math because im very bad at it

we have shown that if [tex]$a_1, a_2, \ldots, a_n$[/tex] are n distinct real numbers, exactly [tex]$n-1$[/tex] multiplications are used to compute their product, regardless of how parentheses are inserted into the product.

What is the principle of mathematical induction?

The principle of mathematical induction is a powerful proof technique used to establish the validity of an infinite sequence of statements.

To prove that exactly [tex]$n-1$[/tex] multiplications are used to compute the product of n distinct real numbers, regardless of how parentheses are inserted into their product, we will use the principle of mathematical induction.

[tex]\textbf{Base Case:}[/tex]

For [tex]$n=2$[/tex], we have two distinct real numbers [tex]a_1$ and $a_2$.[/tex] The product is simply [tex]a_1 \cdot a_2$,[/tex] which requires only one multiplication. Thus, the base case holds true.

[tex]\textbf{Inductive Step:}[/tex]

Assume the statement holds true for [tex]$n=k$[/tex], where [tex]k \geq 2$.[/tex] That is, when multiplying k distinct real numbers, exactly [tex]$k-1$[/tex] multiplications are used.

Now, consider the case for [tex]$n=k+1$[/tex], where we have [tex]$k+1$[/tex] distinct real numbers [tex]$a_1, a_2, \ldots, a_{k+1}$[/tex]. The product can be computed by multiplying [tex]$a_1$[/tex] with the product of the remaining k numbers, which can be denoted as [tex]$(a_2 \cdot a_3 \cdot \ldots \cdot a_{k+1})$[/tex].

By our induction hypothesis, computing the product of k distinct real numbers requires [tex]$k-1$[/tex] multiplications. Therefore, multiplying[tex]$a_1$[/tex] with the product of the remaining [tex]$k$[/tex] numbers requires an additional multiplication, resulting in a total of k multiplications.

Hence, we have shown that if [tex]a_1, a_2, \ldots, a_n$ are $n$[/tex] distinct real numbers, exactly [tex]$n-1$[/tex] multiplications are used to compute their product, regardless of how parentheses are inserted into the product.

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Answer:

a2 + 2al (or) a2 + √a24+h2 a 2 4 + h 2

Step-by-step explanation:

Answer:

SA=L²+2L√(L²/4+H²)

Step-by-step explanation:

Where H is the height and L is the length

Answer:

A

Step-by-step explanation:

Answer:

Yes, it will fit snugly in a 90º corner

Step-by-step explanation:

To do this, we simply need to check if the given sides of the shelf is right-angled.

So, we have:

[tex]Hypotenuse = \sqrt{242[/tex]

[tex]Side\ 1 = Side\ 2 = 11[/tex]

To check for right-angle triangle, we make use of:

[tex]Side\ 1^2 + Side\ 2^2 = Hypotenuse^2[/tex]

This gives:

[tex]11^2 + 11^2 = \sqrt{242}^2[/tex]

[tex]121 + 121 = \sqrt{242}^2[/tex]

[tex]242 = \sqrt{242}^2[/tex]

[tex]242 = 242[/tex]

This shows that the given sides of the shelf is a right-angled triangle.

Hence, it will fit the wall

Answer:

A

Step-by-step explanation:

Answer:

Question A)

[tex]=\sqrt{6}x[/tex]

Question B)

[tex]=3a\sqrt{a}[/tex]

Question C)

[tex]=5\sqrt{2}b^2[/tex]

Step-by-step explanation:

A)

We are given:

[tex]\sqrt{6x^2}\, \text{ where } x\geq 0[/tex]

We can rewrite the expression:

[tex]=\sqrt{6}\cdot \sqrt{x^2}[/tex]

The square root and square will cancel each other out. Thus:

[tex]=\sqrt{6}x[/tex]

B)

We are given:

[tex]\sqrt{9a^3}[/tex]

Rewrite:

[tex]=\sqrt{9}\cdot \sqrt{a^3}[/tex]

Note that the square root of 9 is simply 3. We can also factor the second part:

[tex]=3\cdot \sqrt{a^2\cdot a}[/tex]

Rewriting:

[tex]=3\cdot\sqrt{a^2}\cdot\sqrt{a}[/tex]

Simplify:

[tex]=3a\sqrt{a}[/tex]

C)

We are given:

[tex]\sqrt{50b^4}[/tex]

Rewrite. Note that 50 = 25(2):

[tex]=\sqrt{25}\cdot \sqrt{2}\cdot \sqrt{b^4}[/tex]

Simplify. We can rewrite the factor as:

[tex]=5\cdot \sqrt{2}\cdot \sqrt{(b^2)^2}[/tex]

The square and square root will cancel out. Thus:

[tex]=5\sqrt{2}b^2[/tex]

Answer: 14.6

Step-by-step explanation:

I made a square around the triangle which I then counted the squares, found the Pythagorean theorem, and then added the missing sides together

Answer:

y=13 degrees

Step-by-step explanation:

This is an isosceles triangle, we know this because NO and NM are equal.

In an isosceles triangle, the base angles are congruent. In this case, they are angle NOM and angle NMO.

We also know that the sum of the interior angles of a triangle are equal to 180.

With this information, we can make an equation by gathering all the interior angles:

8y+2(3y-1)=180

Solve for y.

8y+6y-2=180

14y-2=180

14y=182

y=13

Answer:

y=3x+6

Step-by-step explanation:

Just test this on like a graph or something and you will see it works. Let me know if there is a fault in my answer. Thanks! Have a good day.

Answer:

A. 4x¹² + 9x⁷ + 3x³ -x

Step-by-step explanation:

Hi!

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

To write a polynomial in descending order, we write the terms with higher degrees, or exponents, first.

3x³ + 9x⁷ -x + 4x¹²

4x¹² has the highest degree, so it is written as the first term.

⇒4x¹²

9x⁷ has the next highest degree, so it is written next.

⇒4x¹² + 9x⁷

3x³ has the next highest degree, so it is written next.

⇒4x¹² + 9x⁷ + 3x³

-x has the lowest degree, so it is written last.

⇒4x¹² + 9x⁷ + 3x³ -x

4x¹² + 9x⁷ + 3x³ -x

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

Hope I Helped, Feel free to ask any questions to clarify :)

Have a great day!

       -Aadi x

Answer:

A- lll

B- l

C- lV

D- V

E- ll

Step-by-step explanation:

I think that is the answer.

Answer:

Vince is not correct because he should have interpreted the solution as having to save a minimum of $103.55.

Step-by-step explanation:my

my sister which is in college helped me with one.

Answer: its D

Step-by-step explanation:

Answer:

To understand their poems better

Step-by-step explanation:

First you have a general knowledge on why the author made the poem

Second you now have more info to use in order to understand the poems

Third is the same reason why there are people in history books.

Answer:

I think the answers would be :

a . 4/3

b. 1/5

c . - 5/4

hope it helps u ^^

Step-by-step explanation:

the difference is the previous term multiplied by 4 to get the next term

2,8,32,128,512,2048,8192

Hope that helps :)

The frictional torque on the flywheel is `22.58 N.m` in the clockwise direction.

The formula for angular velocity is given by;`ω = (2π / T)`.Where;ω = angular velocity of the object. T = time period (in seconds).`I = 185.0 kg m2` represents the moment of inertia of the flywheel.`ω = 350.0 rev/min = (350.0 * 2π) / 60 = 36.61 rad/s` represents the initial angular velocity of the flywheel.

The flywheel is brought to rest by friction in `5.0 min = 5.0 * 60 = 300 seconds`.

The formula for the angular acceleration is given by;`α = (ωf - ωi) / t`. Where;`α` = angular acceleration of the object.`ωi` = initial angular velocity.`ωf` = final angular velocity of the object.`t` = time taken (in seconds).

At rest, the final angular velocity of the flywheel is zero.

Therefore;`α = (- ωi) / t`.The formula for torque is given by;`τ = I * α`.Where;τ = torque exerted on the object.I = moment of inertia of the object.α = angular acceleration of the object.

Substituting the values;`τ = I * α = 185.0 * (-36.61) / 300 = -22.58 N.m`.

Therefore, the frictional torque on the flywheel is `22.58 N.m` in the clockwise direction.

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Answer:

A.

0.5

Step-by-step explanation:

To solve the given initial value problem using the method of Laplace transforms, we'll take the Laplace transform of both sides of the differential equation. Let's denote the Laplace transform of the function y(t) as Y(s).

The Laplace transform of the second derivative y" is s²Y(s) - sy(0) - y'(0), where y(0) and y'(0) are the initial conditions given.

The Laplace transform of the first derivative y' is sY(s) - y(0).

The Laplace transform of the term 6y is 6Y(s).

The Laplace transform of the term -24e^t can be found using the table of Laplace transforms.

Applying the Laplace transform to the entire differential equation, we get:

s²Y(s) - sy(0) - y'(0) + 5(sY(s) - y(0)) + 6Y(s) - 24/(s-1) = 0

Substituting the initial conditions y(0) = -5 and y'(0) = -19, we have:

s²Y(s) + 5sY(s) + 6Y(s) - 5s + 19 - 24/(s-1) = 0

Now, we can solve this equation for Y(s). Once we find Y(s), we can take the inverse Laplace transform to obtain y(t), the solution to the initial value problem.

Since the given question doesn't specify a particular form for Y(s), I'm unable to provide the exact solution y(t) in terms of e.

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Using binomial probability, the probability that at least 26 people experience flu symptoms is 0.8788 and It indicates a potential adverse reaction to the drug, and further investigation or monitoring may be required to assess the relationship between the drug and flu symptoms.

To estimate the probability that at least 26 people experience flu symptoms, we can use the binomial distribution. The probability of flu symptoms for a person not receiving any treatment is given as 0.021.

Let's denote the number of people treated who experience flu symptoms as X, which follows a binomial distribution with parameters n (number of trials) and p (probability of success).

In this case, n = 1133 (number of people treated) and p = 0.021 (probability of flu symptoms for a person not receiving treatment).

We want to find P(X ≥ 26), which is the probability that at least 26 people experience flu symptoms. To calculate this probability, we need to sum the probabilities of the complementary event (X < 26), which is the probability that fewer than 26 people experience flu symptoms.

Using statistical software or a binomial distribution table, we can calculate the probability as follows:

P(X ≥ 26) = 1 - P(X < 26)

Using the binomial distribution, we can calculate the probability of X < 26 as:

P(X < 26) = Σ[P(X = k)] for k = 0 to 25

Now, let's calculate the probability:

P(X < 26) = Σ[C(n, k) * p^k * (1 - p)^(n - k)] for k = 0 to 25

However, calculating this probability manually for each value of k can be cumbersome. Therefore, I'll use statistical software to estimate the probability.

Using the binomial distribution with parameters n = 1133 and p = 0.021, we find:

P(X < 26) ≈ 0.1212

Therefore, the estimated probability that at least 26 people experience flu symptoms is:

P(X ≥ 26) ≈ 1 - 0.1212 ≈ 0.8788

This suggests that the observed number of people experiencing flu symptoms (26) is higher than what would be expected if the drug had no effect on the likelihood of flu symptoms.

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Answer:

The inequality that can be used to determine the number of guitar lessons Gale could take using her birthday money is:

[tex]25x + 50 \leqslant 200[/tex]

The solution to the inequality is:

[tex]x \leqslant 6[/tex]

She can take at most, 6 guitar lessons

Step-by-step explanation:

Total amount Gale received is $200

For rental equipment, she paid $50

Each guitar lesson will cost $25

Let x represent the number of lessons she could take, then the total amount she will spend must bot exceed $200

The guitar lesson will cost a total of 25x

So, her total spending will be:

[tex]25x + 50[/tex]

This must be at most $200

[tex]25x + 50 \leqslant 20[/tex]

Solving the above inequality:

[tex]25x \leqslant 200 - 50 = 150[/tex]

[tex]x \leqslant \frac{150}{25} = 6[/tex]

Answer:

Yes. See explanation below.

Step-by-step explanation:

The central angle and the degree arc measure of a sector of a circle are  equal. Doubling the arc measure, doubles the central angle measure and vice versa.

Area of the the original sector:

[tex] A_{sector} = \dfrac{n}{360^\circ}\pi r^2 [/tex]

where n = measure of the central angle of the sector

Since the central angle and the arc measure of the sector are equal, changing the arc measure has the same effect as changing the central angle measure.

Let's double the central angle to 2n which is the same as doubling the arc measure.

Area of the sector with a doubled central angle or a doubled arc measure:

[tex] A_{sector} = \dfrac{2n}{360^\circ}\pi r^2 [/tex]

Now we divide the area of the doubled sector by the area of the original sector.

[tex]\dfrac{\frac{2n}{360^\circ}\pi r^2}{\frac{n}{360^\circ}\pi r^2} =[/tex]

Simplify:

[tex]= \dfrac{2n}{n} \times \dfrac{360^\circ \pi r^2}{360^\circ \pi r^2}[/tex]

[tex] = \dfrac{2n}{n} [/tex]

[tex] = 2 [/tex]

The ratio of the areas is 2, so the area of the sector is indeed doubled.

Answer: Yes.

Answer:

$11.25

Step-by-step explanation:

55.50 divided by 6 = 9.25

9.25 + 2 = 11.25

The integral for the area becomes:

A = ∫₀ᵃʳᶜᶜᵒˢ(ᵃ/₂) ∫ₐ(₁+ᶜᵒˢ(θ))³ᵃᶜᵒˢ(θ) r dr dθ

To find the area inside the circle r = 3acos(θ) and outside the cardioid r = a(1 + cos(θ)), we can set up a double integral in polar coordinates.

First, let's find the points of intersection between the two curves. The circle r = 3acos(θ) and the cardioid r = a(1 + cos(θ)) intersect when:

3acos(θ) = a(1 + cos(θ))

Simplifying, we get:

3acos(θ) - a(1 + cos(θ)) = 0

2acos(θ) - a = 0

acos(θ) = a/2

θ = arccos(a/2)

Now, let's set up the integral. We want to find the area inside the circle and outside the cardioid, so the region of integration is defined by:

0 ≤ θ ≤ arccos(a/2)

a(1 + cos(θ)) ≤ r ≤ 3acos(θ)

Evaluating this double integral will give us the desired area inside the circle and outside the cardioid.

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Answer:

True

Step-by-step explanation:

Solutions to quadratic equations are often called

roots / zeros / x- intercepts

Answer:

1m²

Step-by-step explanation:

Answer:

A = 157.3 units²

Step-by-step explanation:

A = 1/2(6.9)(8 x 5.7) = 157.3 units²

Answer: B

Step-by-step explanation:

The value g(x) if f(x) = x2 is g(x) = x2-3, the correct option is C.

Function is a type of relation, or rule, that maps one input to specific single output.

In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) (the dependent variable). In mathematics and the physical sciences, functions are indispensable for formulating physical relationships.

Linear function is a function whose graph is a straight line

We are given that;

The function f(x)=x2

Now,

In the graph of g(x)

Coordinate x=0

Coordinate y=-3

Plugging in the values

g(x) = x2-y

y=3

g(x)=x2-3

Therefore, by the given function the answer will be g(x)=x2-3.

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Answer:

The true statements are -: A f(3)>0 , C g(-1)=0.8 and D g(-1)< f(-1)

Step-by-step explanation:

  Given f(x)= 5- 0.2x and g(x)= 0.2(x+5)

(A)  f(3) > 0

    Putting this value in the given equation -

     f(3) =5 -0.2(3)

      f(3)=5 - 0.6

  f(3) =4.4

Therefore , f(3) > 0  . This is a true statement .

(B)  f(3) > 5

     We got the value of f(3) from the above equation,

     Therefore , f(3) = 4.4 , hence f(3) > 5 option is False.

(C)  g(-1) = 0.8

  Taking the second given equation ,

  g(-1) = 0.2 ( -1+ 5)

  g(-1) = 0.2 (-4)

Therefore,  g(-1) = -0.8 . Hence , this statement is true.

(D) g(-1) < f(-1)

  g(-1) = -0.8

f(-1) =5 - 0.2(-1)

f(-1) = 5 +0.2

f(-1) = 5.2

Therefore , g(-1) < f(-1) , Hence this option is true.

(E) f(0) =g(0)

  f(0) = 5 + 0.2(0)

  f(0) = 5

g(0) = 0.2(0+5)

g(0) =  0.2 +5

g(0) = 1

Therefore , f(0)[tex]\neq[/tex] g(0) , Hence , the option is false .

Hence , true statements about the functions are - A , C , D .  

To solve the equation 7 sin(2x) = 6 for the two smallest positive solutions A and B, we can use algebraic techniques and trigonometric properties.

The solutions A and B are approximately equal to A ≈ 0.287 and B ≈ 1.569, respectively.

To explain the solution, let's begin by rearranging the equation: sin(2x) = 6/7. Since the range of the sine function is between -1 and 1, the equation has solutions only if 6/7 is within this range. We can find the corresponding angles by taking the inverse sine (arcsin) of 6/7. Using a calculator, we find that the arcsin(6/7) is approximately 0.942.

However, this gives us only one of the solutions. To find the other solution, we can use the periodicity of the sine function. We know that sin(θ) = sin(π - θ), where θ is the angle in radians. Therefore, the second solution is π - 0.942, which is approximately 2.199. However, since we're looking for the smallest positive solutions, we need to consider only the values between 0 and 2π. Thus, the two smallest positive solutions are A ≈ 0.287 and B ≈ 1.569.

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To find a function that satisfies the given criteria, we can start by determining the requirements for its derivative, f'(x).

Let's break down the given properties and find the corresponding requirements for f'(x): f(x) is decreasing at x = -6: This means that the slope of the function should be negative at x = -6. Therefore, f'(-6) < 0. f(x) has a local minimum at x = -3: At a local minimum, the slope changes from negative to positive. Thus, f'(-3) = 0. f(x) has a local maximum at x = 3: At a local maximum, the slope changes from positive to negative. Hence, f'(3) = 0.

Now, let's integrate f'(x) to obtain f(x): Integrating f'(x) = -6 < x < -3 will give us a decreasing function on that interval. Integrating f'(x) = -3 < x < 3 will give us an increasing function on that interval. Integrating f'(x) = 3 < x < 6 will give us a decreasing function on that interval. To simplify the process, let's assume that f'(x) is a quadratic function with roots at -6, -3, and 3. We can represent it as: f'(x) = k(x + 6)(x + 3)(x - 3), where k is a constant that affects the steepness of the curve. By setting f'(-3) = 0, we find that k = -1/18.

Therefore, f'(x) = -1/18(x + 6)(x + 3)(x - 3). Integrating f'(x) will give us f(x): f(x) = ∫[-6,x] -1/18(t + 6)(t + 3)(t - 3) dt. Evaluating this integral is a bit complicated. Let's denote F(x) as the antiderivative of f(x): F(x) = ∫[-6,x] -1/18(t + 6)(t + 3)(t - 3) dt. Now, we can find f(x) by differentiating F(x): f(x) = d/dx[F(x)]. To get an explicit equation for f(x), we need to calculate the integral and differentiate the resulting antiderivative. Once you have the equation for f(x), you can plot it on the provided graphing option to verify that it matches the criteria mentioned in the question.

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