An equation of the line passing through (6,−3) having slope −35 is

The probability of selecting a blue marble followed by a purple marble from a jar containing 120 marbles with different colors is approximately 2.2%.

The probability of grabbing a blue marble and then a purple marble is determined by multiplying the probability of grabbing a blue marble first and the probability of grabbing a purple marble second. Here's the calculation:

Probability of selecting a blue marble first: 17/120

Probability of selecting a purple marble second (after one blue marble is already taken out): 18/119

Multiplying the probabilities: (17/120) * (18/119) ≈ 0.022, or 2.2%

According to the discriminant:

The number of real solutions of quadratic equation [tex]y = ax^2 + bx + c[/tex] depends on the discriminant [tex]\Delta = b^2 - 4ac[/tex].

Equation [tex]y = -3x^2 + x + 12[/tex]

Equation [tex]y = 2x^2 - 6x + 5[/tex]

Equation [tex]y = x^2 + 7x - 11[/tex]

Equation [tex]y = -x^2 - 8x - 16[/tex]

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A-Town is conducting a survey using a random sample method by dividing the population into neighborhoods and surveying random households in each, which ensures each household has an equal chance of being selected and provides a representative sample.

A-Town's approach to determining the number of cell phones in each household through a survey is an example of a random sample. The method involves dividing the population into neighborhoods and surveying a random selection of households within each area. This technique helps to ensure that every household has an equal chance of being selected, which should provide a representative sample of the town's population. Such methods are necessary to obtain reliable data that reflects the true distribution and characteristics of the target population without regional bias.

Using a representative sample is crucial in research and polling to measure accurately public opinion or collect data on phenomena. As in the given example where the market research firm surveys 500 randomly selected residents to estimate the percentage of adults with cell phones, the firm employs statistical measures like confidence intervals to infer about the entire population's cell phone ownership based on the sample data. Moreover, the concept of probability sampling is also important for achieving a sample that can be generalized to the whole population with reasonable accuracy.

Case 1: If the letter z is distinctly different

If we have the letter z which is distinctly different than the other "z", then we have five letters in the word F U Z Z Y and the number of arrangements would be = 5*4*3*2*1 =5! = 120 different arrangements

Case 2: If the letter Z is interchangeable

If the letter "Z" is interchangeable with the other "z" then we have 2 letters which cancel out each other that would be represented by 2*1 = 2! = 2

So, the number of arrangements in this case would be 5!/2! = 5*4*3*2*1/2*1 = 60 arrangements

The solution is A = 11/25

The value of the equation A = ( 2x + 1 ) / x² is 11/25

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

A =  ( 2x + 1 ) / x²

when x = 5 , substitute the value of x as 5 in the equation , we get

A = ( 2 ( 5 ) + 1 ) / 5²

The value of A = ( 10 + 1 ) / 25

The value of A = 11/25

Therefore , the value of A = 11/25

Hence , The value of the equation A = ( 2x + 1 ) / x² is 11/25

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The given equation of parabola is

[tex] \frac{1}{32} (y-2)^2 = x-1 [/tex]

Which can also be written as

[tex] x = \frac{1}{32} (y-2)^2 +1 [/tex]

Here vertex (h,k) is (1,2)

And value of a is

[tex] a = \frac{1}{32} [/tex]

Formula of focus is

[tex] (h+ \frac{1}{4a} , k) [/tex]

Substituting the values of h,k and a, we will get

[tex] (1+ \frac{1}{4*(1/32) } , 2} = (1+ 8,2) = (9,2) [/tex]

Therefore the correct option is the last option .

Answer:  The correct option is (D) (9, 2).

Step-by-step explanation:  We are given to find the co-ordinates of the focus for the following parabola:

[tex]\dfrac{1}{32}(y-2)^2=x-1~~~~~~~~~~~~~~~~~~~~(i)[/tex]

We know that the standard form equation of a parabola is

[tex](y-k)^2=4p(x-h),[/tex]

where the co-ordinates of the focus are (h+p, k).

From equation (i), we have

[tex]\dfrac{1}{32}(y-2)^2=x-1\\\\\\\Rightarrow (y-2)^2=32(x-1)\\\\\Rightarrow (y-2)^2=4\times 8(x-1).[/tex]

Comparing the above equation with the standard form equation of the parabola, we get

h = 1, k = 2, and p = 8.

Therefore, the co-ordinates of the focus are

[tex](h+p,k)=(1+8,2)=(9,2).[/tex]  

Thus, option (D) is CORRECT.

Answer:

4 days

Step-by-step explanation:

To express that Jake is younger than Sophie who is 14, we use the inequality j < 14, where j symbolizes Jake's age in years.

The question is asking for an inequality that represents Jake's age compared to Sophie's age. Since we are told that Jake is younger than Sophie and Sophie is 14 years old, we can use an inequality symbol to express this relationship. The appropriate inequality is j < 14, where j represents Jake's age in years. The symbol < means 'less than,' which correctly compares Jake's age to Sophie's, indicating that Jake's age is less than 14 years.

The line of symmetry for the parabola is x = 1.

The line of symmetry for the parabola is x = 1. In the given equation (x-1)^2 = 4(y-1)^2, the x-coordinate of the vertex is 1, which means the line of symmetry is a vertical line passing through x = 1.

Answer:

Step-by-step explanation:

Using a quadratic function, it is found that it will take 5.17 seconds for the javelin to hit the ground.

Considering meters as the measure of height, it is given by:

h(t) = -4.9t² + v(0)t + h(0).

In which:

In this problem, we have that v(0) = 25, h(0) = 1.6, hence:

h(t) = -4.9t² + 25t + 1.6

It hits the ground when h(t) = 0, hence:

-4.9t² + 25t + 1.6 = 0

4.9t² - 25t - 1.6 = 0.

Which is a quadratic function with coefficients a = 4.9, b = -25, c = -1.6. Hence:

[tex]\Delta = (-25)^2 - 4(4.9)(-1.6) = 656.36[/tex]

[tex]x_1 = \frac{25 + \sqrt{656.36}}{9.8} = 5.17[/tex]

[tex]x_2 = \frac{25 - \sqrt{656.36}}{9.8} = -0.06[/tex]

For time we want the positive value, hence, it will take 5.17 seconds for the javelin to hit the ground.

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Answer:

The dolphin is at height 5 feet after 1.40 seconds or 0.228 seconds.

Step-by-step explanation:

Given : The function [tex]h=-16t^2+26t[/tex]models the height h (in feet) of the dolphin after t seconds.

To Find: After how many seconds is the dolphin at a height of 5 feet?

Solution:

Function: [tex]h=-16t^2+26t[/tex]

h is the height

t is the time

Now we are supposed to find After how many seconds is the dolphin at a height of 5 feet

Substitute h = 5

So, [tex]5=-16t^2+26t[/tex]

[tex]16t^2-26t+5=0[/tex]

[tex]16t^2-26t+5=0[/tex] ---1

Using quadratic formula : [tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

General quadratic equation: [tex]ax^2+bx+c=0[/tex]   ---A

On Comparing equation 1 with A

a = 16

b = -26

c=5

Substitute the values in the quadratic formula.

[tex]x = \frac{-(-26)\pm\sqrt{(-26)^2-4(16)(5)}}{2(16)}[/tex]

[tex]x = \frac{26\pm 18.86796}{32}[/tex]

[tex]x = \frac{26+18.86796}{32},\frac{26-18.86796}{32}[/tex]

[tex]x = 1.4021,0.2228[/tex]

Thus the dolphin is at height 5 feet after 1.40 seconds or 0.228 seconds.

The dolphin is at a height of 5 feet at 0.2 second and 1.4 seconds.

Polynomial is an expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Let h represent the height of dolphin at t seconds.

Given the function:

For a height of 5 ft.:

5 = −16t² + 26t

16t² - 26t + 5 = 0

t = 0.2 or t = 1.4

The dolphin is at a height of 5 feet at 0.2 second and 1.4 seconds.

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