Let v,w∈Rn. If |‖v‖=‖w‖, show that v+w and v−w are orthogonal (perpendicular).

Answer:

52

Step-by-step explanation:

Using the Pythagorean theorem

a^2+b^2 = c^2

20^2 + 48^2 = c^2

400 +2304=c^2

2704=c^2

Taking the square root of each side

sqrt(2704) = sqrt(c^2)

52 = c

Answer:

[tex]hypotenuse^{2}[/tex] = [tex]altitude^{2}[/tex] + [tex]base^{2}[/tex]

[tex]hyp^{2}[/tex] = [tex]48^{2}[/tex] + [tex]20^{2}[/tex]

       = 2304 + 400

       = 2704

∴[tex]hyp[/tex] = [tex]\sqrt{2704}[/tex]

       = 52

hope this answer helps you....

Answer:

47

Step-by-step explanation:

^

Answer:

3:1 (Girls:Boys)

6 divided by 2

Answer:

a = 15

t = 5

Step-by-step explanation:

[tex] \frac{12}{a} = \frac{16}{20} \\ \\ 16a = 12 \times 20 \\ \\16 a = 240 \\ \\ a = \frac{240}{16} \\ \\ a = 15 \\ \\ \\ \frac{2}{8} = \frac{t}{20} \\ \\ 8t = 40 \\ \\ t = \frac{40}{8} \\ \\ t = 5[/tex]

Answer:

1) 1.2 minutes

2) 8.3 laps

3) 25.2 minutes

Both the classical method and the p-value method lead to the conclusion that the average distance cars travel per year is less than 20,000 kilometers,

a) t =  -2.564

b)  t = -2.564

To test the claim that the average distance cars travel per year is less than 20,000 kilometers, we can conduct a hypothesis test using the classical method and the p-value method.

a) The steps we need to follow are:

Step 1: Formulate the null and alternative hypotheses:

Null hypothesis (H₀): The average distance cars travel per year is 20,000 kilometers.

Alternative hypothesis (H₁): The average distance cars travel per year is less than 20,000 kilometers.

Step 2: Determine the test statistic:

Since we know the sample size (n = 100), the sample mean (x = 19,000 kilometers), and the sample standard deviation (s = 3900 kilometers), we can use the t-test statistic.

t = (x - μ₀) / (s / √n)

where μ₀ is the assumed population mean under the null hypothesis.

Step 3: Set the significance level:

The significance level is given as 0.05, which means we want to be 95% confident in our conclusion.

Step 4: Calculate the critical value:

Since the alternative hypothesis is one-tailed (less than), we need to find the critical t-value corresponding to a 0.05 significance level and degrees of freedom (df) = n - 1 = 99. From the t-distribution table or calculator, the critical t-value is approximately -1.660.

Step 5: Calculate the test statistic:

t = (19,000 - 20,000) / (3900 / √100)

t = -10 / (3900 / 10)

t ≈ -2.564

Step 6: Compare the test statistic with the critical value:

Since -2.564 is less than -1.660, the test statistic falls in the rejection region. We reject the null hypothesis.

Step 7: Make a conclusion:

Based on the classical method, since the test statistic falls in the rejection region, we conclude that the average distance cars travel per year is significantly less than 20,000 kilometers.

b) The P-value method:

Using the same test statistic t = -2.564 and the degrees of freedom (df) = 99, we can calculate the p-value. The p-value is the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true.

From a t-distribution table or calculator, the p-value corresponding to t = -2.564 and df = 99 is approximately 0.0075 (or 0.75% if multiplied by 100).

Since the p-value (0.0075) is less than the significance level (0.05), we reject the null hypothesis. This suggests strong evidence that the average distance cars travel per year is significantly less than 20,000 kilometers.

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Answer:

____________________________________________________________

   Why?

Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid can never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals.

____________________________________________________________    

   What's a  parallelogram?                                                                                      

 Its a four-sided plane rectilinear figure with opposite sides parallel.

____________________________________________________________

Please don't be afraid to point out errors :)  

____________________________________________________________

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The correct shape which is not a parallelogram is shown in Option D.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

There are four shapes are shown.

Now, We know that;

In a parallelogram, there are two pair of parallel lines.

But In option D;

There are only one pair of parallel lines.

Hence, The correct shape which is not a parallelogram is shown in Option D.

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The solution to the system of differential equations is x₁(t) = e⁻⁵ˣ + 3e³ˣ and x₂(t) = 2e⁻⁵ˣ + 5e³ˣ

Let's solve the given system of differential equations: x₁' = -5x₁ + 0x₂ ...(1) x₂' = -16x₁ + 3x₂ ...(2)

To solve this system, we can rewrite it in matrix form. Let's define the vector X = [x₁, x₂] and the matrix A as:

A = [[-5, 0], [-16, 3]]

The system can then be written as X' = AX, where X' is the derivative of X with respect to time.

Now, let's find the eigenvalues and eigenvectors of matrix A. The eigenvalues are obtained by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix.

A - λI = [[-5 - λ, 0], [-16, 3 - λ]]

det(A - λI) = (-5 - λ)(3 - λ) - 0(-16) = λ² + 2λ - 15 = (λ + 5)(λ - 3)

Setting the characteristic equation equal to zero, we find the eigenvalues: λ₁ = -5 λ₂ = 3

To find the corresponding eigenvectors, we substitute each eigenvalue back into the matrix A - λI and solve the system of equations (A - λI)v = 0, where v is the eigenvector.

For λ₁ = -5: A - (-5)I = [[0, 0], [-16, 8]]

Using Gaussian elimination, we can solve the system of equations to find the eigenvector corresponding to λ₁: -16v₁ + 8v₂ = 0 => -2v₁ + v₂ = 0 => v₁ = (1/2)v₂

Let v₂ = 2, then v₁ = 1. Therefore, the eigenvector corresponding to λ₁ is v₁ = [1, 2].

For λ₂ = 3: A - 3I = [[-8, 0], [-16, 0]]

Solving the system of equations, we find: -8v₁ = 0 => v₁ = 0

Thus, the eigenvector corresponding to λ₂ is v₂ = [0, 1].

Now, let's express the solution of the system in terms of the eigenvalues and eigenvectors.

X(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂

Substituting the eigenvalues and eigenvectors we found earlier, we have: X(t) = c₁e⁻⁵ˣ[1, 2] + c₂e³ˣ[0, 1]

Using the initial conditions, x₁(0) = 1 and x₂(0) = 5, we can find the values of c₁ and c₂.

At t = 0: [1, 5] = c₁[1, 2] + c₂[0, 1] 1 = c₁ 5 = 2c₁ + c₂

Solving these equations, we find: c₁ = 1 c₂ = 3

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Complete Question:

Solve the system of differential equations

x₁' = – 5x₁ + 0x₂

x₂' = – 16x₁ + 3x₂

x₁(0) = 1, x₂(0) = 5

Answer:

(-13,-20)

Step-by-step explanation:

We know that the graph with x-intercept at (7,0) and y-intercept at (0,-7) is y = x-7

and the second equation is y = 2x+6

Therefore, we have two equations.

[tex] \large{ \begin{cases} y = x - 7 \\ y = 2x + 6 \end{cases}}[/tex]

Because both equations are equal (y=y)

x-7=2x+6

-7-6=2x-x

-13=x

We know the value of x, but not y-value. We substitute the value of x to get the value of y.

Substitute in any given equations which I will be substituting in the first equation.

y=-13-7

y = -20

Therefore when x = -13, y = -20

In coordinate form, we can write as (-13,-20).

If you have any questions, feel free to ask.

Answer:

Step-by-step explanation:

1. Write the equation.

y = -2x

y = 5x - 21

2. Substitute the values.

(-2x) = 5x - 21

3. Solve the equation.

-2x = 5x - 21

-5x    - 5x

-7x = -21

4. Both negatives cancel

7x = 21

5. Divide both sides by 7

7x = 21

/7      /7

6. x = 3

7. Check the answer.

-2(3) = 5(3) - 21

-6 = 15 - 21

-6 = -6

Hope this helped,

Kavitha

P.S Sorry for taking so long.

Answer:

3

Step-by-step explanation:

dude kinda obv that its 3

Answer:

answer would be 3

Step-by-step explanation:

A lamina occupies the region of a disk in the first quadrant where 2 ≤ r ≤ 16, and the density at each point is given by the function ρ(r, θ) = 3[tex](r^2).[/tex] Further analysis is required to determine the mass and other properties of the lamina.

The given information describes a lamina occupying a region in the first quadrant of a disk. The radial distance from the origin is limited to the range 2 ≤ r ≤ 16. The density of the lamina at any point within this region is determined by the function ρ(r, θ) = 3[tex](r^2)[/tex], where r represents the radial distance and θ represents the angle in the polar coordinate system.

To fully analyze the lamina, additional calculations are necessary. One important calculation is determining the mass of the lamina, which involves integrating the density function over the given region. By integrating the function ρ(r, θ) = 3[tex](r^2)[/tex] over the appropriate range of r and θ, we can find the total mass of the lamina. Additionally, other properties such as the center of mass or moment of inertia of the lamina could be determined by using appropriate formulas and integration techniques.

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According to the information, we can infer that no, they cannot raise their cumulative average to 2.3 after completing 15 fall semester credits. On the other hand, they would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.

In order to calculate the new cumulative GPA, we need to consider both the current cumulative GPA and the GPA earned in the fall semester. Since the student's current cumulative GPA is 2.0 and they have already accumulated 60 credits, their total grade points earned so far would be 2.0 multiplied by 60, which equals 120 grade points.

To raise the cumulative GPA to 2.3, the student would need a total of 2.3 multiplied by (60 + 15) = 2.3 multiplied by 75 = 172.5 grade points by the end of the fall semester.

Since the student has already accumulated 120 grade points, they would need to earn an additional 52.5 grade points in the fall semester. To calculate the required semester GPA, we divide 52.5 by 15 credits, which gives us a required semester GPA of 3.5.

So, the student would need a semester GPA of 3.5 in order to raise their cumulative average to 2.3 after completing 15 fall semester credits.

They would need an average GPA of 3.25 for their two junior year semesters combined (30 credits) to achieve their goal of a 2.3 cumulative GPA and 90 credits.

Explanation: To calculate the average GPA for the two junior year semesters, we need to consider the total grade points earned and the total number of credits taken.

Currently, the student has accumulated 60 credits and 120 grade points. In order to achieve a cumulative GPA of 2.3 after completing 90 credits, they would need a total of 2.3 multiplied by 90 = 207 grade points.

To calculate the required grade points for the two junior year semesters, we subtract the current grade points (120) from the desired total grade points (207), which gives us 207 - 120 = 87 grade points needed for the junior year.

Since the student plans to take 30 credits during their junior year, they would need to earn 87 grade points in those 30 credits. Dividing 87 by 30 gives us an average GPA of approximately 2.9 for the two junior year semesters.

According to the above, the student would need an average GPA of 3.25 (rounded up) for their two junior year semesters combined to achieve their goal of a 2.3 cumulative GPA and 90 credits.

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Answer:

3. The equivalent fractions are presented as follows;

[tex]\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]

4.  The equivalent fractions are presented as follows;

[tex]\dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]

5. The reason why it works is because [tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{4}[/tex]

Step-by-step explanation:

3. Based on the shaded region of the given figures, the two fractions are equivalent

Mathematically, we can show the equivalency of the two fractions as follows;

[tex]\dfrac{2}{3} =\dfrac{2}{3} \times 1 = \dfrac{2}{3} \times \dfrac{2}{2} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]

Therefore;

[tex]\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} =\dfrac{4}{6}[/tex]

4. From the shaded region of the given figure, the fractions are equivalent

Mathematically, we have;

[tex]\dfrac{2}{3} =\dfrac{2}{3} \times 1 = \dfrac{4}{4} \times \dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]

Therefore;

[tex]\dfrac{2}{3} = \dfrac{4 \times 2}{4 \times 3} =\dfrac{8}{12}[/tex]

5. The amount of salt required by the recipe = 1/4 teaspoon of salt

The measure of the spoon Jonas has = 1/8 teaspoon

When Jonas adds two 1/8 teaspoons, he gets;

[tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1 + 1}{8} = \dfrac{2}{8} = \dfrac{2 \times 1}{2 \times 4} = \dfrac{2}{2} \times \dfrac{1}{4} = 1 \times \dfrac{1}{4} = \dfrac{1}{4}[/tex]

Therefore;

[tex]\dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{4}[/tex]

Two 1/8 measuring teaspoons are equivalent to one 1/4 teaspoons and therefore Jonas is able to get the 1/4 teaspoons of salt the recipe asks for b combining two 1/8 measuring teaspoons of salt he has because 1/8 + 1/8 = 1/4.

Answer:

c is the only one that would result in an integer

Step-by-step explanation:

i hope this helps :)

Option c ∛ 27 would result in an integer.

An integer is the number zero, a positive natural number or a negative integer with a minus sign. The negative numbers are the additive inverses of the corresponding positive numbers.

Given here, ∛27= 3 while the other options ∛60 is not an integer because 60 is not  cubic number similarly for 9 , 18 are not cubic numbers and thus their subsequent cubic roots will not yield an integer.

Hence, Option c ∛ 27 would result in an integer.

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The sum of given series is infinity ∑(n = 1) sin(nx)/(2n - 1).

What is sum of series?

A series' sum is the sum of the words or list of numbers that make up the series. If a series has a sum, it will be one integer (or fraction), such as 0, 1/2, or 99.

As given series is,

f(x) = sinx + sin(2x)/3 + sin(3x)/5+.....

The series function is trigonometric function such as sinx, sin(2x), sin(3x).

The nth term of series is sin(nx).

The coefficient of series is 1, 1/3, 1/5.

The coefficient of nth terms is 1/(2n - 1).

Sum of series is,

= infinity ∑(n = 1) sin(nx)/(2n - 1).

Hence, the sum of given series is infinity ∑(n = 1) sin(nx)/(2n - 1).

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Answer:

48 is the answer

Step-by-step explanation:

9*4 +. 6*2

36 + 12

48

Answer:

Step-by-step explanation:

By triangle sum theorem,

Sum of all angles of a triangle is 180°.

m∠1 + m∠2 + m∠3 = 180°

(m∠1 + m∠3) + m∠2 = 180°

2(m∠1) + 70° = 180° {Given → m∠1 = m∠3]

2(m∠1) = 110°

m∠1 = 55°

Therefore, m∠1 = m∠3 = 55°

Answer:

If you just solve normally, you will get x=2 and x=-3. But if you plug these in to check your work, you will find that they are wrong. Your answer is no solution

Step-by-step explanation:

ln(2x+3)+ln(x-2)=ln(x^2-2x)

Rule: log(a) + log(b) = log(a*b)

ln( (2x+3)(x-2) ) = ln(x^2-2x)

Rule: If log(a) = log(b) then a = b

(2x+3)(x-2) = x^2 - 2x

2x^2-x-6=x^2-2x

x^2+x-6=0

Using Quadratic Formula:

x = 2 and x = -3

But, plugging these numbers back into the original equation is false!

Answer

6 and 2/3 percent of an hour OR 6.666.... hour

I don’t know if you have to round or not but if it does just round

Step-by-step explanation:

Well 4 minutes of an hour is basically 4/60

4/60=1/15

1/15=x/100

solve the proportion by cross multiplying

100=15x

x=6.66666666

That is yoru percent

Answer:

80 tsp.

Step-by-step explanation:

400 divided by 5 is 80, so 80 tsp's.

Answer:

80 tsp.

Step-by-step explanation:

400 mL = 80 tsp

Answer:

E

Step-by-step explanation:

Answer:

Each cereal bar weighs 16.8 oz

Step-by-step explanation:

Multiply 12 times 1.4 to get 16.8

Add 16.8 and 1.75 to get 18.55

Hope this helps! Pls mark brainliest

Answer:

area = 164.52 cm²

Step-by-step explanation:

area = (12x6) + (12x3x0.5x2) + (3.14)(0.5)(6²) = 164.52 cm²