The table shown below gives the approximate enrollment at the University of Michigan every fifty years. How many more students were enrolled at the University of Michigan in 1950 than in 1900?

The curvature of r(t) = 3t, t2, t3 at the point (3, 1, 1) is 0.178

In this question we need to find the curvature of r(t) = <3t, t^2, t^3> at the point (3, 1, 1).

The derivative of r(t) with respect to t is:

r'(t) = <3, 2t, 3t^2>

The point (3, 1, 1) corresponds to t = 1,

and r′(1) = <3, 2(1), 3(1)^2>

r'(1) = <3, 2, 3>

|r'(1)| = √(3^2 + 2^2 + 3^2)

|r'(1)| = √(9 + 4 + 9)

|r'(1)| = √(22)

|r'(1)| = 4.69

Now, r''(t) = <0, 2, 6t>

and r′'(1) = <0, 2, 6(1)>

r''(1) = <0, 2, 6>

r′(1) × r′′(1) = <0, 4, 18>

|r′(1) × r′′(1) |

= √(0^2 + 4^2 + 18^2)

= √(16 + 324)

= √340

= 18.44

So, the curvature of r(t) would be,

κ(1) = |r′(1) × r′′(1)| ÷ |r′(1)|^3

k(1) = 18.44 / 4.69³

k(1) = 0.178

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In order to solve a quadratic equation with one unknown, we need to factor it. If at first glance we can't find easy roots, we do a discrimination.

If this value is less than zero, Δ[tex]< 0,[/tex] we determine that it has no solution belonging to the set of real numbers. We will find a conjugate and two complex roots. ([tex]a+bi[/tex])

Let's use the following formulas to find the roots.

Complex Solutions:

Answer:

g = 109 degree

h = 71 degree

m = 71 degree

k = 84 degree

Step-by-step explanation:

g= 109 degrees because it is a vertical angle to the 109-degree angle, that is why it is 109 degree

h= 71 degrees because they are supplementary angles which means they add up to 180 degrees. We see that one is 109 degrees, so the other must be 71 degrees.

k = 84 degrees because it is a vertical angle to the 84-degree angle, which is why it is 84 degrees.

m= 71 degrees because m and g must add up to 180 degrees, and we know that g is 109 degrees, so m is 71 degrees!

The common difference in the sequence 7, 12, 17, 22, 27,… is that the difference between two numbers increases by five in the ascending order.

A sequence can be referred to or considered as the set of observations, which usually consists of numbers arranged in such a way that they described a common characteristic between them. A sequence can be exhaustive or inclusive, depending upon the numbers being used in the continuity of the sequence. It can also be said that a sequence is a logical arrangement.

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Complete question

what is the common difference in the sequence 7, 12, 17, 22, 27, ...?

Answer:

m∠D = 49°
m∠E = 41°
m∠F = 90°

Step-by-step explanation:

        We know a triangle's angles add up to 180, so we can create an equation and solve for x. We also know the little box represents 90 degrees.

Given:

        90° + 5x° + 14° + 3x° + 20° = 180°

Combine like terms:

        8x° + 124° = 180°

Subtract 124 from both sides of the equation:

        8x° = 56°

Divide both sides of the equation by 8:

        x = 7

        Now, we will plug this value of x back into the angles to solve. Again, the little box represents that Angle F is equal to 90 degrees.

m∠D = 5x° + 14° = 5(7)° + 14° = 49°
m∠E = 3x° + 20° = 3(7)° + 20° = 41°
m∠F = 90°

Using the arithmetic progression, If a theatre has 30 rows of seats there are 22 seats in the first row,21 in the second row, 51 in the third row, then the total number of people that the theatre hold is 2400

The total number of rows = 51

Number of seats in the first row = 22

Number of seats in the second row = 21

Number of seats in the third row = 51

Common difference= Second term - first term

= 26-22

= 4

The given sequence is in arithmetic progression

Substitute the values in the equation

= 15[44+29×4]

= 15[44+116]

= 15×160

= 2400

Hence, using the arithmetic progression, if a theatre has 51 rows of seats  and there are 22 seats in the first row,21 in the second row, 51 in the third row, then the total number of people that the theatre hold is 2400

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What we should do is plot each part of the graph individually.

Step 1: Plot f(x) = 2x + 6 part

We know it starts at x = -5 (noninclusive) and ends at x = -2 (inclusive). Therefore, we can draw a line between the two points (-5, -4) and (-2, 2), with an open point at x = -5 and a closed point at x = -2

Step 2: Plot f(x) = 1 part

We know it starts at x = -2 (noninclusive) and ends at x = 2 (inclusive). Therefore, we can draw a line between the two points (-2, 1) and (2, 1), with an open point at x = -2 and a closed point at x = 2

Step 3: Plot f(x) = -3x + 10 part

We know it starts at x = 2 (noninclusive) and ends at x = 5 (inclusive). Therefore, we can draw a line between the two points (2, 4) and (5, -5), with an open point at x = 2 and a closed point at x = 5

You should end up with a graph that looks like the attached image!

The given function f(x) has been plotted in the graph piecewise with proper endpoints.

A graph is a diagram that shows the fluctuation of one variable in relation to one or more other variables.

In order to plot the graph, we need to find out y's values corresponding to x's value

As per the given piecewise function,

f(x) = [tex]\left \{ {{2x+6 \ \ \ \ -5 < x\leq -2}} \right.[/tex]

f(x) = [tex]\left \{ {{1 \ \ \ \ -2 < x\leq 2}} \right.[/tex]

f(x) = [tex]\left \{ {{-3x+10 \ \ \ \ 2 < x\leq 5}} \right.[/tex]

The function f(x) = 2x + 6 is a linear line with slope 2 and y-intercept 6 shown in red line -5 is excluded while - 2 is included.

The function f(x) = 1 is a constant horizontal line shown in the blue line that -2 excludes while  2 is included.

The function f(x) = -3x + 10 is a linear line with slope -3 and y-intercept 10 shown in the green line 2 is excluded while - 5 is included.

Hence "With the appropriate endpoints, the supplied function f(x) has been piecewise displayed in the graph".

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Solution is 225/42 or 5.35

According to question

5 6/7 + (-2 1/3) + 1 5/6

As it is a mixed fraction we will first solve it

To solve it denominator will get multiplied with the whole number and the solution will be get added to the numerator and the new numerator will get formed.

∴ 41/7 +(-7/3) + 11/6

⇒246-98+77/42.                              (LCM of 7,3,6 is 42)

⇒225/42

∴5.35

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Answer: 5x + 10y = or > 75

Step-by-step explanation:

make the number of cookies sold  = x, and the number of cupcakes sold = y, since we don't know the number.

each cookie is $5 so it is 5 times x, and each cupcake is $10 so it is 10 times y.

add those together, and the sum must equal 75. If you have already learned this, it would be an equal to or greater than sign. if not then just use an equal sign.

Answer:

1/ We use the SSS method to prove that triangle CBD = triangle ABD

2/ We use the AA method to prove that <CBD equal < ABD

Step-by-step explanation:

1/ SSS, or side side side method, is the method we use to see if triangles are the same or not, and we know that the sides of both triangles are equal; this proves that both triangles are equal!

2/ AA, or Angle Angle method, is the method we use to see if the angles of triangles are the same or not, and we know that there are marks at <C and <A, these marks prove that both angles are equal! So, <CBD equal <ABD

Answer:

Step-by-step explanation:

The greatest common divisor of two integers a and b is a divisor common to both integers that is greater than any other common divisor. The least common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b.

To find the smallest possible value of a+b, we can start by trying to find the smallest possible values of a and b that satisfy the given conditions.

Since the greatest common divisor of a and b is 5, we know that both a and b must be divisible by 5. We can write a = 5x and b = 5y, where x and y are positive integers.

The least common multiple of a and b is 55, so we know that both a and b must be factors of 55. We can write 55 as the product of its prime factors: 55 = 5 * 11. Since a is divisible by 5, we know that a must be equal to 5 or 25. Similarly, since b is divisible by 5, we know that b must be equal to 5 or 25.

There are four possible pairs of values for a and b: (5,5), (5,25), (25,5), and (25,25). The smallest of these is (5,5), which gives us a+b = 5+5 = 10. Therefore, the smallest possible value of a+b is 10.

I think the answer be 12 and -18 D

When rotated through 180° anticlockwise or clockwise about the origin, the new position of the above points is.

Rotate shape A [tex]$180^{\circ}$[/tex] about point (1,0) is (-1,0)

How do you rotate a shape about a certain point?

Move each point on a shape the specified number of degrees in a circle centered on the point of rotation to rotate it. Make sure that every new point is located the same amount of time away from the point of rotation as its corresponding original point.

There are 360 degrees in a circle's complete revolution. We discovered that a rotation of 270 degrees is virtually a full circle. We learned that a whole rotation of a circle is divided in half by 180 degrees. 90 degrees, or half of 180, are simple to identify as angles.

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The remainder of the polynomial using the remainder theorem and factor theorem is 6.

Apply the remainder theorem,

When we divide a polynomial

f(x) by (x − c)

f(x) = (x − c)q(x) + r

f(c) = 0 + r

Here,

f(x)=(x−c)q(x)+rf(c)=0+r

and (x−c) is (x−(−2))

Therefore,

f(−2) = [tex](-2)^{4} - (-2)^3 - 3(-2)^2 + 4(-2) + 2[/tex]

= 16 + 8 − 12 − 8 + 2

= 6

Hence, the remainder of the polynomial using the remainder theorem is 6.

Whereas using the factor theorem and doing synthetic division, we get,  

x = -2 is a zero of f(x), and x+2 is a factor of f(x). To factor f(x), we divide

the coefficients of the polynomial as follows -

         -2   |   1  -1    -3     4      2

                      -2   6    -6      4

        -----------------------------------------

                    1   -3   3    -2     6

Hence, we get that 6 is the remainder when ([tex]x^4-x^3-3x^2+4x+2[/tex]) ÷ (x+2), using the factor theorem.

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The complete question is -

Find the remainder using the Remainder Theorem and Factor Theorem using the synthetic division of the given polynomial, [tex]x^4-x^3-3x^2+4x+2[/tex] ÷ (x+2)

Answer:

62.5% or in decimal form 0.625

Step-by-step explanation:

The value of x for the expression 5log(3x) = 15 is 6.7.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the expression is 5log(3x) = 15. The value of x for the expression will be calculated as,

5log(3x) = 15

log(3x) = 15 / 5

log(3x) = 3

3x = e³

x = 20 / 3

x = 6.7

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Reflection is a type of rigid transformation in which an object is flipped about a reference point or line to produce the required image. The three statements that apply are:Vertex B is the midpoint of AA'The length of CD is equal to the length of C'D'Vertex C and vertex C are located at the same point

Rigid transformation is a process in which the size, orientation or position of a given object can be changed by applying some principles. Types of rigid transformation are: reflection, translation, dilation, and rotation.

In the given question, the required statements that apply are:

Vertex B is the midpoint of AA'The length of CD is equal to the length of C'D'Vertex C and vertex C are located at the same point

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Amount of each monthly statement is 1400.

According to question ,

Cash price of refrigerator =34500

Deposit money=6500

Outstanding money=28000

Rate of interest=10%

Time period=24 months or 2 years

∴Simple interest =(principal ×rate of interest×time)÷100

⇒(28000×10×2)÷100

⇒5600

∴The total amount to be paid in 2 years is 28000+5600 =33600

∴Amount to be paid in one month =33600÷24=1400

∴Amount of each monthly statement is 1400.

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Answer:

39 children, 201 adults

Step-by-step explanation:

Let c = number of children

a = number of adults

c + a = 240

1.50c + 2.25a = 510.75

--------------------------------

1.50c + 1.50a = 360

1.50c + 2.25a = 510.75

-------------------------------

.75a = 150.75

a = 201, c = 39

a steep line line has  a  bigger absolute greater slope than a flat line

At [tex]$5 \%$[/tex] level, the data is not strong enough to establish the claim of toothpaste advertisement.

It is recommended to switch to new toothpaste.

(a)

The null and alternative hypotheses are,

[tex]$$\begin{aligned}& H_0: \mu=3 \\& H_1: \mu < 3\end{aligned}$$[/tex]

Here, [tex]$\mu$[/tex] is the hypothesized mean.

Let the sample size be [tex]$n=2500$[/tex].

Let the sample mean be [tex]$\bar{x}=2.95$[/tex].

Let the population standard deviation be [tex]$\sigma=1$[/tex].

Since, the population standard deviation is known, one sample z-test is used to test the claim.

The test statistic is,

[tex]$$\begin{aligned}z & =\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}} \\= & \frac{2.95-3}{\frac{1}{\sqrt{2500}}} \\& =-2.5\end{aligned}$$[/tex]

So, the test statistic is, [tex]$z=-2.5$[/tex].

Let the level of significance be [tex]$\alpha=0.05$[/tex].

Find the critical value.

Using the standard normal distribution area tables, the left tail critical value is,

[tex]$$z_{0.05}=-1.645$$[/tex]

z score table attached at the end of solution.

The test statistic value is falls in the rejection region. Reject the null hypothesis. Therefore, it can be concluded that this does not support the null hypothesis at 5 percent level. So, at [tex]$5 \%$[/tex] level, the data is not strong enough to establish the claim of toothpaste advertisement.

(b)

The value of test statistic is also less than [tex]$-Z_{0.1}=-2.33$[/tex].

So, the claim cannot be established even at 1 percent level. It is recommended to switch to new toothpaste.

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Answer: (a) Obtain a point estimate for the population proportion of adults in the country who believe that televisions are a luxury they could do without.

The point estimate for the population proportion is 514 / 1012 = 0.507.

(b) Verify that the requirements for constructing a confidence interval about p are satisfied.

The sample is stated to be a simple random sample.

The value of n * p = 1012 * 0.507 = 514.7796 is greater than or equal to 10.

The value of n * (1 - p) = 1012 * (1 - 0.507) = 497.2204 is greater than or equal to 10.

The population proportion is less than or equal to 5% of the population size.

Therefore, the requirements for constructing a confidence interval are satisfied.

(c) Construct and interpret a 95% confidence interval for the population proportion of adults in the country who believe that televisions are a luxury they could do without.

The 95% confidence interval for the population proportion is 0.507 +/- 1.96 * sqrt((0.507 * (1 - 0.507)) / 1012) = 0.507 +/- 0.0245 = (0.4825, 0.5315).

We are 95% confident that the proportion of adults in the country who believe that televisions are a luxury they could do without is between 0.4825 and 0.5315.

(d) Is it possible that a supermajority (more than 60%) of adults in the country believe that television is a luxury they could do without? Is it likely?

It is possible that a supermajority of adults in the country believe that television is a luxury they could do without because the 95% confidence interval contains 60%. However, it is not likely because the confidence interval is relatively narrow and the point estimate is far from 60%.

(e) Use the results of part (c) to construct a 95% confidence interval for the population proportion of adults in the country who believe that televisions are a necessity.

The 95% confidence interval for the population proportion of adults who believe that televisions are a necessity is 1 - 0.5315 = 0.4685 to 1 - 0.4825 = 0.5175. Therefore, the 95% confidence interval for the population proportion of adults who believe that televisions are a necessity is (0.4685, 0.5175).

Step-by-step explanation:

False. An objective function is necessary for both maximization and minimization problems.

An objective function is an equation that represents the goal of an optimization problem. It is used to evaluate the performance of a certain set of decisions. Regardless of whether the goal is to maximize or minimize an outcome, the objective function is a necessary component of an optimization problem. In a maximization problem, the objective function is used to determine the highest possible output that can be achieved. In a minimization problem, the objective function is used to determine the lowest possible output that can be achieved. Therefore, an objective function is necessary for both maximization and minimization problems.

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No, the statistical notion of an outlier as an infrequently observed value is not meaningful for this data.

Since the points are uniformly distributed on the interval [0,1], they should all be observed with the same frequency. Therefore, there are no outlier values.The statistical notion of an outlier is an infrequently observed value, meaning that it is an observation that is far away from the rest of the data. For a set of points that are uniformly distributed on the interval [0,1], however, all the points should be observed with the same frequency since they are evenly spaced. Therefore, there is no outlier value and the statistical notion of an outlier is not meaningful for this data.

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one of two or more numbers that multiply together to form a specific product; a number that divides into another number with no remainder is a factor.

A component is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, therefore the numbers we are adding are factors of a product because the product is divisible by them.

Which of these four main types of factoring are they? The Grouping technique, the Differences in Two Squares, the Total or Difference in Cubes, as well as the Greatest Common Factors (GCF) are the four primary methods of factoring.

A factor is a value that completely divides another number. Consequently, the values we are multiplication are factors if multiplying two whole numbers results in a product.

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To find the amount of extra money Robert will have after two years if he compounds continuously instead of collecting simple interest monthly, we need to calculate the compound interest on the investment.

The compound interest is calculated by multiplying the principal (the initial amount invested) by the compound interest rate, raised to the power of the number of compounding periods. In this case, the compound interest rate is 1.9%/12 = 0.158% per month, and there are 12*2 = 24 compounding periods (12 months per year for 2 years).

The compound interest is therefore calculated as follows:

Compound interest = $20,000 * e^(0.158%*24) - $20,000

= $20,000 * e^0.3832 - $20,000

= $20,000 * 1.454 - $20,000

= $4,080 - $20,000

= -$15,920

So the amount of extra money Robert will have after two years if he compounds continuously instead of collecting simple interest monthly is -$15,920. Therefore, the correct answer is [A] $800.

By applying the algebra concept, we can conclude that the movie that Mario and his friend watched starting at 7 PM

The use of algebra in everyday life.

Algebra is a branch of mathematics that uses numbers, symbols, or letters to solve math problems. In everyday life, algebra is often used for financial planning, traveling time calculation, and so on.

In the above problem, we can take some notes as follows:

Departure time (x) = 5:15 PM

Travel time (t) = 1 hour 20 minutes

Arrival time (y) = 25 minutes

If z is the playing time of the movie, then z can be expressed mathematically in the following way:

z = x + t + y

= 5:15 + 1 hour 20 minutes + 25 minutes

= 6:60 PM or generally written 7:00 PM

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To find the function, we need to find the values of x and y.

In the aforementioned example, x and y are the independent and dependent variables, respectively. A function is an equation with just one solution for every value of y in the equation. A function pairs each input of a particular type with exactly one output. Instead of y, it is typical to name functions f(x) or g(x).

As we know,

fx = 3x to the power 3

f(x) =

[tex]3x {}^{3 } \: dx \: + c \\ \\ fx = 3x { }^{4} \div 4 + c \\ \\ \\ \\ 3x {}^{3} = - 81 \\ \\ x \: = 243 \\ [/tex]

Hence x is 242

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a. The number of choices possible for each digit of the code is of: 10.

b. The total number of 4-digit passwords that Jess can create is of: 10,000.

c. The number of different ways to create even passwords is of: 5,000.

The Fundamental Counting Theorem states that if there are n independent trials, each with [tex]n_1, n_2, \cdots, n_n[/tex] possible results, the total number of outcomes is calculated by the multiplication of the number of outcomes for each trial, according to the rule presented as follows:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

Each digit can be between 0 and 9 inclusive, hence the number of outcomes possible for each digit of the password is given as follows:

10.

The password is composed of four digits, hence the total number of passwords possible is of:

N = 10 x 10 x 10 x 10 = 10,000.

Considering that Jess would like an even number, there are only five outcomes possible for the last digit (0, 2, 4, 6 or 8), hence the number of outcomes is of:

N = 10 x 10 x 10 x 5 = 5,000.

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Add missing statements and reasons.

    Statement                            Reason