Properties of Definite Integrals A definite integral is a function that assigns a value to the area under a curve on a given interval.

The coordinate of the parabola's focus in the following equation, x2=6y, is (0,3/2)

What are Directrix and Focus?

What are a parabola's focus and directrix? Typically speaking, parabolas are the graphs of quadratic functions. They can alternatively be thought of as the collection of all points whose separation from the focus is the same as their separation from a certain line (the directrix).

What are the focus's coordinates?

The focus's coordinates f(0,a)

The equation that we have given is

x^2 = 6y

We are aware that the parabola's general equation is,

x^2 = 4ay

The axis of symmetry is a line that splits the parabola in half and runs through the focus.

We must ascertain the worth of

6y = 4ay

6 = 4a

a = 6/4 = 3/2

The focus' coordinate (0,3/2) is as a result.

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The acceleration of gravity of the satellite, as compared to the acceleration of gravity on the surface of Mars is (D) 1/16.

The term Acceleration defined as the gravity at the surface of Mars is given by the formula

Gₙ = GM/R²

where

M refers mass of Mars

R refers radius of Mars

Here we need to find if we take a point which is at height 3 times the radius of Mars from its surface.

While we take the height 3 times the radius of Mars from its surface.

The it can be written as,

=> Gₙ = GM/(3R + R)²

When we expand it then we get.

=> Gₙ = GM/16R²

Therefore, the resulting gravity on the surface of the mars is 1/16.

So, the option (D) is correct.

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Rounding to seven decimal places, the probability that one individual would be selected by all witnesses is 0.0000128.

Six different witnesses chose the same individual from a lineup in a homicide case. There were 5 guys in the lineup. The likelihood that each of the six witnesses would choose the same person if the identifications were done based just on guesswork.

P = 1 ÷ 5 is the likelihood that a witness will choose a certain individual at random.

The likelihood that all seven witnesses would select the same subject is P = [tex](\frac{1}{5})^{7}[/tex]= 0.0000128.

Simply put, the probability is the likelihood that something will occur. We can discuss the probabilities of different outcomes to determine how likely they are if we are unclear about how an event will turn out.

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The Local maxima and minima will be at X = 0 , 4 , -5 and point of inflation will be at [tex]\frac{(-1 + \sqrt{61})}{3},[/tex] [tex]\frac{(-1 - \sqrt{61})}{3}[/tex].

The curve of a function has peaks and troughs called maxima and minima. A function may have any number of peaks and minima. Calculus allows us to determine any function's maximum and lowest values without ever consulting the function's graph. Maxima will be the curve's highest point within the specified range, while minima will be its lowest.

Extrema is the result of maxima and minima combined. The graph in the graphic below shows several peaks and falls. We obtain the function's maximum and lowest values at x = a and 0 respectively, and at x = b and c respectively. The valleys are the minima and all the peaks are the maximum.

The peaks and minima of the function that appear in a certain interval are known as local maxima and minima. The value of a function at a place in a certain interval for which the values of the function near that point are always smaller than the value of the function at that point would be considered a local maximum. Local minima, on the other hand, would be the value of the function at a location where the values of the function nearby are higher than the value of the function at that location.

When the concavity of a function's graph (or picture) changes, a point of inflection is identified. We need to determine where the second derivative of the function switches from negative to positive, or vice versa, in order to determine this in algebra. So, we calculate the provided function's second derivative.

The fuction is :

[tex]3x^{4} + 4x^{3} -120x^{2} + 120[/tex]

So for local Maxima and minima first derivatives should be zero

Therefore,

[tex]\frac{d}{dx}\\[/tex] ([tex]3x^{4} + 4x^{3} -120x^{2} + 120[/tex] ) = 0

⇒ [tex]12x^{3} + 12x^{2} - 240x = 0[/tex]

⇒ [tex]x^{3} + x^{2} - 20x = 0[/tex]

⇒[tex]x(x^{2} + x - 20) = 0 \\[/tex]

⇒x(x-4)(x+5) = 0

For x = 0, 4 , -5 there is a local maxima or minima.

Now for Point of Inflation second derivative is zero

Therefore,

[tex]\frac{d^{2} }{dx^{2} }[/tex] ( [tex]3x^{4} + 4x^{3} -120x^{2} + 120[/tex]) = 0

⇒  [tex]\frac{d}{dx}[/tex][tex]12x^{3} + 12x^{2} - 240x = 0[/tex]

⇒ [tex]36x^{2} + 24x - 240 = 0[/tex]

⇒[tex]3x^{2} + 2x - 20 = 0[/tex]

⇒ x = [tex]\frac{(-1 + \sqrt{61})}{3},[/tex] [tex]\frac{(-1 - \sqrt{61})}{3}[/tex]

So the point of inflations are [tex]\frac{(-1 + \sqrt{61})}{3},[/tex] [tex]\frac{(-1 - \sqrt{61})}{3}[/tex].

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Since the probabilities lie exclusively between 0 and 1 and the sum of probabilities is equal to 1 , therefore the yes , the distribution is discrete probability distribution.

As the probabilities range from 0 to 1, and the sum of probability equals 1.

Discrete probability distribution is a sort of probability distribution that provides all potential values of a discrete random variable along with the related probabilities. In other words, a discrete probability distribution describes the possibility of each potential value of a discrete random variable occurring.

Some common discrete probability distributions are geometric distributions, binomial distributions, and Bernoulli distributions. This page defines a discrete probability distribution, as well as its formulas, types, and numerous instances.

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For sketching the graph we need to know the coordinate of the vector which are r(1) = ⟨−1, 2⟩, r′(1) = ⟨1, 2⟩

Given:-

x = t − 2,

y = t² + 1 ⇒ t = x + 2

⇒ y = (x + 2)2 + 1

⇒ y = x² + 4x + 5

For find r′(t) we have to ,

r′(t) = ⟨1, 2t⟩

To sketch the position vector r(t) and the tangent vector r′(t) for t = 1.

r(1) = ⟨−1, 2⟩, r′(1) = ⟨1, 2⟩

See the above figure. The red vector is r(1) and the blue one is r′(1)

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Answer: The degree of a polynomial is the highest exponent of the variable in the polynomial. The leading coefficient of a polynomial is the coefficient of the term with the highest degree.

In the expression "n + 8", the highest degree of the variable is 1 and the coefficient of the term with the highest degree is 1. Therefore, the degree of the polynomial is 1 and the leading coefficient is 1.

So, the degree of the polynomial is 1 and the leading coefficient is 1.

Note: If the expression had included terms with higher degrees of the variable, such as "n^2 + 3n + 8", the degree of the polynomial would have been 2 and the leading coefficient would have been 1.

The slope of the given line is 13 and the y - intercept of the line is 9.

What is equation of line?

A straight line's general equation is y = mx + c, where m denotes the gradient and y = c denotes the point at which the line crosses the y-axis. On the y-axis, this value c is referred to as the intercept.

Consider, the given equation of line,

C(m) = 13m + 9

We have to find the slope and y - intercept of the given equation.

From the given equation the slope of the equation is 13 and the y - intercept is 9.

Consider, the general equation of line

y = mx + c

where,

m is the slope of line and

c is the y - intercept of the line.

Compare the given equation C(m) = 13m + 9 with, y = mx + c

⇒  m = 13 and c = 9

Hence, the slope of the given line is 13 and the y - intercept of the line is 9.

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The probability of those that favors building the health center is 0.196

The probability is the measure of the likelihood of an event to happen. It measures the certainty of the event. The formula for probability is given by; P(E) = Number of Favorable Outcomes/Number of total outcomes.

In a recent survey, 70% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 8 of them favor the building of the health center.

According to the scenario, 70% of the community favoured in building a health centre.

Hence , p=0.7.

In order to find the probability, binomial theorem will be used as follows :

= 0.196

Hence, The probability is 0.196

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By using the least-squares regression line the equation will be y = 1.518x + 0.305.

Let us find the value of slope and y-intercept that suits the data y = mx + b

And then , we have to find the value of x² ,∑x,∑y,∑x²,∑xy also N

Then, we have to calculate the slope m

m = n∑(xy) - ∑x ∑y / n ∑(x²) - (∑x)²

= 5 X 236 -26 X 41 / 40 - 676

= 249 / 164

= 1.5183

Then , we hace to calculate the intercept b

b = ∑y - m∑x / n

= 41 - 1.5183 X 26 / 5

= 0.3049

After this we have to assemble it in the equation of line

y =mx + b

Therefore ,

y = 1.518x + 0.305

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Answer:

$28.89

Step-by-step explanation:

he subscription is on sale for 25% off, so the cost is $36 - (25% * $36) = $36 - $9 = $27.

Then, a 7% tax is added at checkout, so the total cost is $27 + (7% * $27) = $27 + $1.89 = $28.89.

Therefore, the total cost is $28.89.

Answer:

Step-by-step explanation:

36 will be 27 and  7% of 27 should be added 1,89 so 28,89

The quadratic equation y = (2 / 5) · x² + (2 / 5) · x - 8 contains the points (- 5, 0), (0, - 8) and (4, 0).

In this problem we find the case of a parabola, parabolae are defined by quadratic equations of the form:

y = a · x² + b · x + c

Where:

First, write the points associated with the parabola:

(x₁, y₁) = (- 5, 0), (x₂, y₂) = (0, - 8), (x₃, y₃) = (4, 0)

Second, create a system of linear equations:

25 · a - 5 · b + c = 0

c = - 8

16 · a + 4 · b + c = 0

Third, resolve the system by numerical methods:

(a, b, c) = (2 / 5, 2 / 5, - 8)

Fourth, write the resulting quadratic equation:

y = (2 / 5) · x² + (2 / 5) · x - 8

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Answer:

1,2,-5,-1,3

Step-by-step explanation:

the coefficients are the numbers before the variables (x and y)

if there is nothing infront of the variable then the coefficient is 1

if there is only a negative sign infront of the variable then the coefficient is -1

Answer:

y = 5

Step-by-step explanation:

the line is horizontal and has always y = 5

Answer:

86.3

Step-by-step explanation:

86.345

step 1

identify their place values

8 — Tens

6 — Units

.3 — tenth( 1 decimal place)

.34 — Hundredth (2 decimal place)

.345 — Thousandth ( 3 decimal place)

to one decimal place is

= 86.3

remember 4 is a round down number so .345 can't actually affect the 3

i hope this answered helped.

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Answer:

148 degrees

Step-by-step explanation:

b/c a line is 180 degrees and the angle given is 32 so 180- 32= 148

and the line is bisected at point p so it is 148 because of alternate interior angles.

Answer: an=-2(an-1), a1=3.5

Step-by-step explanation: took the quiz and got it right

An = -3.5(-2)ⁿ⁻¹ and -3.5  are the Nth and first terms of the above sequence, respectively.

Every term in a geometric series is obtained by multiplying the term before it by the same number. Aₙ = a₁ rⁿ⁻¹ is the general phrase for it. The common ratio is denoted by the number r.

Given a₂ = 7 and a₅ = −56 for a geometric sequence.

From the general formula of geometric sequence's nth term

Aₙ = a₁ rⁿ⁻¹

Thus

A₂ = 7 = a₁ r²⁻¹ = a₁ r

A₅ = -56  =  a₁ r⁵⁻¹ = a₁ r⁴

comparing both equations

A₂/ A₅ = a₁ r /  a₁ r⁴

=>  -7/56 = 1/r³

=> r³ = -8

=> r = -2

putting this value of r in equation 1

a₁ = -7/2

Hence, for the nth term of the sequence

Aₙ = -7/2(-2)ⁿ⁻¹

Therefore, the Nth term and the first term of the given sequence are Aₙ = -3.5(-2)ⁿ⁻¹ and -3.5 respectively.

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Answer:

There are 3 blue fish in the aquarium.

Step-by-step explanation:

Let the total number of fish be denoted by T, total number of blue fish be B and total number of red fish be R

We can write, T = B + R

Since the aquarium has one fewer redfish than blue fish, R = B - 1

So,

T = B + (B - 1) = 2B - 1

As 60% of total fish is blue, we can write

60% of T = B

or, (60/100) x T = B

or, 3/5 x T = B

or, 3/5 x (2B - 1) = B

or, 6B - 3 = 5B

or, 6B - 5B = 3

or B = 3

Therefore, there are 3 blue fish in the aquarium.

No. of red fish, R = B - 1 = 3 - 1 = 2

Answer:

x = 3+y

Step-by-step explanation:

is your ans .................

The mean of the given data is 12.4

The mean can be calculated only for numeric variables, no matter if they are discrete or continuous. It's obtained by simply dividing the sum of all values in a data set by the number of values.

Given that, a data, 10, 12, 11, 13, 14, 12, 13, 14, 13, and 12

Arranging the data in ascending order, 10, 11, 12, 12, 12, 13, 13, 13 14, 14

The sum is = 10+12+11+13+14+12+13+14+13+12 = 124

Total number of data = 10

Mean = sum of the numbers/ total numbers of data

Mean = 124/10

Mean = 12.4

Hence, the required mean is 12.4

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The relationship between the temperature and the coffee sales is if the temperature decreases then the number of coffee sales decreases.

So, option a is correct.

The SI unit of temperature according to the International System of Units is Kelvin, indicated by the sign K. The Kelvin scale is widely recognized or used in research and engineering. However, the global standard for measuring temperature is the Celsius or Fahrenheit scale. The unit of measurement known as temperature can be expressed on a number of scales, including the Fahrenheit and Celsius systems. Temperature depicts the direction in which heat energy spontaneously moves, i.e., from a hotter (body with a higher temperature) to a colder (body with a lower temperature) (one at a lower temperature).

The relationship between the temperature and the coffee sales is if the temperature decreases then the number of coffee sales decreases.

Hence, option a is correct.

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Answer:

Do u like m&m because I do

Answer: 6,250

Step-by-step explanation: As the equation states there are 25 boxes with 250 packs of ketchup take the 25 boxes and multiply the 250 packets

250x25=6,250

Answer:

Step-by-step explanation:

To find f(4), you can substitute 4 for x in the equation f(x) = 3x + 4. This gives you:

f(4) = 3(4) + 4

= 12 + 4

= 16

Therefore, f(4) = 16.

Mathematically, the balance on Howard Gessner's American Express card on September 1 is $360.63.

The ending balance on September 1 is determined using the mathematical operations of addition, subtraction, and multiplication (for the finance charge).

The finance charge is the interest and other fees paid for borrowing credit.

The finance charge is always based on a percentage, called the annual or monthly percentage rate (APR or MPR).

In this situation, the monthly percentage rate is applied to the unpaid balance (multiplication) at the end of August, to determine the finance charge, which is added to the unpaid balance to compute the September 1 balance.

August 1 Balance =        $543.27

Purchases                        201.25

Returns                            (89.22)

Payments                      (300.00)

Unpaid balance           $355.30

Finance charge                  5.33 ($355.30 x 1.5%)

September 1 Balance $360.63

Thus, using mathematical operations, the balance on September 1 is $360.63.

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Answer: x=22

Step-by-step explanation:

Based on the given figure, we can tell that 2x and 44 are vertical angles. Vertical angles are congruent to each other, so we can set them equal to each other.

2x=44           [divide both sides by 2]

x=22

Therefore, x=22.

Answer:

f(x) = 5/2x

Step-by-step explanation:

The parent function for the given equation is f(x) = 1/2x. When this function is vertically stretched by a factor of 5, the resulting function will be f(x) = 5/2x. This means that for any value of x, the output of the transformed function will be 5 times greater than the output of the parent function.

For example, if x = 2, then f(x) = 1/2 * 2 = 1 for the parent function, and f(x) = 5/2 * 2 = 5 for the transformed function. Similarly, if x = 3, then f(x) = 1/2 * 3 = 1.5 for the parent function, and f(x) = 5/2 * 3 = 7.5 for the transformed function.

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Answer:

No, Franco's calculation is incorrect.

Step-by-step explanation:

No, Franco's calculation is incorrect. The correct answer is 9.37 x 0.1 = 0.937.

When multiplying a decimal number by a decimal number, the decimal point in the product should be placed so that the number of decimal places in the product is equal to the sum of the number of decimal places in the factors. In this case, the number 9.37 has two decimal places, and the number 0.1 has one decimal place. Therefore, the product of 9.37 x 0.1 should have a total of 2 + 1 = 3 decimal places.

To obtain the correct answer, we need to multiply 9.37 x 0.1 using standard long multiplication:

Copy code

9.37

x 0.1

0.937

Therefore, the correct answer is 0.937, not 93.7 as Franco stated.

Answer:

No. This is incorrect.

Step-by-step explanation:

9.37 × 1 = 9.37

Times by 1 the number stays the same.

If you multiply by 0.1 the decimal will move one place and the 9.37 should get smaller! The sentence in the problem shows 93.7 which is bigger.

9.37 × 10 = 93.7

9.37 × 0.1 = 0.937