Which expressions are polynomials?

Select each correct answer.


6x² + 5x

−7x^2+5/3x

−x^2 +5x

x^2+5x ^1/5

Answer: The length of the parking lot should be increased 20 ft.

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Thanks.

Answer:

b on edge

Step-by-step explanation:

We have a coordinate plane that is placed over an empty lot. So you and your friend are set at that coordinate system, so I'll give you the representation of each statement and the distance that the snowball travels from your friend's hand to you.

1. You face the positive y-axis and your friend faces the negative y-axis. 

This statement is represented in Figure 1. So you are the Red square and your friend is the Blue one. The arrows upon the squares mean that you are facing the positive y-axis and your friend the negative one. 

2. You run 20 feet forward, then 15 feet to your right. At the same time, your friend runs 16 feet forward,  then 12 feet to her right.

This is shown in Figure 2. So, at the coordinate system, you move 20 feet upward and 15 feet to your right, that is, you walk from the origin to the point [tex]P_{1}(15,20)[/tex] first moving through the positive y-axis and next through the positive x-axis. On the other hand, your friend moves 16 feet downward and 12 feet to her right, that is, she walks from the origin to the point [tex]P_{2}(-12,-16)[/tex] first moving through the negative y-axis and next through the negative x-axis. 

3. She stops and hits you with a snowball.

This statement is represented in Figure 3. So the snowball has been drawn in gray. The line from your friend to you is the distance the snowball runs.

4. Distance the ball runs.

We can get this answer by using the Distance Formula, that is:

[tex]d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^2} \\ d=\sqrt{[15-(-12)]^{2}+[20-(-16)]^2} \\ \boxed{d=45ft}[/tex]

Answer with explanation:

In the given right triangle

 [tex]\sin 60^{\circ}=\frac{\text{Perpendicular}}{\text{Hypotenuse}}\\\\ \frac{\sqrt{3}}{2}=\frac{\text{Perpendicular}}{2\sqrt{3}}\\\\ \text{Perpendicular}}=2\sqrt{3} \times\frac{\sqrt{3}}{2}\\\\\text{Perpendicular}}=\sqrt{3} \times\sqrt{3} \\\\\text{Perpendicular}}=3[/tex]

→Side opposite to 60° angle = 3 units

Option C : 3 Units

Answer:

the length of the hypotenuse = 32.45 centimeters

Step-by-step explanation:

A right triangle has legs that are 18 centimeters and 27 centimeters long

In a right angle triangle , to find hypotenuse we use Pythagorean theorem

[tex]c^2= a^2+b^2[/tex]

where a  and b are the length of two legs

Given a= 18  and b = 27

Lets find out C , plug in all the value in the formula

[tex]c^2= 18^2+27^2[/tex]

[tex]c^2=324 +729= 1053[/tex]

c^2 = 1053

now take square root on both sides

c= 32.45

So the length of the hypotenuse = 32.45 centimeters

Answer:  The other endpoint of the segment is (-1, -11).

Step-by-step explanation:  Given that the midpoint of a line segment is (6, -6) and one endpoint is (13, -1).

We are to find the co-ordinates of the other endpoint.

Let (a, b) be the co-ordinates of the other end-point.

Then, according to the given information, we have

[tex]\left(\dfrac{a+13}{2},\dfrac{b+(-1)}{2}\right)=(6,-6)\\\\\\\Rightarrow \left(\dfrac{a+13}{2},\dfrac{b-1}{2}\right)=(6,-6).[/tex]

Equating the x and y co-ordinates on both sides of the above, we get

[tex]\dfrac{a+13}{2}=6\\\\\\\Rightarrow a+13=12\\\\\Rightarrow a=12-13\\\\\Rightarrow a=-1[/tex]

and

[tex]\dfrac{b-1}{2}=-6\\\\\\\Rightarrow b-1=-12\\\\\Rightarrow b=-12+1\\\\\Righatrrow b=-11.[/tex]

Thus, the other endpoint of the segment is (-1, -11).

To find the shortest distance from Robert's home to his office, we use the Pythagorean theorem with the distances traveled north and east to calculate the length of the hypotenuse, which is approximately 7.2 kilometers.

Robert leaves his home and drives 6 km due north and then 4 km due east. To determine the shortest distance from Robert's home to his office, we can use the Pythagorean theorem. This scenario forms a right-angled triangle where the two sides are the north-bound and east-bound legs of his journey, and the hypotenuse is the shortest distance.

Step 1: Label the lengths of the two sides adjacent to the right angle as 'a' and 'b', where 'a' is the 6 km north-bound leg and 'b' is the 4 km east-bound leg.

Step 2: Apply the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides:

c² = a² + b²

Step 3: Substitute the known values into the theorem:

c² = 6² + 4²

Step 4: Calculate the squares of the sides and sum them:

c² = 36 + 16

Step 5: Sum the squares gives us:

c² = 52

Step 6: Take the square root of both sides to find 'c':

c = √52

Step 7: Calculate the square root which approximately equals:

c = 7.2 km

Therefore, the shortest distance from Robert's home to his office is approximately 7.2 kilometers.

Sides of a parallelogram are equal to [tex]\boldsymbol{6}[/tex] inches and [tex]\boldsymbol{8}[/tex] inches if ratio of sides is [tex]3:4[/tex] and perimeter is equal to [tex]28[/tex] inches.

A basic quadrilateral with two pairs of parallel sides is known as a parallelogram.

Ratio of two sides of a parallelogram [tex]=\boldsymbol{3:4}[/tex]

Let the sides be [tex]3x,4x[/tex].

Perimeter of a parallelogram [tex]=\boldsymbol{28}[/tex] in.

Perimeter of a parallelogram is equal to sum of all the sides.

[tex]3x+4x+3x+4x=28[/tex] in.

                       [tex]14x=28[/tex] in.

                           [tex]x=2[/tex] in.

So, sides are equal to [tex]3(2),4(2)[/tex] that is sides are equal to [tex]\boldsymbol{6}[/tex] inches and [tex]\boldsymbol{8}[/tex] inches.

Find out more information about parallelogram here:

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Given that csc θ = 14 and cot θ < 0, we find that sin θ = 1/14 and cos θ must be negative. We use the identity sin² θ + cos² θ = 1 to solve for the exact value of cos θ, selecting the negative solution. The remaining trigonometric functions are then found using these values.

Given that the cosecant of theta (csc θ) is 14 and cotangent of theta (cot θ) is less than zero, we can find the other trigonometric values. We begin by recalling that cosecant is the reciprocal of the sine function, so sin θ = 1/14. Subsequently, we are told cot θ < 0, which means either the cosine or the sine (or both) must be negative.

Since cot θ is negative and we know sin θ is positive (since csc θ is positive), then we can conclude that cos θ must be negative. However, the exact value of cos θ is not readily identifiable from these properties alone.

To find the trigonometric value of cos θ, we can utilize the identity sin² θ + cos² θ = 1. Substituting our known sin θ value, we solve for cos θ. This gives us two possible solutions for cos θ, either positive or negative. As previously deduced, we select the negative solution for cos θ. The remaining trigonometric functions can then be found given these values:

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The question involves a mathematical reading rate scenario where Naoya calculates how much of a book he has left to read. Since the numbers provided are unrealistic, an example with plausible figures is used to illustrate the process of determining total reading time and daily reading goals.

The question deals with a reading rate calculation problem involving Naoya, who has read part of a book and wants to figure out how much more he needs to read. We can figure out the total number of pages in the book by considering the pages he has read at the rate of 555,555 pages per hour over 444 hours, and adding the remaining 350,350,350 pages he has left to read. However, it appears there are typographical errors in the question with the repetition of numbers, which should likely be simplified to realistic figures before we can calculate the total number of pages in the book.To apply the concept efficiently, let's take an example with realistic numbers similar to the approach mentioned for Marta. Suppose Marta reads at a rate of 48 pages per hour and she needs to finish a 497-page novel. We divide the total page count (497) by her hourly rate (48 pages/hour) to find the total hours needed, which is approximately 10.35 hours or roughly 10 to 11 hours.

If Marta wishes to finish the novel over two weeks, she would divide her total reading time by the number of days she plans to read, ensuring she allocates enough time each day to reach her goal. Similar reading strategies can be applied whether balancing act, early bird, or taking the approach of reading a certain number of pages each day to make a larger task more doable.

Final answer:

πr typically represents an expression involving the mathematical constant π and the radius of a circle (r). In mathematics, this is commonly part of formulas to calculate properties of circles, such as area or circumference.

Explanation:

When you see πr in mathematics, it typically represents the expression involving the mathematical constant π (pi), which is the ratio of a circle's circumference to its diameter, and r, which stands for the radius of the circle.

The actual value of π is approximately 3.14, but it is an irrational number, meaning it extends to infinity without repeating. In many equations and formulas, πr could be part of a larger expression to calculate properties related to circles, such as the area or circumference.

For example, the formula for the circumference of a circle is 2πr, and the formula for the area of a circle is πr2. Without additional context, πr could be part of these or other mathematical expressions related to circles. It is not to be confused with the Greek letter rho, which can also be represented as p and is used in different ways in physics and other sciences.

The simplified form of i^13 is -i.

The simplified form of i^13 is -i.

To find the simplified form, remember that i^4 = 1, so i^13 = i^(4*3+1) = i^(4*3) * i = (i^4)^3 * i = 1^3 * i = i.

Therefore, the simplified form of i^13 is -i.

The probability of rolling exactly three 2s in five rolls of a six-sided die is 32.15%, calculated using the binomial probability formula. The correct answer from the provided options is C) 32.15%.

The student is asking about the probability of achieving a specific number of successes in a series of independent events, which is a problem that can be solved using the binomial probability formula. In this case, a success is defined as rolling a 2 on a six-sided die. The probability of rolling a 2 (success) on a single roll is \(\frac{1}{6}\), and the probability of not rolling a 2 (failure) is \(\frac{5}{6}\).

Therefore, the probability of rolling exactly three 2s in five rolls can be calculated by the formula:

\(P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\)

where \(n\) is the number of trials, \(k\) is the number of desired successes, and \(p\) is the probability of a single success.

Substituting the values:

\(P(X=3) = \binom{5}{3} \cdot \left(\frac{1}{6}\right)^3 \cdot \left(\frac{5}{6}\right)^{5-3}\)

\(P(X=3) = 10 \cdot \left(\frac{1}{6}\right)^3 \cdot \left(\frac{5}{6}\right)^2\)

\(P(X=3) = 10 \cdot \frac{1}{216} \cdot \frac{25}{36}\)

\(P(X=3) = \frac{250}{7776}\)

\(P(X=3) \approx 0.03215 \text{ or } 3.215\%\)

So the correct answer from the provided options is C) 32.15%.

The chances of rolling exactly three successes when rolling a 6-sided die 5 times, where rolling a 2 is considered a success, is approximately 2.143%.

To find the chances of rolling exactly three successes, we need to use the concept of binomial probability. In this case, the probability of rolling a 2 (success) is 1/6, and the probability of not rolling a 2 (failure) is 5/6. We can use the formula for binomial probability: P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success in one trial.

For this problem, n=5 (since the die is rolled 5 times), k=3 (we want exactly three successes), and p=1/6 (probability of rolling a 2). Plugging these values into the formula:

P(X=3) = (5 choose 3) * (1/6)^3 * (5/6)^(5-3)

Simplifying, we get:

P(X=3) = 10 * (1/6)^3 * (5/6)^2 = 10 * (1/216) * (25/36) = 250/11664 ≈ 0.02143 ≈ 2.143%

To find x, we need to solve the quadratic equation 5x^2 - 100 = 0 using the quadratic formula.

To find x, we need to solve the quadratic equation 5x^2 - 100 = 0 using the quadratic formula. The formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.

In this case, a = 5, b = 0, and c = -100. Plugging these values into the quadratic formula, we get:

x = (-0 ± √(0^2 - 4 * 5 * -100)) / (2 * 5)

Simplifying further, we have:

x = (√2000) / 10

Therefore, x equals approximately ±14.14.

Final answer:

Among the options provided, only D.) 97 is a prime number because it only has two divisors: 1 and itself. All other options have more than two divisors and thus are not prime.

Explanation:

To determine which number is prime, we must recall that a prime number is a number that has only two distinct positive divisors: 1 and itself. Now, let's evaluate the options given:

A.) 49 is 7 times 7, so this is not a prime number.

B.) 27 is 3 times 9, hence this is not a prime number either.

C.) 14 is 2 times 7, which means it is not a prime number.

D.) 97 does not have any divisors other than 1 and itself, so it is a prime number.

So, the correct answer is D.) 97, since it fulfills the conditions for being a prime number.

Answer:

8*6*6

Step-by-step explanation:

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The situation can be represented as three inequalities: g + p < 120, g < 45, p > 30, where g is the time spent in the gym and p is the time spent in the pool.

The situation you described can be represented as a system of inequalities. Let's denote gym time as g and pool time as p. Then, the system of inequalities would be the following:

These inequalities represent the constraints on how you can divide your time between the gym and the pool. Any solution to this system would be a pair of numbers (g, p) that satisfy all three inequalities, meaning it's a valid way for you to divide your time.

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The correct system of inequalities is Option 1: x + y < 120 (representing the total time constraint), x < 45 (reflecting the condition of spending less time in the gym), and y > 30 (representing the condition of spending more time in the pool). This corresponds to Option 1 in the given systems of inequalities.

The correct system of inequalities that represents your time allocation between the gym and the pool is Option 1. Let's define x as the time you spend in the gym and y as the time you spend in the pool. According to the given conditions and constraints, the total time, which is the sum of x and y, should be less than 120 minutes (x + y < 120). Furthermore, you want to spend less than 45 minutes in the gym (x < 45) and more than 30 minutes in the pool (y > 30). These three inequalities jointly form a system that accurately represents your situation at the gym and pool.

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The complete question is given below:

You have less than 120 minutes to spend in the gym and in the pool. You want to spend less than 45 minutes in the gym and more than 30 minutes in the pool. Which system represents the situation?

Option 1:

x + y < 120

x < 45

y > 30

Option 2:

x + y = 120

x = 45

y = 30

Option 3:

x + y <= 120

x < 45

y > 30

Option 4:

x + y < 120

x <= 45

y >= 30