A ball is thrown upward from the top of a building. The function below shows the height of the ball in relation to sea level, f(t), in feet, at different times, t, in seconds:f(t) = −16t2 + 48t + 100The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is _____feet per second.

The height, in feet, of the grasshopper above the ground after t seconds is modeled by the function , which is

[tex] h(t) =-t^2+\frac{4}{3}t+ \frac{1}{4} [/tex]

When the grasshopper land on the ground,

[tex] h(t)=0 [/tex]

that is,

[tex] -t^2+\frac{4}{3}t+ \frac{1}{4}=0 [/tex]

Multiplying whole equation by 12 to get rid of denominators 3 and 4

[tex] -12t^2+16t +3=0 [/tex]

[tex] (3-2t)(6t+1)=0
\\
3-2t=0 , 6t+1=0
\\
t= \frac{3}{2} , \frac{-1}{6} [/tex]

And time cant be negative, so the correct option is

[tex] t = \frac{3}{2}=1.5 seconds [/tex]

Correct option is C .

The correct option is (D) 0.7 seconds.To determine when the grasshopper lands on the ground based on its height function, solve the quadratic equation and round to the nearest tenth for the time in seconds.

The height of the grasshopper above the ground is given by the function h(t) = -t² + 4/3t + 1/4. To find when the grasshopper lands on the ground, set h(t) = 0 and solve for t.

By setting -t² + 4/3t + 1/4 = 0 and solving the quadratic equation, you'll find the solutions for t. Round to the nearest tenth to get the time when the grasshopper lands.

The correct answer is 0.7 seconds (D).

Answer:

Opposite sides of a parallelogram are congruent.

Step-by-step explanation:

The correct answer is A. Diagonals of a parallelogram bisect opposite angles.

The theorem states that the diagonal of a parallelogram divides it into two congruent triangles. This means that the two triangles have the same corresponding side lengths and angles. Since the diagonal is a common side to both triangles, the opposite angles must also be congruent.

Option B, "Consecutive sides of a parallelogram are congruent," is not always true. For example, in a rectangle, which is a type of parallelogram, the consecutive sides are not congruent.

Option C, "Consecutive angles of a parallelogram are congruent," is also not always true. For example, in a rectangle, which is a type of parallelogram, the consecutive angles are not congruent.

Option D, "Consecutive angles of a parallelogram are supplementary," is not always true. For example, in a square, which is a type of parallelogram, the consecutive angles are not supplementary.

Therefore, the only statement that follows from the given theorem is A. Diagonals of a parallelogram bisect opposite angles.

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When a is greater than 0 then the parabola opens upwards and if the value of 'a' is less than 0 then the parabola opens downwards and from the given equation a < 0 that is why the graph of the quadratic equation is a parabola that opens downward.

Given :

Quadratic Function  ---  [tex]y = -x^2-10x + 1[/tex]

The graph of the quadratic equation is in the shape of the parabola. The generalized quadratic equation is given by:

[tex]ax^2 + bx +c = 0[/tex]

Now, compare the given equation ([tex]y = -x^2-10x + 1[/tex]) with the generalized quadratic equation which is given above.

[tex]a = -2[/tex]

b = -10

c = 1

According to the rules of the graph, when a is greater than 0 then the parabola opens upwards and if the value of 'a' is less than 0 then the parabola opens downwards.

Therefore, the correct option is c).

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The statement that holds true with a 95% confidence level is  true population mean of the number of automobiles in households of students is between 2.6524 and 2.7476."

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

To determine which of the following statements holds true, we need to use the confidence interval formula:

Confidence Interval = sample mean ± (critical value x standard error)

where the critical value is determined by the confidence level and the standard error is calculated using the formula:

standard error = standard deviation / sqrt(sample size)

With a 95% confidence level, the critical value is 1.96. Plugging in the values given in the question, we get:

standard error = 0.7 / sqrt(800) = 0.0247

Confidence Interval = 2.7 ± (1.96 x 0.0247) = (2.6524, 2.7476)

Therefore, the statement that holds true with a 95% confidence level is:

"The true population mean of the number of automobiles in households of students is between 2.6524 and 2.7476."

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To find the length of a side of the pen, we can use the Pythagorean Theorem. The area of the pen can be calculated by multiplying the length of a side by itself.

To find the length of a side of the pen, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, the diagonal of the pen acts as the hypotenuse.

Let's let s represent the length of a side of the pen. Using the Pythagorean Theorem, we have:

s^2 + s^2 = 84^2

Simplifying the equation, we get:

2s^2 = 7056

Dividing both sides of the equation by 2, we get:

s^2 = 3528

Taking the square root of both sides, we get:

s ≈ 59.46 meters

To find the area of the pen, we can multiply the length of a side by itself. So, the area is:

s * s = 59.46 * 59.46 = 3536.8516 square meters

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Answer:

C) 288π units³

Step-by-step explanation:

When the height of an oblique cylinder is given, the formula for volume is

V = π×r²×h

Our radius is 6 and our height is 8:

V = π(6²)(8) = π(36)(8) = 288π

The volume of the cylinder is [tex]\boxed{288\pi {\text{ unit}}{{\text{s}}^3}}[/tex].Option (C) is correct.

Further explanation:

Given:

Theheight of the cylinder is [tex]{\text{8 units}}[/tex] and the radius of the cylinder is [tex]{\text{6 units}}.[/tex]

The options of the volume are given as follows.

A. [tex]{\text{216}}\pi {\text{uni}}{{\text{t}}^3}[/tex]

B. [tex]324\pi {\text{uni}}{{\text{t}}^3}[/tex]

C. [tex]288\pi {\text{uni}}{{\text{t}}^3}[/tex]

D. [tex]432\pi {\text{uni}}{{\text{t}}^3}[/tex]

Explanation:

The formula for the volume of the cylinder can be expressed as,

[tex]\boxed{{\text{Volume}} = \pi {r^2}h}[/tex]

‘r” is the radius of the cylinder and “h” is the height of the cylinder.

The volume of the right cylinder can be obtained as follows,

[tex]\begin{aligned}{\text{Volume}} &= \pi {r^2}h\\&= \pi\times {\left( 6 \right)^2} \times 8\\&= \pi\times 36 \times 8\\&=288\pi \\\end{aligned}[/tex]

The volume of the cylinder is [tex]36\pi {\text{ unit}}{{\text{s}}^3}.[/tex]

The volume of the cylinder is [tex]\boxed{288\pi {\text{ unit}}{{\text{s}}^3}}[/tex].Option (C) is correct.

Learn more:

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Mensuration

Keywords: Cylinder, surface area of the cylinder, volume of the cylinder, right cylinder, height, radius depth.

The probability of advancing to the seventh round by randomly guessing on a game show with three answer choices per question is 1/729.

The probability of advancing to the seventh round if the contestant randomly guesses the answer to each multiple-choice question with three answer choices is calculated by raising the probability of guessing one question correctly to the power of six, since the contestant needs to guess correctly six times in a row to reach the seventh round.

The probability of guessing one question correctly is 1/3, as there are three possible choices and only one correct answer. Thus, the probability of guessing six questions correctly in a row is (1/3)⁶.

Simplifying this, we get:

(1/3)⁶ = 1/729.

Therefore, the probability is 1/729.

Commission is typically a percentage of sales paid to an agent or employee, and it can be calculated by multiplying the percentage rate by the sale amount. Real estate agencies and stock brokers have different structures for commission, which can significantly affect agents' earnings and motivation. Properly designed sales commission structures balance salary and commissions for optimum sales and customer satisfaction.

Commission is a fee paid to an employee or agent for conducting a transaction or service, typically a percentage of the sale made. Real estate agencies often pay agents a combination of salary and commission, with the intent to motivate them to achieve higher sales and provide satisfactory service to clients. For instance, agencies like RE/MAX pay almost 100% commission, attracting agents who are confident in their sales abilities and prefer a high-reward structure, though this may come at a cost to agency-wide cooperation.

The calculation of commission generally involves multiplying the sale amount by the commission percentage agreed upon. If a real estate agent sells a house for $200,000 and has a commission rate of 5%, their commission would be $200,000 * 0.05 = $10,000. While most agencies operate on average commission rates of about 50%, RE/MAX's near-100% commission structure can be very lucrative for highly skilled agents.

In the stock market, since May 1, 1975, when the SEC abolished fixed commissions, commission rates have been competitive and are typically lower for 'round lots' (buying or selling 100 shares) compared to 'odd lots'. It is also a common practice for firms to add a flat brokerage charge to the transaction.

To design an effective compensation plan incorporating commission along with a fixed salary, companies must carefully balance the two to ensure that they achieve an optimum balance between sales volume, profitability, and customer satisfaction. This balance is critical to attract and retain talented agents and salespeople who are motivated to perform well.

It includes a pledge of collateral.

i believe the answer is d

Answer:

Option A

Step-by-step explanation:

To find nth term we use the given options.

[tex]a_n= 8(\frac{1}{2})^{n-1}[/tex]

Lets pick any point from the graph and plug it in the given option

LEts pick (1,8) and (2,4)

[tex]a_n= 8(\frac{1}{2})^{n-1}[/tex], n=1, an=8

[tex]8= 8(\frac{1}{2})^{1-1}[/tex]

[tex]8= 8(\frac{1}{2})^{0}[/tex]

8=8 True

LEts check with (2,4), n=2 and an=4

[tex]4= 8(\frac{1}{2})^{2-1}[/tex]

4=4  True

Average rate of change from n=1 to n=3= [tex]\frac{a_3-a_1}{3-1} =\frac{2-8}{2} =-3[/tex]

Average rate of change from n=1 to n=3 is -3

Answer:

7, 24, 25

Step-by-step explanation:

7^2=49

24^2 = 576

25^2 = 625

576 + 49 = 625.

therefore, its a right angle

Answer with Step-by-step explanation:

6x – 2y = 10

2x + 3y = 51

Solving the first equation above for y:

6x-2y=10

⇒ 2y=6x-10

dividing both sides by 2

y=3x-5

Putting value of y in second equation

2x + 3(3x-5) = 51

2x+9x-15=51

11x=51+15

11x=66

⇒ x=6

y=3×6-5

=18-5

= 13

Hence, solution is: x=6 and y=13

and  Solving the first equation above for y gives: y=3x-5

Answer:
D) 4x - 17
E) -b - 6a
F) -2(7x + 34)

Explanation:
For question D, we divide each term in the first binomial by 3x and simplify:
(12x² / 3x) - (51xy / 3x)
12 / 3 = 4, x² / x = x         51 / 3 = 17, x / x = 1, y / 1 = y
4x - 17(1)y
4x - 17y is left as a quotient

For question E, we do distribute -3a to the binomial in the numerator and do the same, divide each term in the binomial by -6a and simplify:

-3a(-2b - 12a) = -3a(-2b) - 3a(-12a) = 6ab + 36a²

(6ab + 36a²) / -6a =
(6ab / -6a) + (36a² / -6a)
6 / -6 = -1, a / a = 1, b / 1 = b         36 / -6 = -6, a² / a = a
-1(1)(b) + (-6)(a)
-b - 6a is left as a quotient

For question F, we distribute the 4 and 3x to their respective binomials and then combine like terms where applicable:
4(x² - 8x) + 3x(-6x - 12)
4(x²) + 4(-8x) + 3x(-6x) + 3x(-12)
4x² - 32x - 18x² - 36x
4x² - 18x² - 32x - 36x
-14x² - 68x, both terms have a greatest common factor (GCF) of -2 which can be divide out of them to further simplify:
-2(7x + 34)