lionel wanted to share 4 muffins with his 5 friends. He separated the muffins into 6 equal parts​

Answer:

The number of banana splits sold was [tex]28[/tex]

Step-by-step explanation:

Let

x-----> the number of sundaes sold

y-----> the number of banana splits sold

we know that

[tex]2x+3y=156[/tex] -----> equation A

[tex]x=y+8[/tex] ----> equation B

substitute equation B in equation A and solve for y

[tex]2(y+8)+3y=156[/tex]

[tex]2y+16+3y=156[/tex]

[tex]5y=156-16[/tex]

[tex]5y=140[/tex]

[tex]y=28[/tex]

Answer:

Part 5) [tex]x=50\°[/tex]

Part 6) [tex]x=15\°[/tex]

Step-by-step explanation:

Part 5) we know that

[tex](2x-10)\°+90\°=180\°[/tex] -----> by consecutive interior angles (supplementary angles)

solve for x

[tex]2x=180\°-80\°[/tex]

[tex]2x=100\°[/tex]

[tex]x=50\°[/tex]

Find the value of the labeled angle

[tex](2x-10)\°=2(50\°)-10\°=90\°[/tex] ----> is a right angle

Verify the answer

we know that

In a quadrilateral the sum of the internal angles must be equal to 360 degrees

so

[tex](2x-10)\°+90\°+(180-x)\°+x\°=360\°[/tex]

[tex](2x+260)\°=360\°[/tex]

substitute the value of x

[tex]2(50\°)+260\°=360\°[/tex]

[tex]360\°=360\°[/tex] ------> is true, therefore the value of x is correct

Part 6) we know that

[tex](8x+10)\°+(4x-10)\°=180\°[/tex] -----> by consecutive interior angles (supplementary angles)

solve for x

[tex]12x=180\°[/tex]

[tex]x=15\°[/tex]

Find the value of each labeled angle

[tex](8x+10)\°=8(15\°)+10\°=130\°[/tex]

[tex](4x-10)\°=4(15\°)-10\°=50\°[/tex]

[tex]130\°[/tex] and [tex]50\°[/tex] are supplementary angles

Answer:

Step-by-step explanation:

The question is incomplete. p(x 16) is actually [tex]P(X\geq 16)[/tex] ; p(x 15) is actually [tex]P(X\leq 15)[/tex] and e(x) is [tex]E(X)[/tex]

Wherever a random variable X can be modeled as a binomial random variable we write :

X ~ Bi (n,p)

Where ''n'' is the number of Bernoulli experiments taking place (whose variable is called binomial random variable).

And where ''p'' is the success probability.

In a Bernoulli experiment we define which event will be a ''success''

In order to calculate the probabilities for the variable X we can use the following equation :

[tex]P(X=x)=f(x)=(nCx).(p^{x}).(1-p)^{n-x}[/tex]

Where ''[tex]P(X=x)[/tex]'' is the probability of the variable X to assume the value x.

Where ''[tex]nCx[/tex]'' is the combinatorial number define as :

[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]

In our question

X ~ Bi (20,0.70)

Now let's calculate the probabilities :

[tex]f(12)=P(X=12)=(20C12).(0.70)^{12}.(1-0.70)^{20-12}=0.1144[/tex]

[tex]f(16)=P(X=16)=(20C16).(0.70)^{16}.(1-0.70)^{20-16}=0.1304[/tex] (I)

[tex]P(X\geq 16)[/tex] ⇒

[tex]P(X\geq 16)=P(X=16)+P(X=17)+P(X=18)+P(X=19)+P(X=20)[/tex] (II)

[tex]P(X=17)=(20C17).(0.70)^{17}.(1-0.70)^{20-17}=0.0716[/tex] (III)

[tex]P(X=18)=(20C18).(0.70)^{18}.(1-0.70)^{20-18}=0.0278[/tex] (IV)

[tex]P(X=19)=(20C19).(0.70)^{19}.(1-0.70)^{20-19}=0.0068[/tex] (V)

[tex]P(X=20)=(20C20).(0.70)^{20}.(1-0.70)^{20-20}=0.0008[/tex] (VI)

Using (I), (III), (IV), (V) and (VI) in (II) :

[tex]P(X\geq 16)=0.1304+0.0716+0.0278+0.0068+0.0008=0.2374[/tex]

Now :  

[tex]P(X\leq 15)[/tex]

[tex]P(X\leq 15)=1-P(X\geq 16)[/tex]

[tex]P(X\leq 15)=1-0.2374=0.7626[/tex]

Finally,

[tex]E(X)=[/tex] μ (X)

[tex]E(X)[/tex] is the mean of the variable X

In this case, X is a binomial random variable and its mean can be calculated as

[tex]E(X)=(n).(p)[/tex]

In the question :

[tex]E(X)=(20).(0.70)=14[/tex]

The binomial experiment with n = 20 and p = 0.70 indicates that the probabilities are;

f(12) ≈ 0.1144

f(16) ≈ 0.1304

P(X ≥ 16) ≈ 0.2375

P(X ≤ 15) ≈ 0.7265

E(X) = 14

A binomial experiment is a statistical experiment that consists of a specified number of independent trials, in which each the trials has only two possible outcomes. The probability of success is the same for all trials and the trials are independent, such that the outcome of one trial does not affect the outcome of the other trials.

In a binomial experiment with n = 20 and p = 0.7, the probability of exactly k successes in n independent trials can be found from the following probability mass function.

f(k) = [tex]_nC_k[/tex] × [tex]p^k[/tex] × [tex](1 - p)^{(n-k)}[/tex]

Where [tex]_nC_k[/tex] is the binomial coefficient, which can be calculated an [tex]_nC_k[/tex] = n!/(k!·(n - k)!)

Using the formula the probabilities can be calculated as follows;

f(12) = ₂₀C₁₂ × 0.70¹² × 0.3⁸ = 125970 × 0.70¹² × 0.3⁸ ≈ 0.1144

f(16) = ₂₀C₁₆ × 0.70¹⁶ × 0.3⁴ = 4845 × 0.70¹⁶ × 0.3⁴ ≈ 0.1304

P(X ≥ 16) = f(16) + f(17) + f(18) + f(19) + f(20) ≈ 0.2375

P(X ≤ 15) = 1 - P(X ≥ 16) ≈ 1 - 0.2375 = 0.7625

The expected value of the binomial random variable X is; E(X) = n·p, where n is the number of trials and p is the probability of success on a single trial.

E(X) = n·p = 20 × 0.7 = 14

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Answer:

  d. 11.3 cm

Step-by-step explanation:

The radius of the circle is the length CP, which can be found using the Pythagorean theorem. Since CQ ⊥ PR, you know that Q is the midpoint of PR and PQ = 8 cm.

Then the Pythagorean theorem tells you ...

  CP² = CQ² +PQ² = (8 cm)² + (8 cm)² = 128 cm²

  CP = √128 cm = 8√2 cm

  CP = 11.3 cm

Answer:

[tex]\large\boxed{d.\ 11.3\ cm}[/tex]

Step-by-step explanation:

Look at the picture.

CQ is a perpendicular bisector of PR. Therefore QR = QP.

[tex]PR=16\ cm\to QP = 16 cm : 2 = 8\ cm[/tex]

The segment CP is a radius of the given circle.

We have the right triangle CQP. Use the Pythagorean theorem:

[tex]CQ^2+QP^2=CP^2[/tex]

Substitute CQ = 8 cm and QP = 8 cm.

[tex]CP^2=8^2+8^2\\\\CP^2=64+64\\\\CP^2=128\to CP=\sqrt{128}\\\\CP\approx11.3\ cm[/tex]

Answer:

  It depends

Step-by-step explanation:

The shape will have a number of sides equal to the number of faces it intersects (counting the base). Depending on the shape of the hexagonal base, the symmetry of the peak, and where the slicing is done, the vertical slice could have 3, 4, 5, 6, or 7 sides, so could be anything from a triangle to a heptagon.

Answer:

It took 2 hours by plane

Step-by-step explanation:

let bus is represented by x and plane is represented by y

the total distance of bus and plane will be represented by equation:

40 x + 600 y = 1320           (1)

the total time taken by bus and plane will be represented by:

x + y = 5                              (2)

solving equations (1) and (2) simultaneously:

Multiply by 40 on both sides in eq (2)  and then subtract (1) and (2)

40 x + 600 y = 1320

40 x + 40 y = 200

_______________

          560 y = 1120

                y   = 1120/ 56

                y= 2

since plane is represented by y so, the time they spent travelling by plane is 2 hours.

Final answer:

By setting up and solving a system of equations based on the total travel time and distance, we find that the time spent travelling by plane is 2 hours.

Explanation:

To determine the time spent travelling by plane, we start by setting up two equations based on the given information.

Let x represent the time spent on the bus and y represent the time spent on the plane. The total travel time is given as 5 hours, and the total distance travelled is 1320 km.

Therefore, we have two equations:

We can solve this system of equations by first expressing x in terms of y from the first equation:

x = 5 - y

Substituting x in the second equation:

40(5 - y) + 600y = 1320

Solving for y, we find that the time spent travelling by plane is 2 hours.

Answer: 14 cars 4 left over

Step-by-step explanation:

74/5=14.8

14 x 5 = 70

4 wheels left over

Answer:

A) 0.3336; B) 0.8531

Step-by-step explanation:

For part A,

We use the z-score formula for an individual score:

[tex]z=\frac{X-\mu}{\sigma}[/tex]

Our X value is 332, our mean, μ, is 335, and our standard deviation, σ, is 7:

z = (332-335)/7 = -3/7 ≈ -0.43

Using a z table, we see that the area under the curve less than this (the probability that X is less than this value) is 0.3336.

For part B,

We use the z-score formula for the mean of a sample:

[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]

Our X-bar value is 332, our mean, μ, is 335, our standard deviation, σ, is 7, and our sample size, n, is 6:

z = (332-335)/(7÷√6) = 3/2.8577 ≈ 1.05

Using a z table, we see that the are under the curve to the left of this, or the probability less than this, is 0.8531.

Using the Z-score formula, the probability of a single beer bottle having less than 332 ml is 33.36%, and the probability of a 6-pack having a mean amount less than 332 ml is 14.69%.

To solve this problem, we can use the Z-score. The Z-score is the number of standard deviations a particular value is from the mean in a normal distribution. The formula for the Z-score is (X-µ)/σ, where X represents the value of interest, µ represents the population mean, and σ represents the standard deviation.

So, let's calculate the Z-score:
a) We use the formula Z = (X-µ)/σ = (332-335)/7 = -0.43 (rounded to 2 decimal places). To find the probability that a bottle of beer contains less than 332 ml, we refer to the standard Z-table, which gives us approximately 0.3336. Therefore, there is a 33.36% chance a randomly selected beer bottle contains less than 332 ml of beer.

b) For a 6-pack, the standard deviation decreases because it is now σ/√n (with n being the size of the sample, in this case, 6). The new standard deviation is 7/√6 = 2.86 ml (rounded to 2 decimals). Using the same Z-score formula, Z= (332-335)/2.86= -1.05, and referring to the Z-table, the probability is approximately 0.1469. This means there's about a 14.69% chance that a randomly selected 6-pack will have a mean amount of less than 332 ml.

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Answer: Defualt

Step-by-step explanation: Dan

➷ Find the multiplier:

28/100 = 0.28

1 - 0.28 = 0.72

Multiply the total amount by this multiplier:

40,000 x 0.72 = 28,800

She will make $28,800

➶ Hope This Helps You!

➶ Good Luck (:

➶ Have A Great Day ^-^

↬ ʜᴀɴɴᴀʜ ♡

Answer:

She will make 28,800 dollars after tax.

Step-by-step explanation:

just subtract 28 percent of 40,000.

Or even simpler just follow peachy's instructions cause she/he did her crud right. a percentage is the same as a decimal. 1 percent is 0.01. since 28 percent is 0.28 we subtract 0.28 from one, because 1 is 100 percent. Also,all of this is the same as subtracting 28 percent of 40,000 from 40,000 1 - 0.28= 0.72, and multiply 40,000 by 0.72.

All credit on this part is peachy's thank her/his answer and give her/him brainliest. :)

btw why i say him/her, he/she, and her/his is because I dot want to assume gender

Answer:

The value of x is 9

Step-by-step explanation:

* It first we must to talk about the rational expression

- The rational expression is defined for all values of x except

  the values make the denominator = zero

Ex: If the rational expression is a/b, then b ≠ 0 to be defined

* In any rational expression we must to avoid any values

 make denominator = zero

* If we want to make rational expression equal to zero then we

  must put the numerator equal to zero

* Now lets look to our problem

∵ The rational expression (x - 9)/(x - 4)(x + 4)

- To make this rational expression equal to zero, we must

  equate the numerator by zero

∴ x - 9 = 0 ⇒ add 9 to both sides

∴ x = 9

* The value of x is 9

* Remember ⇒ x must not equal to 4 or -4 to avoid make

  this rational expression undefined

Answer:

w√26

Step-by-step explanation:

A rectangle is a four sided shape with 4 perpendicular angles. It has two pairs of parallel sides which are equal in distance: width and length. The width here is w and the length is 5w or 5 times the width. A diagonal can be drawn between opposite corners that splits the triangle into two equal right triangles. The distance of this diagonal is found using the Pythagorean Theorem a² + b² = c². In the rectangle a = w and b = 5w. Substitute these values and simplify using a square root operation.

w² + (5w)² = c²

w² + 25w² = c²

26w² = c²

√26w² = c

w√26 = c

Answer:

[tex]1\frac{2}{3}\ kg[/tex]

Step-by-step explanation:

we know that

To find out how much the guests ate, multiply the total amount of kg of mashed potatoes by the 4/12 fraction

so

[tex]5(\frac{4}{12})=\frac{20}{12}\ kg[/tex]

convert to mixed number

[tex]\frac{20}{12}=\frac{12}{12}+\frac{8}{12}=1\frac{8}{12}\ kg[/tex]

simplify

[tex]1\frac{8}{12}=1\frac{2}{3}\ kg[/tex]

The guests consumed 1.67 kilograms of the 5 kilograms of mashed potatoes that the chef had made.

To solve this problem, we need to multiply the total amount of mashed potatoes made by the fraction that the guests consumed.

Given that the chef cooked 5 kilograms of mashed potatoes and the guests ate 4/12 (which simplifies down to 1/3) of this amount, we multiply these two together.

So, the calculation would be 5 × 1/3 = 1.67 kilograms.

Therefore, the guests ate 1.67 kilograms of mashed potatoes.

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Answer:

6√3 cm

Step-by-step explanation:

The hypotenuse of a right triangle is 12 centimeters, and the shorter leg is 6 centimeters then the other leg is 6√3

The side length of the canvas is best found by using the quadratic formula

because the equation is prime. Solving the equation produces two

approximate measurements, and one must be discarded for being

unreasonable.

I took the test and this was correct.

Answer:

x=2

Step-by-step explanation:

The formula for inverse variation is

xy = k

We know x = 4 and y = 8

4*8= k

32 = k

xy = 32

We want to find x when y = 16

x*16 = 32

Divide each side by 16

16x/16 = 32/16

x =2

Answer:

xy=32

16x=32

x=2

Answer:

13

Step-by-step explanation:

Answer:

Their average rate of speed is 53 miles per hour.

Step-by-step explanation:

Given : A family on a vacation drives 123 miles in 2 hours then gets stuck in traffic and goes 4 miles in the next 15 minutes. The remaining 191 miles of the trip take [tex]3\frac{3}{4}[/tex] hours.

To find : What was their average rate of speed to the nearest tenth of a mile per hour ?

Solution :

We know, [tex]\text{Speed}=\frac{\text{Distance}}{\text{Time}}[/tex]

Total distance traveled by family on vacation is

D= 123 miles + 4 miles + 191 miles = 318 miles

Total time taken by family on vacation is

T= 2 hours + 15 minutes + [tex]3\frac{3}{4}[/tex] hours

T= 2 hours + [tex]\frac{15}{60}[/tex] hours + [tex]3\frac{3}{4}[/tex] hours

T= [tex]2+ \frac{1}{4}+ \frac{15}{4}[/tex] hours

T= [tex]\frac{8+1+15}{4}[/tex] hours

T= [tex]\frac{24}{4}[/tex] hours

T= 6 hours

Substitute the value in the formula,

[tex]\text{Speed}=\frac{318}{6}[/tex]

[tex]\text{Speed}=53[/tex] miles per hour.

Therefore, Their average rate of speed is 53 miles per hour.

Answer:

See below  

Step-by-step explanation:

7)

AT² = 11² + 4² = 121 + 16 = 137

AT = √137

sinA = DT/AT = 11/√137 = (11√137)/137

cosA = AD/AT = 4/√137 = (4√137)/137

tanA = DT/AD                 = 11/4

sinT = AD/AT = 4/√137 = (4√137)/137

cosT = DT/AT = 11/√137 = (11√137)/137

tanT = AD/DT                 = 4/11

9)

AT² = 8² + 3² = 64 + 9 = 73

AT = √73

sinA = LT/AT = 8/√73 = (8√73)/73

cosA = AL/AT = 3/√73 = (3√73)/73

tanA = LT/AL                = 8/3

sinT = AL/AT = 3/√73 = (3√73)/73

cosT = LT/AT = 8/√73 = (8√73)/73

tanT = AL/LT                = 3/8

11)

    6² =  4² + RT²

  36 = 16  + RT²

RT² = 20

RT =√20 = √(4× 5) = 2√5

 sinA = RT/AT = (2√5)/6 = (√5)/3

cosA = AR/AT = 4/6        = 2/3

tanA = RT/AR = (2√5)/4 = (√5)/2

 sinT = AR/AT = 4/6        = 2/3

cosT = RT/AT = (2√5)/6 = (√5)/3

tanT = AR/RT = 4/(2√5)  = (2√5)/5

Answer:

Step-by-step explanation:

see attached

a)

[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]

b)

[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]

c)

[tex]\tan \dfrac{7\pi}{4}=-1[/tex]

We are asked to find the value of:

a)

[tex]\sin \dfrac{7\pi}{4}[/tex]

We know that:

[tex]\dfrac{7\pi}{4}=2\pi-\dfrac{\pi}{4}[/tex]

Hence, we have:

[tex]\sin (\dfrac{7\pi}{4})=\sin (2\pi-\dfrac{\pi}{4})\\\\\\\sin (\dfrac{7\pi}{4})=-\sin (\dfrac{\pi}{4})[/tex]

Since,

[tex]\sin (2\pi-\theta)=-\sin \theta[/tex]

Hence, we have:

[tex]\sin (\dfrac{7\pi}{4})=-\dfrac{1}{\sqrt{2}}[/tex]

b)

[tex]\cos \dfrac{7\pi}{4}[/tex]

[tex]\cos (\dfrac{7\pi}{4})=\cos (2\pi-\dfrac{\pi}{4})\\\\\\\cos (\dfrac{7\pi}{4})=\cos (\dfrac{\pi}{4})[/tex]

Since,

[tex]\cos (2\pi-\theta)=\cos \theta[/tex]

Hence, we have:

[tex]\cos \dfrac{7\pi}{4}=\dfrac{1}{\sqrt{2}}[/tex]

c)

[tex]\tan \dfrac{7\pi}{4}[/tex]

[tex]\tan (\dfrac{7\pi}{4})=\tan (2\pi-\dfrac{\pi}{4})\\\\\\\tan (\dfrac{7\pi}{4})=\tan (\dfrac{\pi}{4})[/tex]

Since,

[tex]\tan (2\pi-\theta)=-\tan \theta[/tex]

Hence, we have:

[tex]\tan \dfrac{7\pi}{4}=-1[/tex]

Answer: The Slope-Intercept Form equation is y=mx+b

i think the answer is y=3x+2 but i hope this helps

The equation of the line in Slope-intercept form is y = -3x + 2.

The correct choice is option C. y=-3x+2.

Let's, find the equation of the line in slope-intercept form, we need to determine the slope (m) and the y-intercept (b). The slope represents the steepness of the line, and the y-intercept is the point where the line crosses the y-axis.

From the graph, we can see that the line passes through the points (-1, 5) and (1, -1). Using the slope formula, we can calculate the slope as follows:

m = (y2 - y1) / (x2 - x1)

m = (-1 - 5) / (1 - (-1))

m = -6 / 2

m = -3

Now that we know the slope, we can plug it into the slope-intercept form of the equation and solve for b:

y = mx + b

y = -3x + b

We can find the value of b by substituting the coordinates of one of the points on the line into the equation. For example, we can use the point (-1, 5):

5 = -3(-1) + b

5 = 3 + b

b = 5 - 3

b = 2

Therefore, the equation of the line in slope-intercept form is y = -3x + 2.

option C. is correct choice y=-3x+2.

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Answer:

Check out lesson 3.09, It'll help ;)

Answer:

This would probely be 266.

Step-by-step explanation:

There are three kids so I would divide them by three. Or multiply 266 by three. Of course there would still be 2$ left.

Answer:

Share of Eli, Freda and Geoff will be $180, $260, $360 respectively.

Step-by-step explanation:

Eli, Freda and Geoff were given $800 to share in the ratio of their age.

Their ages were 9 years, 13 years and 18 years respectively.

Ratio of their ages will be 9 : 13 : 18

Now share of Eli will be = [tex]\frac{9}{9+13+18}\times 800[/tex]

= [tex]\frac{9\times 800}{40}[/tex]

= $180

Share of Freda will be = [tex]\frac{13}{9+13+18}\times 800[/tex]

                                     = $260

Share of Geoff will be = [tex]\frac{18}{9+13+18}\times 800[/tex]

                                     = $360

Therefore, share of Eli, Freda and Geoff will be $180, $260, $360 respectively.

Answer:

Step-by-step explanation:

We must calculate the lateral area of a cylinder.

The formula is:

[tex]A=2\pi rH[/tex]

r - radius

H - height

We have H = 6 cm and r = 3.5.

Substitute:

[tex]A=2\pi(3.5)(6)=42\pi\ cm^2[/tex]

Use [tex]\pi\approx\dfrac{22}{7}[/tex]

[tex]A\approx42\left(\dfrac{22}{7}\right)=(6)(22)=132\ cm^2[/tex]

Answer: ❤️Hello!❤️He asks every tenth customer who checks into the hotel. Hope this helps! ↪️ Autumn ↩️

Answer:

B

Step-by-step explanation:

Since the d value is changed, we're talking about a vertical transformation. Since d > 0, the graph is shifted up.

now, let's recall that a geometric sequence is one that uses some "r" common ratio to get the next term, by simply multiplying the current term by it.

[tex]\bf \begin{array}{|cl|ll} \cline{1-2} term&value\\ \cline{1-2} a_1&\underline{\qquad }\\&\\ a_2&6\\&\\ a_3&\underline{6(r)}\\&\\ a_4&6(r)(r)\\&\\ &54\\ \cline{1-2} \end{array}\qquad \implies \begin{array}{llll} 54=6r^2\implies \cfrac{54}{6}=r^2\implies 9=r^2\\\\ \sqrt{9}=r\implies 3=r \end{array} \\\\[-0.35em] ~\dotfill\\\\ a_1=6\div 3\implies a_1=2~\hfill a_3=6(3)\implies a_3=18[/tex]

and of course, the next term or a₅ = 54(3) --> a₅ = 162.

The first five terms are 2, 6, 18, 54, 162

Given,

The second term is 6.

The fourth term is 54.

We need to find the first five terms of a geometric sequence.

A sequence where each term after the first is found by multiplying the previous one with a common ratio.

The sequence is given by:

a, ar, ar^2, ar^3, ar^4, ar^5,...

The nth term is given by:

a_n = ar^(n-1)

We have,

Second term = 6

a_2 = ar^(2-1)

6 = ar^1

6 = ar

a = 6/r _____(1)

Fourth term = 54

a_4 = ar^(4-1)

54 = ar^3

a = 54/r^3 ______(2)

From (1) and (2)

6/r = 54/r^3

r^3/r = 54/6

r^2 = 9

r = 3

Putting in (1)

a = 6/r

a = 6/3

a = 2

We have,

a = 2 and r = 3

Find the first five terms of a geometric sequence.

It is given by:

a, ar, ar^2, ar^3, ar^4

2, 2x3, 2x9, 2x27, 2x81

2, 6, 18, 54, 162

The first five terms are 2, 6, 18, 54, 162

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An example of dependent events is drawing a red marble from the same jar without replacing the first marble, which changes the probabilities for the second draw. This is known as sampling without replacement, which contrasts with independent events where the item is replaced and the probabilities remain unchanged.

An example of dependent events is drawing a red marble out of the same jar, without replacing the first marble. This creates a dependency because removing the first marble affects the outcome chances for the subsequent draw. If the jar starts with four blue and three white marbles and you draw one blue marble and do not replace it, the jar then contains three blue and three white marbles. Therefore, the probability of drawing another blue marble has changed from the original draw.

Dependent events are associated with sampling without replacement, which means once an item is drawn, it is not put back into the population from which it was taken, altering the probabilities of subsequent draws. In contrast, independent events, such as drawing and replacing a marble, do not affect the subsequent probabilities, since the composition of the jar remains unchanged with each draw.

Answer:

84 feet to the nearest foot.

Step-by-step explanation:

We have a right angled triangle with adjacent side (A) = 100 and you want to find the height of the tower, the opposite side (O).

A = 100 , O = ? so we need the tangent , (from SOH-CAH-TOA).

tan 40 = O/ 100

O = 100 tan 40

= 83.9 feet.

Final answer:

To determine the height of the tower, we can use the tangent of the angle of elevation, 40 degrees, multiplied by the distance, 100 feet, which results in approximately 84 feet.

Explanation:

To find the height of a tower with an angle of elevation of 40°, observed from 100 feet away, you can use trigonometric functions. Specifically, the tangent function, which is defined as the ratio of the opposite side (the height of the tower we're looking for) to the adjacent side (the distance from the tower).

We have:

The angle of elevation (θ) = 40°

The distance from the tower (adjacent side) = 100 feet

The height of the tower can be calculated as:

height = tan(θ) × adjacent side
= tan(40°) × 100 feet

Using a calculator, we find:

height = tan(40°) × 100
= 0.8391 × 100
≈ 84 feet (to the nearest foot)

Therefore, the tower is approximately 84 feet high.

In spherical coordinates, we set

[tex]x=\rho\cos\theta\sin\varphi[/tex]

[tex]y=\rho\sin\theta\sin\varphi[/tex]

[tex]z=\rho\cos\varphi[/tex]

so that the volume element under this transformation becomes

[tex]\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=|\det\mathbf J|\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

The region [tex]E[/tex] is given by the set

[tex]\left\{(\rho,\theta,\varphi)\mid0\le\rho\le1,0\le\theta\le2\pi,0\le\varphi\le\dfrac\pi3\right\}[/tex]

so that the integral is

[tex]\displaystyle\iiint_Ey^2z^2\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi/3}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=1}\rho^6\sin^2\theta\sin^3\varphi\cos^2\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

[tex]\displaystyle=\left(\int_0^{\pi/3}\sin^3\varphi\cos^2\varphi\,\mathrm d\varphi\right)\left(\int_0^{2\pi}\sin^2\theta\,\mathrm d\theta\right)\left(\int_0^1\rho^6\,\mathrm d\rho\right)[/tex]

[tex]=\dfrac{47}{480}\cdot\pi\cdot\dfrac17=\dfrac{47\pi}{3360}[/tex]

The volume of the solid using the triple integral [tex]\mathbf{\iiint_E y^2z^2 dV \ \ is \ \ = \dfrac{47 \pi}{3360}}[/tex]

Given that;

Then the spherical coordinates can be expressed as:

where;

Now, the expression for the solid E in the spherical coordinates can be computed as:

[tex]\mathbf{E = \Big \{ ( \rho , \theta, \phi )\Big| 0 \leq \rho \leq 1, 0 \leq \phi \leq \dfrac{\pi}{3}, 0 \leq \theta \leq 2 \pi \Big \} }[/tex]

and the volume of the solid using the triple integral is calculated as:

[tex]\mathbf{\iiint_E y^2z^2 dV = \iiint _E \ y^2 z^2 \ dx dy dz }[/tex]

[tex]\mathbf{\implies \iiint _E \ y^2 z^2 \ dx dy dz = \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \int ^1_0 \ (\rho sin \phi sin \theta )^2 ( \rho cos \phi )^2 \rho^2 sin \phi d \rhod \phi d \theta }[/tex]

[tex]\mathbf{\implies \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi sin^2 \theta \ \int^1_0 \ \rho^6 d \rho d \phi d \theta }[/tex]

[tex]\mathbf{\implies \int ^{2 \pi}_{0} \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi sin^2 \theta \Big [ \dfrac{\rho^7}{7} \Big]^1_0 \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta \int ^{\dfrac{\pi}{3}}_{0} \ sin^3 \phi cos^2 \phi \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta \int ^{\dfrac{\pi}{3}}_{0} \ sin \phi( 1- cos^2 \phi)cos^2 \phi \ d \phi d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{cos^5 \phi}{5}- \dfrac{cos ^3 \phi}{3} \Bigg ] ^{\dfrac{\pi}{3}}_{0} d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{cos^5 \dfrac{\pi}{3}}{5}- \dfrac{cos ^3 \dfrac{\pi}{3}}{3}- \dfrac{cos^5 0}{5}+ \dfrac{cos^3 0}{3} \Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{2}{15}- \dfrac{17}{480} \Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{(480\times 2) -(15\times 17)}{15\times 480}\Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{1}{7} \int ^{2 \pi}_{0} sin ^2 \theta\Bigg [\dfrac{705}{15\times 480}\Bigg ] d \theta }[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \int ^{2 \pi}_{0} 2sin ^2 d \theta}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \int ^{2 \pi}_{0} (1-cos 2\theta) \ d \theta}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \Bigg [\theta - \dfrac{sin 2 \theta }{2}\Bigg] ^{2 \pi}_{0}}[/tex]

[tex]\mathbf{\implies \dfrac{47}{6720} \Bigg [2 \pi\Bigg] }[/tex]

[tex]\mathbf{\iiint_E y^2z^2 dV = \dfrac{47 \times 2 \pi}{6720}} }[/tex]

[tex]\mathbf{\iiint_E y^2z^2 dV = \dfrac{47 \pi}{3360} }[/tex]

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