2. Write an equation that
gives your own "recipe" for
making a skateboard.

Every subgroup of Z is cyclic and is generated by a unique nonnegative integer. This property makes Z a particularly simple and well-behaved group for studying group theory.

Group theory is a branch of abstract algebra that studies mathematical structures called groups. Group theory provides a powerful framework for studying symmetry, structures, and transformations in various areas of mathematics, physics, chemistry, and other fields of science. It has applications in cryptography, coding theory, crystallography, quantum mechanics, and many other areas of science and engineering.

The set of integers subject to addition, or Z, is an example of an abelian group in group theory. A subset that also forms a group by addition is referred to as a subgroup of Z. It turns out that every Z subgroup is cyclic, i.e., created by a single element.

Let H be a subgroup of Z to be more precise. H is closed under addition since Z is an additive group, and it includes the identity element (0). We can demonstrate that H is produced by a, which means that every element in H can be written as a multiple of a, by looking at the lowest positive integer in H (if any), indicated as a. Thus, H becomes a cyclic subgroup with an as its generator.

Moreover, this generator a is exclusive to H. Assume that H contains a second positive integer b that likewise yields H. We can demonstrate that the set of multiples of a and b is indeed the same set, which suggests that a and b produce the same subgroup H. As a result, a = b, and H has a unique generator.

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The pair made the following investments: The combined interest for the first year was $477 from a $2550 savings account earning 6% simple interest annually and a $3600 development plan earning 9% simple interest annually.

Borrowers are required to pay the lender's basic interest as a fee in exchange for a loan.

Compound interest is not taken into account in the calculation; just the initial principal is taken into account.

Simple interest applies to all loans, not just specific ones.

Also highlighted is the type of interest that banks provide their customers on their savings accounts.

So, we presumptively have $x invested in the savings account.

Consequently, $ was invested in the development plan.(6150 - x).

S.I. = (Prt)/100 is the formula for simple interest on a sum of money (P), compounded at a rate (r), over a time period (t).

We figure out the interest on both investments:

Contribution to a savings account:

Amount invested (P) = $x.

Rate of interest (r) = 6%.

Time for investment (t) = 1 year.

As a result, interest was received= (Prt)/100 = $(x*6*1)/100 = $ (6x)/100

We are aware that $477 in interest has been paid in total.

Thus, we can write the following equation using the interest from both investments added:

(6x)/100 + {9(6150 - x)}/100 = 477

or, 6x + 55350 - 9x = 47700,

or, 55350 - 47700 = 3x,

or, 3x = 7650,

or x = 7650/3 = 2550.

As a result, investments in:

Savings account = $x = $2550.

Development plan = $(6150 - x) = $(6150 - 2550) = $3600.

Therefore, the pair made the following investments: The combined interest for the first year was $477 from a $2550 savings account earning 6% simple interest annually and a $3600 development plan earning 9% simple interest annually.

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Correct question:

In investing $6,150 of a couple's money, a financial planner put some of it into a savings account paying 6% annual simple interest. The rest was invested in a riskier mini-mall development plan paying 9% annual simple interest. The combined interest earned for the first year was $477. How much money was invested at each rate?

The range of [tex]f(x) = \frac{1}{3x+4}[/tex] is all real numbers except for 0 i.e. {y ∈ R, y ≠ 0} and there is no x-intercept in f(x) = 1/(3x - 4)

The range of [tex]f(x) = \frac{1}{3x+4}[/tex] is all real numbers except for 0.

To see this, note that the denominator of [tex]f(x) = \frac{1}{3x+4}[/tex], can take any real value except -4/3, since division by zero is undefined.

Therefore, f(x) can take any value except 0, which means the range of f(x) is {y ∈ R, y ≠ 0}

The reciprocal of a is 1/a

From the list of options, we have

f(x) = 1/(x + 3)

The reciprocal is x + 3 and this is a linear function

So, the reciprocal of a function that is linear is f(x) = 1/(x + 3)

The reciprocal function of the form f(x) = 1/x has a vertical asymptote at x = 0.

Similarly, the reciprocal function of the form f(x) = 1/(ax + b) has a vertical asymptote at x = -b/a.

Comparing the given functions with the above form, we can see that:

f(x) = 1/(x + (5/2)) is of the required form with a = 1 and b = 5/2.

So, it has a vertical asymptote at x = -(5/2)/1 = -5/2.

Therefore, the function with a vertical asymptote at x = -5/2 is option (a), f(x) = 1/(x + (5/2)).

The correct answer is (d) All of the above, as all functions a, b and c have a horizontal asymptote at y = 0, when solved by graphs

As x approaches (5/3)^+ (from the right-hand side), the function f(x) approaches positive infinity.

So, the true statement is (b) f(x) -> oo

Given that

f(x) = 1/(3x - 4)

So, we have

1/(3x - 4) = 0

This gives

1 = 0

This means that there is no x-intercept.

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Answer:

Katherine needs 1 unit of insulin for 15 g of carbohydrates.

Step-by-step explanation:

To find out how many units of insulin Katherine should inject for her lunch which has 60 g of carbohydrates, we can use the following proportion:

1 unit : 15 g = x units : 60 g

where x is the number of units of insulin Katherine should inject.

To solve for x, we can cross-multiply:

1 unit * 60 g = 15 g * x units

60 = 15x

x = 4

Therefore, Katherine should inject 4 units of insulin for her lunch.

a) The 95% confidence interval for a sample size of 34 is given as follows: (17.26, 20.34).

b) The 95% confidence interval for a sample size of 61 is given as follows: (17.67, 19.63).

c) Increasing the sample size, the margin of error decreases.

The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the following rule:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

In which the variables of the equation are presented as follows:

The margin of error is defined as follows:

[tex]M = t\frac{s}{\sqrt{n}}[/tex]

From this we get that it is inversely proportional to the square root of the sample size, hence if we increasing the sample size, the margin of error will decrease.

The mean and the standard error are given as follows:

[tex]\mu = 18.8, s = 4.4[/tex]

For a sample size of 34, we have 33 degrees of freedom, hence, using a t-table, the critical value is given as follows: 2.0345.

Hence the bounds of the interval are given as follows:

For a sample size of 61, we have 60 degrees of freedom, hence, using a t-table, the critical value is given as follows: t= 2.

Hence the bounds of the interval are given as follows:

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The coefficient of the squared term in the parabola's equation is: -5

We are given the following information in the question:

Vertex of parabola: (2,-4)

Also, (x = 3, y = -1) lies on the parabola.

The general equation of parabola is of the form:

y = a(x - h)² + k

where:

(h, k) is the vertex of the parabola

Putting h = 2, k = -4 in the equation, we get:

y = a(x - 2)² - 4

Now, putting x = -3 and y = -3 in the above equation, we get:

-3 = a(-3 - 2)² - 4

a = 1/25

The second parabola is of the form:

x = a(y + 4)² + 2

At -3, -3:

-3 = a + 2

a =-5

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The end behavior of f(x) is described as follows:

The end behavior of a function is given by the limit of the function is the input x goes to either negative infinity or positive infinity.

The function for this problem is given as follows:

f(x) = 4(5/7)^x.

It is an exponential function with base with absolute value less than 1, hence the end behavior is given as follows:

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The new angular speed, ω₂, is four times the initial angular speed, ω₁.

To solve this problem, we can use the conservation of angular momentum. The initial and final angular momenta should be equal:

I₁ω₁ = I₂ω₂

Where I₁ and I₂ are the initial and final moments of inertia, respectively.

Assuming the particles are connected in a circular arrangement, the moment of inertia can be calculated using:

I = n * m * r²

Where n is the number of particles and r is the distance from the center of rotation.

Now we have:

(n * m * L²)ω₁ = (n * m * (0.5L)²)ω₂

Simplifying the equation, we get:

ω₂ = 4ω₁

So the new angular speed, ω₂, is four times the initial angular speed, ω₁.

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A system of small particles, each with mass m, is connected by light strings of length L. The system is rotating at an initial angular speed ω₁. If the strings are shortened to 0.5L, what is the new angular speed, ω₂?

The values of x and y in the given figure are 10.5 and 15 respectively in the quadrilateral

We have to find the values of x and y in quadrilateral

4y+14+7y+1=180

Combine like terms

11y+15=180

Subtract 15 from both sides

11y=180-15

11y=165

Divide both sides by 11

y=15

The value of y is 15

Now 7x+1+105=180

7x+106=180

Subtract 106 from both sides

7x=180-106

7x=74

Divide both sides by 7

x=10.5

Hence, the values of x and y in the given figure are 10.5 and 15 respectively

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Answer:

7034529

Step-by-step explanation:

Assuming you mean to multiply 0.75 by 9379372, the result would be:

0.75 x 9379372 = 7034529

Answer:

the function is decreasing.

the slope is -3/4 and the y-intercept is 7.

the function is linear and continuous.

y = -3/4 x + 7 3 this function.

Step-by-step explanation:

"the function is linear and discrete" is false. there are no interruptions or steps in the curve. it is a continuous curve.

the usual function form of a line is

y = ax + b

"a" being the slope, "b" being the y-intercept.

that is why the other statements are false.

4. The Federal Reserve System

5. about 10 percent

The Fed is the renowned central bank of the United States, famous for their diverse and crucial functions which includes:

Providing short-term loans to banks: The Federal Reserve serves as a lender during critical times, acting seamlessly to help out other financial institutions in order meet the necessities required to stabilize within our economic system.

Govern through banking services provided: The account management of U.S government lies on this exceptional institution. It proficiently sponsors numerous activities such as payment processing and providing governmental services amongst others.

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To find MR, we take the derivative of the demand curve with respect to Q:

MR = dP/dQ = 900 - 0.06Q

To find MC, we need to take the derivative of the quadratic cost function with respect to Q:

C(Q) = 500 + 10Q + 0.005Q^2

MC = dC/dQ = 10 + 0.01Q

To find AC, we divide the total cost by the quantity:

AC = C(Q)/Q = (500 + 10Q + 0.005Q^2)/Q = 500/Q + 10 + 0.005Q

To find P, we use the demand curve and solve for Q:

P = 900 - 0.03Q

Q = (900 - P)/0.03

Explanation:

To find P, we use the demand curve and solve for Q:

P = 900 - 0.03Q

Q = (900 - P)/0.03

To find MR, we take the derivative of the demand curve with respect to Q:

MR = dP/dQ = 900 - 0.06Q

Now we can graph MC, AC, P, and MR:

|

P  |        /MR

  |       /

  |      /

  |     /

  |    /

  |   /

  |  / AC

  | /

MC |/

  |------------- Q

Note:

The graph is a rough sketch and not drawn to scale. The MC curve is upward-sloping, the AC curve is U-shaped, the P curve is downward-sloping, and the MR curve lies below the P curve and is also downward-sloping.

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X + 1 = 0, so x = -1, and x - 24 = 0, so x = 24. Consequently, the two answers to the equation are x = -1 and x = 24.

It is a method for illustrating a notion, a connection, or a behaviour that often involves two or more variables. Equations can be used to explain several events, including economic models and physical principles. They can also be applied to problem-solving and forecasting. Symbols, integers, and mathematical operations like addition, subtraction, multiplication, and division are used to create equations.

An essential tool for factoring quadratic equations is the Zero Factor Property. Finding two terms, one of which is a factor of the constant term and the other of which is a factor of the coefficient of the squared term, whose product is equal to the constant term, is the fundamental principle behind the Zero Factor Property. With this knowledge, the given quadratic equation can be factored into two parts that equal zero.

One must first factor a quadratic equation into two factors that are equal to zero in order to solve it using the zero factor property. To do this, one needs determine the constant term's factors as well as the coefficient of the squared term's factors. The coefficient of the squared term is always obtained by multiplying the factors, which are always different integers.

The constant term's factors are always sums of numbers that result in the constant term. Once the elements are known, the equation can be factored into two factors that are equal to zero each.

Consider the equation [tex]x^2+5x-24=0[/tex], for instance. The factors of the constant term are -1 and 24, while the factors of the squared term's coefficient are 1 and 5. The equation then reads as (x + 1)(x - 24) = 0.

At this stage, the Zero Factor Property can be used to find x. Since all factors must be equal to zero, x can be solved for by setting all factors to zero. Thus, x + 1 = 0, resulting in x = -1, and x – 24 = 0, resulting in x = 24. Thus, x = -1 and x = 24 are the two answers to the problem.

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Complete questions as follows-

How do I use the zero factor property when solving a quadratic equation by factoring?

Required purchase price is $120 and advertised price is $300.

How much did Danielle purchase the antique dresser for?

Let's assume that Danielle purchased the antique dresser for $x.

According to the problem, she advertised it for 250% of her purchase price, which means she advertised it for,

250% of x = 2.5x

She sold the dresser for 15% less than the advertised price of 2.5x, so she sold it for,

(1 - 0.15) × 2.5x = 2.125x

We know that the selling price was $255, so we can set up the equation,

2.125x = 255

Solving for x,

x = 120

Therefore, Danielle purchased the antique dresser for $120.

What was her initial advertised price?

To find the initial advertised price, we can plug x = 120 into the expression 2.5x,

2.5x = 2.5 × 120 = 300

So, the initial advertised price was $300.

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The correct answer is:

[tex]h1(t) = -16t^2 + 121[/tex] is a vertical translation of [tex]h2(t) = -16t^2 + 225.[/tex]

The y-intercept of h1 is 104 ft less than that of h2.

What is a y-intercept?

A function or relationship's graph meets the y-axis of the coordinate system at a point known as a y-intercept, sometimes known as a vertical intercept.

The height function for an object dropped from a height of h0 is given by:

[tex]h(t) = -16t^2 + h0\\\\h1(t) = -16t^2 + 121\\\\h2(t) = -16t^2 + 225[/tex]

To find the y-intercept of h1, we set t = 0 in the equation[tex]h1(t) = -16t^2 + 121:[/tex]

[tex]h1(0) = -16(0)^2 + 121 = 121[/tex]

Therefore, the y-intercept of h1 is 121 feet.

To find the y-intercept of h2, we set t = 0 in the equation [tex]h2(t) = -16t^2 + 225:\\\\h2(0) = -16(0)^2 + 225 = 225[/tex]

Therefore, the y-intercept of h2 is 225 feet.

The difference in their y-intercepts is:

225 - 121 = 104

Therefore, the y-intercept of h1 is 104 feet less than that of h2.

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Answer:

B, C, E

Step-by-step explanation:

B:

[tex](\frac{3}{4})^2-\frac{6}{16} \\= \frac{9}{16} - \frac{6}{16} = \frac{3}{16}[/tex]

C:

[tex](\frac{3}{8})^2 / \frac{3}{4} \\= \frac{9}{64} / \frac{3}{4} = \frac{9}{64} *\frac{4}{3} = \frac{3}{16}[/tex]

E:

[tex]2*\frac{3}{4} - (\frac{3}{4})^2 - \frac{3}{4} \\= \frac{6}{4} - \frac{9}{16} - \frac{3}{4} = \frac{3}{4} - \frac{9}{16} = \frac{12}{16} - \frac{9}{16} = \frac{3}{16}[/tex]

Answer:

B, C, E

Step-by-step explanation:

B:

[tex](\frac{3}{4})^2-\frac{6}{16} \\= \frac{9}{16} - \frac{6}{16} = \frac{3}{16}[/tex]

C:

[tex](\frac{3}{8})^2 / \frac{3}{4} \\= \frac{9}{64} / \frac{3}{4} = \frac{9}{64} *\frac{4}{3} = \frac{3}{16}[/tex]

E:

[tex]2*\frac{3}{4} - (\frac{3}{4})^2 - \frac{3}{4} \\= \frac{6}{4} - \frac{9}{16} - \frac{3}{4} = \frac{3}{4} - \frac{9}{16} = \frac{12}{16} - \frac{9}{16} = \frac{3}{16}[/tex]

The box plot that represents the data is given as follows:

Bottom plot.

A box and whisker plots shows these five features from a data-set, listed as follows:

The ordered data-set for this problem is given as follows:

2, 2, 4, 7, 7, 14, 16, 18.

The minimum and maximum values are given as follows:

The data-set has an even cardinality, hence the median is given by the mean of the two middle elements as follows:

Median = (7 + 7)/2 = 7.

(which removes the top two options).

The quartiles are given as follows:

Meaning that the last plot is correct. (the one that is already marked).

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We can eulerize this graph in the most efficient way possible by making these vertices have even degree through  adding  an edge between them.

We can eulerize a graph by ensuring  to add edges to the graph in a way that preserves its connectivity while ensuring that all vertices have even degree.

A good method to do this is by  adding  edges between pairs of vertices that have odd degree until all vertices have even degree.

In the scenario above,  we can see that all vertices have odd degree except for the last vertex.

Hence , in order to make this vertex have even degree, we can add an edge from it to any other vertex with odd degree.

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Based on the data, the appropriate 95% confidence interval is (0.039, 0.361), which is closest to the answer option (0.038, 0.362).

A specific type of ordinary dispersion with a mean of 0 and a standard deviation of 1 is known as a standard typical conveyance.

We may use the following calculation to create a 95% confidence interval for the variation in the proportion of students taking an AP class in each region:

(East - West) ± z*√[(East proportion)(1 - East proportion)/East sample size + (West proportion)(1 - West proportion)/West sample size]

where z* denotes the critical value of the standard normal distribution for the desired level of confidence, East proportion denotes the percentage of East students who took at least one AP class, West proportion denotes the percentage of West students who took at least one

AP class, East sample size denotes the sample size of East students, and West sample size denotes the sample size of West students.

Using the given data, we have East proportion = 20/50 = 0.4, West proportion = 16/80 = 0.2, East sample size = 50, and West sample size = 80.

The crucial value z* for a 95% confidence level can be calculated using a conventional normal distribution table or calculator and is roughly 1.96.

after entering each value into the formula:

(0.4 - 0.2) ± 1.96√[(0.4)(0.6)/50 + (0.2)(0.8)/80]

= 0.2 ± 0.161

As a result, the answer choice (0.038, 0.362) comes the closest to the right 95% confidence interval based on the data, which is (0.039, 0.361).

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The area of the right triangle is 1702 sq units and the area of the isosceles triangle is 539.39 sq units

The area of the right triangle is calculated as

Area = 1/2 * Base * Height

So, we have

Area = 1/2 * 37 * 92

Area = 1702

The area of the isosceles triangle is calculated as

Area = 1/2s²sin(c)

So, we have

Area = 1/2 * 37² * sin(52)

Evaluate

Area = 539.39

Hence, the value of the area is 539.39 sq units

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Answer:

B. chocolate-peanut butter cups and chocolate-caramel bites

Step-by-step explanation:

Chocolate-peanut butter cups cost $11.46 per pound, so one-half pound costs $5.73.

Chocolate-caramel bites cost $9.62 per pound, so one-half pound costs $4.81.

$5.73 + $4.81 = $10.54, so B is the correct answer.

The graph of the functions (A) y=x+5 (B) y=-(x+5)  and  (C) y=|x+5| are added as an attachment

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

To mark all points on the same coordinate plane, we simply plot the graphs of the three functions on the same graph

Using the above as a guide, we have the following:

The graph of the functions (A) y=x+5 (B) y=-(x+5)  and  (C) y=|x+5| are added as an attachment

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The best description of the equation is no real solutions. Option C

To solve the equation 3(x + 12) + 5x = 8x - 36, we can simplify the left-hand side by distributing the 3, and then combine like terms:

3x + 36 + 5x = 8x - 36

8x + 36 = 8x - 36

36 = -36

This equation is a contradiction, since there is no value of x that will make 36 equal to -36. Therefore, there are no real solutions to this equation.

The answer is (C) no real solutions.

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By performing the division, the jogging track is 0.45 miles long.

Division is a mathematical operation that involves the splitting of a quantity into equal parts or groups.

The problem asks us to convert a length of 792 yards to miles. To do this, we need to use a conversion factor to relate yards to miles. We know that there are 1760 yards in a mile, so we can use this relationship to convert the length in yards to miles.

To convert yards to miles, we divide the length in yards by the number of yards in a mile. This is because we want to cancel out the units of yards and be left with the corresponding number of miles.

So, 792 yards / 1760 yards/mile gives us the length of the jogging track in miles. We can simplify this expression by performing the division to get the result of 0.45 miles. Therefore, the jogging track is 0.45 miles long.

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Answer:

32

Step-by-step explanation:

m∠S = tan⁻¹(5/8) = 32⁰

Using the combination formula, it is found that there are 3300 ways to choose the songs.

The order in which the musics are chosen is not important, hence the combination formula is used to solve this question.

What is the combination formula?

��,�Cn,x is the number of different combinations of x objects from a set of n elements, given by:

��,�=�!�!(�−�)!Cn,x=x!(n−x)!n!

For the rock songs, we have three from a set of eleven, hence:

�11,3=11!3!8!=165C11,3=3!8!11!=165

For the alternative songs, we have four from a set of five, hence:

�5,4=5!4!1!=5C5,4=4!1!5!=5

For the rap songs, we have three from a set of four, hence:

�4,3=4!3!1!=4C4,3=3!1!4!=4

Since the songs are independent, the total number of ways is given by:

�=165×5×4=3300T=165×5×4=3300

hope this helps you

Using the combination formula, it is found that there are 3300 ways to choose the songs.

The order in which the musics are chosen is not important, hence the combination formula is used to solve this question.

What is the combination formula?

��,�Cn,x is the number of different combinations of x objects from a set of n elements, given by:

��,�=�!�!(�−�)!Cn,x=x!(n−x)!n!

For the rock songs, we have three from a set of eleven, hence:

�11,3=11!3!8!=165C11,3=3!8!11!=165

For the alternative songs, we have four from a set of five, hence:

�5,4=5!4!1!=5C5,4=4!1!5!=5

For the rap songs, we have three from a set of four, hence:

�4,3=4!3!1!=4C4,3=3!1!4!=4

Since the songs are independent, the total number of ways is given by:

�=165×5×4=3300T=165×5×4=3300

hope this helps you

The solution is : The measure of angle ADC is 115°.

Here, we have,

Let A, B, C represent the vertex angle measures of ΔABC.

Then the sum of angles in ΔADC is ...

 A/2 +∠ADC + C/2 = 180°

Multiplying by 2 gives ...

 A + C + 2×∠ADC = 360°

From the given information, we know ...

 A + C + 50° = 180° . . . . . sum of angles in ΔABC

This lets us write an expression for (A +C):

 A + C = 130° . . . . . . . . subtract 50° from the previous equation

Substituting for A+C, we get ...

 130° + 2×∠ADC = 360°

 2×∠ADC = 230° . . . . . . . . . subtract 130°

 ∠ADC = 115° . . . . . . . . . . . divide by 2

The measure of angle ADC is 115°.

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complete question:

The measure of an angle ABC is 50°,AD bisects angle BAC, and DC bisects angle BCA. The measure of angle ADC is what?​

A square prism has the following characteristics:

Number of faces: 6

Number of vertices: 8

Number of edges: 12

A cuboid has the following characteristics:

Number of faces: 6

Number of vertices: 8

Number of edges: 12

What is meant by square prism?

A square prism is a three-dimensional geometric shape with two parallel square bases and rectangular faces connecting the two bases. It has 8 vertices, 12 edges, and 6 rectangular faces.

What is meant by cuboid?

A cuboid is a three-dimensional geometric shape with six rectangular faces, where opposite faces are equal in size and shape. It has 8 vertices, 12 edges, and 6 rectangular faces. A cuboid is also known as a rectangular prism.

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