, V₂, Ve, Vo, Vi, Vs.

A graph consists of two main components: nodes and edges. Nodes, also known as vertices, represent the entities or objects in the graph.

For example, in a social network graph, each node could represent a person, while in a transportation network, nodes could represent cities or intersections.

Edges, also called arcs or links, represent the relationships or connections between the nodes. They can be directed or undirected, indicating the nature of the relationship.

For example, in a directed graph, the edges have a specific direction, while in an undirected graph, the edges are bidirectional.

(a) To determine if a graph with the specified properties exists, we can check if the degree sequence is graphical. The degree sequence is the list of degrees of each vertex in non-increasing order.

Degree sequence: 5, 3, 3, 3, 2

To check if this degree sequence is graphical, we can use the Havel-Hakimi algorithm:

1. Arrange the degree sequence in non-increasing order: 5, 3, 3, 3, 2.

2. Start with the highest degree (5) and subtract 1 from it. Decrease the next highest degrees (3, 3, 3) by 1 as well.

  New degree sequence: 4, 2, 2, 2, 2.

3. Repeat step 2 until the degree sequence becomes non-increasing or contains negative numbers.

After applying the algorithm, we obtain the following degree sequence: 4, 2, 2, 2, 2. Since this is a non-increasing sequence where all degrees are positive, a graph with the specified properties does exist.

(b.1) Breadth First Search (BFS) starting from vertex A with a queue ADT will visit nodes in the following order: A, B, C, D, E, F.

(b.2) Depth First Search (DFS) starting from vertex A with a stack ADT will visit nodes in the following order: A, B, D, E, F, C.

(c) The ordering of vertices produced by the topological sort algorithm given in class starting from vertex V₁ when it is run on the following direct acyclic graph (represented by its adjacency list, in-degree form) is: V₁, V, V₂, Ve, Vo, Vi, Vs.

Justification: V₁ is the starting point of the algorithm, so it is visited first. Then, since V and V₂ have no incoming edges, they are visited next. Next, we visit Ve, since its incoming edge is from V. We then visit Vo, since its incoming edge is from both V and Vi. Finally, we visit Vi and Vs, since their incoming edges are from V₂ and Vo, respectively.

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4x-7 =5

4x=5+7

x= 12/4

x=3

Answer: A Geometric sequence is a type of sequence in which each subsequent term after the first term is determined by multiplying the previous term by a constant (not 1), which is referred to as the common ratio.

Step-by-step explanation: Hope this was helpful

Step-by-step explanation:

The sequence

a, ar, ar^2, .....ar^n-1,...

where a is the first term and any two consecutive numbers differ by a factor r, is called a Geometric progression (G.P.).

Geometric progression also known as Geometric sequence.

The equation for the line tangent to the graph of f(x) = x^2 - x at the point (3, 6) is y = 5x - 9.the tangent line to the graph of y = x^3 - 12x + 2 is horizontal at x = -2 and x = 2.

The derivative of f(x) = 20x^(1/2) - (1/2)^(x^20) is f'(x) = 10/x^(1/2) + (1/2)^(x^19) * ln(1/2) * (x^20).

To find the values of x where the tangent line to the graph of y = x^3 - 12x + 2 is horizontal, we need to find the x-values where the derivative is equal to zero.

Differentiating y = x^3 - 12x + 2 with respect to x gives y' = 3x^2 - 12.

Setting y' = 0 and solving for x, we have 3x^2 - 12 = 0. Simplifying further, we get x^2 - 4 = 0. Factoring the quadratic equation, we have (x + 2)(x - 2) = 0. So, x = -2 and x = 2.

Therefore, the tot tangent line the graph of y = x^3 - 12x + 2 is horizontal at x = -2 and x = 2.

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Answer:

a)

b)

Explanation:

a) Let b represent the number of striped bass one is allowed to catch.

Since we're only allowed to catch at most 3 striped basses, that is, the number of striped basses can be less than or equal to 3 but not more, we can go ahead and write the inequality as seen below;

See below the graph of the above inequality on a number line;

b) Let l represent the length of each striped bass

Since each striped bass must not be less than 18 inches long, that is, each striped bass can

be more than or equal to 18 inches in length but not less, we can go ahead and write the inequality as seen below;

Graphing on a number line, we'll have;

Answer:

7 Groups of 1/2 in 3 1/2

To factor the expression 25a² - 120a + 144, we can use the factoring technique called "quadratic trinomial factoring." Here are the steps:

Look for common factors among the coefficients. In this case, the greatest common factor (GCF) of 25, -120, and 144 is 1. Therefore, we cannot factor out any common factors. Identify the sign of the middle term (-120a). Since it is negative, we need to find two numbers whose product is 144 (the coefficient of the constant term) and whose sum is -120 (the coefficient of the middle term).

The two numbers that satisfy these conditions are -6 and -24. Therefore, we can rewrite the middle term (-120a) as -6a - 24a.
Now, we group the terms and factor by grouping. The expression becomes:
25a² - 6a - 24a + 144.
We can factor by grouping:
(25a² - 6a) - (24a - 144).
Now, we factor out the common factors from each group:
a(25a - 6) - 24(25a - 6).
Notice that we have a common binomial factor, (25a - 6), in both terms. We can factor it out:
(25a - 6)(a - 24).
Therefore, the factored form of the expression 25a² - 120a + 144 is (25a - 6)(a - 24).

The expression 25a² - 120a + 144 can be factored into (25a - 6)(a - 24).
To factor the expression 25a² - 120a + 144, we first check if there are any common factors among the coefficients. In this case, there are no common factors. Next, we need to identify the sign of the middle term (-120a). Since it is negative, we need to find two numbers whose product is 144 (the coefficient of the constant term) and whose sum is -120 (the coefficient of the middle term).

The two numbers that satisfy these conditions are -6 and -24. We then group the terms and factor by grouping, resulting in (25a² - 6a) - (24a - 144). We factor out the common factors from each group, giving us a(25a - 6) - 24(25a - 6). Finally, we notice that we have a common binomial factor, (25a - 6), in both terms, so we factor it out to get the final answer: (25a - 6)(a - 24).
The expression 25a² - 120a + 144 can be factored into (25a - 6)(a - 24).

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Step-by-step explanation:

3(x-2)+1

= 3x-6+1

= 3x-5

Answer:

Step-by-step explanation:

3(x-2)+1=

Then distribute, and now you get:

3x-6+1=

Now combine like terms, and now you get:

3x-5=

There is nothing much to do because there isn't a answer for it

Answer:

1/4000

1 milimeter on paper means 4000 in real