find f. f '(t) = 16 1 + t2 , f(1) = 0

The situation for which the explicit formula of the arithmetic sequence would be used instead of the recursive formula is given as follows:

Find the 106th term when the first term is of 2 and the common difference is of 5.

An arithmetic sequence is a sequence of numbers in which the difference between two consecutive terms is always constant, called common difference d.

The recursive formula is given as follows:

f(n + 1) = f(n) + d.

The first term f(1) is given.

Hence, the recursive sequence should be used to calculate a small number of terms, such as at most 15 terms.

For a large number of terms, the explicit formula should be used, which is given as follows:

[tex]a_n = a_1 + (n - 1)d[/tex]

In which [tex]a_1[/tex] is the first term.

Hence, in this problem, the 106th term should be calculated with the explicit formula, meaning that the first option is the correct option.

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The equation with value V indicating the decrease in value of iPhone 13 with years t is V = [tex] {1000 (1 - 0.02)}^{t} [/tex].

The equation to be used for the calculation of decreased value is -

A = [tex] {P (1 - r)}^{t} [/tex]. In this formula, A is amount, P is principal, r is rate and t represents time. Based on the information in question, A will be replaced with V. So, the required formula will be -

V = [tex] {P (1 - r)}^{t} [/tex]

Now keeping the values in formula for numerical equation -

V = [tex] {1000 (1 - 0.02)}^{t} [/tex]

Since two values are unknown here, the equation can not be solved further.

Therefore, the required equation is V = [tex] {1000 (1 - 0.02)}^{t} [/tex].

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The 4th term of an AP is 6 if the um of the 8th and 9th term is -72, The 4th term of an AP is -18.

The sum of 8th and 9th term of an AP is given as -72.

We can use the formula for the sum of any n terms of an AP,

Sn = n/2[2a + (n-1)d]

Where,

a = first term

d = common difference

n = number of terms

Therefore,

-72 = 8/2[2a + (8-1)d]

-72 = 4[2a + 7d]

-18 = 2a + 7d

Subtracting 7d from both the sides,

-18 - 7d = 2a

Dividing both the sides by 2,

(-18 - 7d)/2 = a

a = -18/2 - (7d)/2

a = -9 - (7d)/2

For the 4th term of an AP,

a4 = a + (4-1)d

=-9 - (7d)/2 + 3d

=-9 + 6d

=-9

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The solution to the maximization problem P(x, y) = 7x + 9y - 5 is 33.4

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

Objective function:  P(x, y) = 7x + 9y - 5

Subject to: x ≥ 0

y ≥ 0

2x + 3y ≤ 12

3x + 2y ≤ 12

x + y ≥ 2

Next, we plot the graph of the constraints

The vertices of the feasible region are

(0, 2), (2, 0) and (2.4, 2.4)

Substitute (0, 2), (2, 0) and (2.4, 2.4) in P(x, y) = 7x + 9y - 5

So, we have

P(0, 2) = 7(0) + 9(2) - 5 = 13

P(2, 0) = 7(2) + 9(0) - 5 = 9

P(2.4, 2.4) = 7(2.4) + 9(2.4) - 5 = 33.4

Hence, the maximum value is 33.4

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Answer:

y = (17/4)x + (3/4)

Step-by-step explanation:

Formula we use,

→ y = mx + b

Now the slope-intercept form is,

→ y = mx + b

→ 4y - 5x = 12x + 3

→ 4y = 12x + 5x + 3

→ y = (17x + 3)/4

→ y = (17/4)x + (3/4)

Thus, answer is y = (17/4)x + (3/4).

Answer:

y = [tex]\frac{17}{4}[/tex] x + [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

4y - 5x = 12x + 3 ( add 5x to both sides )

4y = 17x + 3 ( divide through by 4 )

y = [tex]\frac{17}{4}[/tex] x + [tex]\frac{3}{4}[/tex] ← in slope- intercept form

Answer:

- When x = 0, f(x) = -5

- Each time x increases by 1, f(x) increase by 3

- f(x) = 3x - 5

Step-by-step explanation:

m = slope of the graph

m = [tex]\frac{y_{2}-y_{1}}{x_{2}-x{1}}[/tex]

m  = [tex]\frac{4- (-5)}{3-0}[/tex]

m = [tex]\frac{9}{3}[/tex]

m = 3

[tex]y - y_{1} = m ( x - x_{1})[/tex]

[tex]y - (-5) = 3 ( x - 0 )[/tex]

[tex]y = 3x - 5[/tex]

The correct option e) 94.29%, for the probability that among the next 166 responses there will be at most 43 correct answers.

If we use a normal distribution to simulate any binomial distribution with p = 0.24 for n = 166 results, then:

There should be 43 correct responses, or

μ = np = 166*0.24

μ = np = 34.86

The number of accurate responses has a standard deviation of

σ = √(np(1-p))

σ = √(34.86*0.76)

σ = 5.14

43 correct responses have a z-value;

z = (x - μ)/σ

z = (43 - 34.86)/5.14

z = 1.58

The likelihood that 43 accurate responses are provided is:

N(1.58) = 0.9429

Thus, the probability that among the next 166 responses there will be at most 43 correct answers is 94.29%.

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The complete question is-

Use the normal distribution to approximate the desired probability. A certain question on a test is answered correctly by 24.0 percent of the respondents. Estimate the probability that among the next 166 responses there will be at most 43 correct answers.

a) 74.56870% b) 74.30203% c) 75.20203% d) 94.95%  e) 94.29%

Answer:

We all must have heard the word “exponential” before. What does this word mean? Exponential means to become more and more rapid in growth. However, in mathematics, it represents a mathematical expression that has one or more exponents. Hence, we know it as an exponential form.

We can say that a thing may increase with an exponential rate when the increase becomes quicker and quicker as the thing which is talked about becomes larger.

Now let us learn more about the exponential form of numbers which is greatly used and applied in mathematics.

The iterated integral by converting to polar coordinates [tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] is 7a^4/12.

In the given question, we have to evaluate the iterated integral by converting to polar coordinates.

The given integral is

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex]

Let polar coordinate.

x = rcosθ, y = rsinθ, x^2 + y^2 = r^2

So dxdy = rdrdθ

r from 0 to a and θ from π/2 to π

Now the integral is

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\int_{\pi/2}^{\pi}\int_{0}^{a}7(r\cos\theta)^2(r\sin\theta)drd\theta[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\int_{\pi/2}^{\pi}\int_{0}^{a}7(r^2\cos^2\theta)(r\sin\theta)drd\theta[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\int_{0}^{a}r^3\cos^2\theta\sin\theta drd\theta[/tex]

Now integrating the first integral

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\left[\frac{r^4}{4}\right]_{0}^{a}\cos^2\theta\sin\theta d\theta[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]7\int_{\pi/2}^{\pi}\left[\frac{a^4}{4}-\frac{0^4}{4}\right]\cos^2\theta\sin\theta d\theta[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]\frac{7a^4}{4}\int_{\pi/2}^{\pi}\cos^2\theta\sin\theta d\theta[/tex]

Let cosθ = t

So -sinθ = dt/dθ

So sinθdθ = -dt

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\int_{\pi/2}^{\pi}t^2dt[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\left[\frac{t^3}{3}\right]_{\pi/2}^{\pi}[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{4}\left[\frac{\cos^3\theta}{3}\right]_{\pi/2}^{\pi}[/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = [tex]-\frac{7a^4}{12}\left[\cos^3\pi-\cos^3(\pi/2)\right][/tex]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = -7a^4/12[(-1)^3 - 0]

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex] = 7a^4/12

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The complete question is:

Evaluate the iterated integral by converting to polar coordinates.

[tex]\int_{0}^{a}\int_{-\sqrt{a^2-y^2}}^{0}7x^2ydxdy[/tex]

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ D(\stackrel{x_1}{10}~,~\stackrel{y_1}{7})\qquad F(\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ x +10}{2}~~~ ,~~~ \cfrac{ y +7}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE E}}{(1~~,~~-1)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ x +10}{2}=1\implies x+10=2\implies \boxed{x=-8} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ y +7}{2}=-1\implies y+7=-2\implies \boxed{y=-9}[/tex]

Answer: 1402

Step-by-step explanation:

Im assuming u mean [tex]6k^{4} +2k^{3} -6[/tex]

sub in the k value when k = -4

[tex]6(-4)^{4} +2(-4)^{3} -6[/tex]

1536−128−6=1402

Answer: 32%

Step-by-step explanation:

To find the relative frequency, you divide the specific term by the sum of all terms.

start by adding all of the colors together, 5 + 6 + 6 + 8 = 25

then divide the blue number by the sum, 8/25 = 0.32

than multiply that by 100 to get its percentage, 0.32(100) = 32%

Answer: no

Step-by-step explanation:

use pythagorean theorem.

you can use a^2+b^2=c^2

the biggest value is always c(the hypotenuse) -> c=20

a and b would be 15 and 17(doesn't matter which is which)

plug it into the equation -> 15^2+17^2=20^2

225+289=400

514≠400

so no, it's not a right triangle

The range of the given function f(x) = √x + 7 is y ≥ 7. So, option B is correct.

The given function is f(x) = √x + 7

Since it is a square root function, its domain is D = {x ≥ 0}

Then, substitute x = 0 in the function, and we get

f(0) = √0 + 7 = 7

⇒ y =7

f(1) = √1 + 7 = 8

⇒ y =8

So, for x ≥ 0, the output values form a set {y ≥ 7}

Therefore, the range of the given function is y ≥ 7.

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Answer: hope it helps

Step-by-step explanation:

STEP

1

:

Equation at the end of step 1

 ((2•5x2) -  2x) -  3  = 0

STEP

2

:

Trying to factor by splitting the middle term

2.1     Factoring  10x2-2x-3

The first term is,  10x2  its coefficient is  10 .

The middle term is,  -2x  its coefficient is  -2 .

The last term, "the constant", is  -3

Step-1 : Multiply the coefficient of the first term by the constant   10 • -3 = -30

Step-2 : Find two factors of  -30  whose sum equals the coefficient of the middle term, which is   -2 .

     -30    +    1    =    -29

     -15    +    2    =    -13

     -10    +    3    =    -7

     -6    +    5    =    -1

     -5    +    6    =    1

     -3    +    10    =    7

     -2    +    15    =    13

     -1    +    30    =    29

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step

2

:

 10x2 - 2x - 3  = 0

STEP

3

:

Parabola, Finding the Vertex:

3.1      Find the Vertex of   y = 10x2-2x-3

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 10 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   0.1000  

Plugging into the parabola formula   0.1000  for  x  we can calculate the  y -coordinate :

 y = 10.0 * 0.10 * 0.10 - 2.0 * 0.10 - 3.0

or   y = -3.100

Answer:

a=82°

b=32°

Step-by-step explanation:

x is the reflection of b

y is the reflection of a

Answer:

Step-by-step explanation:

I graphed these equations on desmos, the solution is the point where they intercept. The point where they intercept or the solution is (4, 0)

Answer: at 4 seconds it will hit the ground. The maximum height is reached at 2 seconds due to the symmetry of the quadratic.

Step-by-step explanation:

The function that matches the graph is f(x) = (17/24)(x + 2)²(x + 6).

A graph is a diametrical representation of any function between the dependent and independent variables.

The graph is easy to understand the behavior of the graph.

As per the given graph,

The function is symmetric at x = -2 so it will be an even degree function as (x + 2)².

The function is not symmetric about x = -6 so it will have an odd degree as (x + 6)¹.

The function will be f(x) = C(x + 2)²(x + 6).

The constant will be C = 17/(2² × 6) = 17/24

Hence "The function that matches is f(x) = (17/24)(x + 2)²(x + 6)".

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Answer:

$31

Step-by-step explanation:

x=cost before tax

$32.55 = x + x * 5%  ==> solve for x

$32.55 = x + x * 5/100  

$32.55 = x + 5x/100

$32.55 = 100x/100 + 5x/100 ==> make common denominators

$32.55 = (100x+5x)/100

$32.55 = 105x/100  ==> simplify

100 * $32.55 = 100 * 105x/100  ==> multiply by 100 on both sides to remove

                                                         fractions

$3255 = 105x

x = $3255/105

x = $31

(a) When he is 10 feet from the base of the light, the rate at which the length of his shadow changing is 7.5 ft/s

(b) When he is 10 feet from the base of the light, the rate at which the tip of his shadow changes is 2.5 ft/s.

We have, the height of man, CD = 6 feet

Height of light , AB = 16 feet

Rate of walk by man, dx/dt = 5 feet per second.

a) we have to calculate the rate at which the length of his shadow is moving when he is 10 feet from the base of the light.

let x feet be the distance between the light and man and y feet be the distance between the light and man's shadow(E) i.e BC = x ft. and BE = y ft.

By using ratio of similar triangles,

16/y = 6/(y - x)

On cross multiplication,

16(y - x) = 6y

16y - 15x = 6y

16y - 6y = 15x

10y = 15x

Dividing by 3 on both sides,

2y = 3x

On differentiating both sides with respect to t,

2(dy/dt) = 3(dx/dt)

dy/dt = (3/2)(dx/dt)

dy/dt = (3/2)(5)

dy/dt = 15/2

dy/dt = 7.5 ft/s

Therefore, the rate at which the length of his shadow is changing is 7.5 ft/s.

b) We have to calculate the rate at which the length of his shadow is changing when he is 10 feet from the base of the light.

Length of shadow, CE = y - x

Rate of change of shadow is calculated by differentiating length with respect to time.

d(CE)/dt = dy/dt - dx/dt

d(CE)/dt = 7.5 - 5

= 2.5 ft/s

Therefore, the rate at which the length of his shadow is changing is 2.5 ft/s.

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Answer:

1. 38%

2. 0.4

3. 5/8

Step-by-step explanation:

Hope its right

Answer:

Least 38%

Middle 0.4

Greatest 5/8

Step-by-step explanation:

you can make them all % by multiplying by 100

so

38% (given)

0.4 × 100 = 40%

5/8 × 100 = 62.5%

The time will 90% of the population have heard the rumor are by the time of 3:48 PM.

There are commonly 3 forms of populace pyramids made out of age-intercourse distributions-- expansive, constrictive and stationary.Population is described as all nationals gift in, or briefly absent from a country, and extraterrestrial beings completely settled in a country.

This indicator indicates the variety of humans that typically stay in an area. Growth prices are the yearly adjustments in populace as a result of births, deaths and internet migration at some stage in the year.3000–240=1260

1260 folks heard in four hours=315 folks consistent with hour

90% of populace= 3000 x 0.9= 2700 folks

2700–1500=1200 folks

1200/315=3.eight hours past midday or 3.forty eight pm.

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To solve this problem, we need to use the z-score formula to convert the given values to standard units. This will allow us to use the normal distribution table to find the percentage of pregnancies that fall within a certain range of lengths.

First, let's find the z-score for a pregnancy lasting more than 274 days:

z = (274 - 266) / 16 = 0.5

Next, let's use the normal distribution table to find the percentage of pregnancies that are longer than 274 days. Since the z-score we calculated is positive, we need to look in the right tail of the distribution. From the table, we find that the percentage of pregnancies that are longer than 274 days is approximately 30.85%.

Now let's find the z-score for a pregnancy lasting less than 246 days:

z = (246 - 266) / 16 = -1.25

To find the percentage of pregnancies that are shorter than 246 days, we need to look in the left tail of the distribution. From the table, we find that the percentage of pregnancies that are shorter than 246 days is approximately 8.38%.

Therefore, the percentage of children that are born from pregnancies lasting more than 274 days is approximately 30.85%, and the percentage of children that are born from pregnancies lasting less than 246 days is approximately 8.38%.

To find the percentage of children who are born from pregnancies lasting more than 274 days or less than 246 days, we need to use the z-score formula to convert the lengths of pregnancies to standard units and the standard normal distribution to find the percentage of children who fall within the given range of pregnancy lengths.

The z-score formula is:

z = (x - mu) / sigma

where z is the z-score, x is the raw score, mu is the mean, and sigma is the standard deviation.

In this case, we are given that mu = 266 days and sigma = 16 days. We can use this information to find the z-score for the lower and upper bounds of the pregnancy length range, 246 days and 274 days, respectively.

For the lower bound of the pregnancy length range, we have:

z = (246 - 266) / 16

= -20 / 16

= -1.25

For the upper bound of the pregnancy length range, we have:

z = (274 - 266) / 16

= 8 / 16

= 0.5

Now we can use the standard normal distribution to find the percentage of children who have a z-score between -1.25 and 0.5. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1, so the percentage of children who fall within the given pregnancy length range is the same as the percentage of children who have a z-score within this range.

To find the percentage of children who have a z-score between -1.25 and 0.5, we can use a standard normal distribution table to look up the z-scores and find the corresponding probabilities. The standard normal distribution table shows the percentage of values that fall within a given range of z-scores, so to find the percentage of children who have a z-score between -1.25 and 0.5, we need to look up the percentage of values that fall within this range.

According to the standard normal distribution table, the percentage of values that fall within the range of z-scores -1.25 to 0.5 is approximately 0.38. Therefore, the percentage of children who are born from pregnancies lasting more than 274 days or less than 246 days is approximately 38%, to the nearest hundredth.

Answer: S=19cm

Step-by-step explanation:

area = s^2

s(s-5)=266

s^2-5s=266

s^2-5s-266=0

(s-19)(s+14)=0

s=19cm

19 x 19 = 266

The original dimensions of the square cardboard sheet is 19 x 19.

ax² + bx + c = 0 is a quadratic equation, which is a second-order polynomial equation in a single variable, a≠0. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem. The answer could be simple or complicated.

Given:

Let the length of each side of the original square cardboard be x cm.

After cutting off a strip of 3cm wide its dimension becomes

length = x cm and breadth= (x−5) cm

So, Area = x² - 5x

266 = x² - 5x

x² - 5x - 266= 0

x² - 19x + 14x - 266= 0

x(x - 19) + 14 (x - 19)= 0

(x- 19)( x + 14) = 0

x= 19, -14.

Hence, the original dimensions of the square cardboard sheet is 19 x 19.

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Answer: 40 tickets

Step-by-step explanation:

Divide 2,800 by 7

    40

7|  2800|

   28

   0

You get 40. The jazz band would need to sell 40 tickets to meet its goal.

Answer:

[tex]0.5\sqrt{16a^2}:\quad 2a[/tex]

Step-by-step explanation:

[tex]\sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0\\\\\sqrt{16a^2}=\sqrt{16}\sqrt{a^2}\\=0.5\sqrt{16}\sqrt{a^2}\\\\\sqrt{16}\\\mathrm{Factor\:the\:number:\:}\:16=4^2\\=\sqrt{4^2}\\\\\sqrt{4^2}=4\\=4\\=0.5\cdot \:4\sqrt{a^2}\\\sqrt{a^2}=a\\=0.5\cdot \:4a\\=2a[/tex]

The correct option is: As the sample size increases, the variability of the sampling distribution decreases. is the FALSE statement.

The probability distribution of the a statistic that is acquired by repeated sampling of a particular population is called the sampling distribution.

Thus, it is incorrect to say that as the sample size increases, the variability of the sampling distribution decreases.

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Answer:

The correct answers would be

B, C, and D.

Step-by-step explanation:

B) ΔABD ~ ΔCBA

∠B is congruent to ∠B (itself) by the Reflexive Property.

C) ΔDAB ~ ΔDCA

I took the quiz and got this one right though I'm not 100% sure how it works.

D) ΔBCA ~ ΔACD

∠C is congruent to ∠C through the Reflexive property.

I hope this helps :)

I tried my best

The similarity statement that would be true for the given figure is; A.

∆ DAB ~∆ DAC

Similar triangles are those triangles that have the same proportionate side lengths as well as related angle measurements.

Two triangles are said to be similar if their corresponding angles and sides have the same ratio. Similar triangles will possess the same form but may or may not have the same size.

The SAS similarity rule states that If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then it means that the two triangles are similar.

The Side-Side-Side (SSS) similarity rule states that If all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.

In this question, when we analyze the triangles well, it is clear that;

DA is congruent to itself by reflexive property of congruence.

Thus, we can say that ΔDAB is similar to ΔDAC,

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