find one positive and one negative angle coterminal with an angle of 166 Question 4 options:

526°, –194°

516°, –14°

526°, –76

256°,-76

[tex]\boxed{a_{n}=-3+8(n-1)}[/tex]

The explicit formula for the general nth term of the arithmetic sequence is given by:

[tex]a_{n}=a_{1}+d(n-1) \\ \\ \\ Where: \\ \\ a_{n}:nth \ term \\ \\ n:Number \ of \ terms \\ \\ a_{1}:First \ term \\ \\ d:common \ difference[/tex]

Here we know that:

[tex]a_{1}=-3 \\ \\ a_{10}=69[/tex]

So, our goal is to find the common difference substituting into the formula:

[tex]a_{10}=a_{1}+d(10-1) \\ \\ 69=-3+d(9) \\ \\ Solving \ for \ d: \\ \\ 9d=69+3 \\ \\ 9d=72 \\ \\ d=8[/tex]

Finally, we can write the explicit formula as:

[tex]\boxed{a_{n}=-3+8(n-1)}[/tex]

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The explicit formula for the general nth term of the arithmetic sequence is an = 8n - 11.

To find the explicit formula for the general nth term of the arithmetic sequence with a1 = -3 and a10 = 69, we can use the formula for the nth term of an arithmetic sequence, which is:

an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.

First, we find the common difference by using the 10th term:

69 = -3 + (10 - 1)d

72 = 9d

d = 8

Now we have the common difference, so we can write the formula for the nth term:

an = -3 + (n - 1)(8)

Simplifying:

an = 8n - 11

This is the explicit formula for the general nth term of the given arithmetic sequence.

Answer: The company has to record a loss of $2,000

Step-by-step explanation:

The accounting for bonds retired early would require the company to pay out cash to remove the bonds payable from its balance sheet. To determine the gain or loss, if the cash paid is less than the carrying value of the bond, then a gain is determined, if the cash paid is more than the carrying value of the bond, then a loss is determined.

From the question the company pays $100,000 which is more than the carrying value of $98,000. Therefore, the company would record a loss $2,000.

Answer:

$21,000

Step-by-step explanation:

Given,

Net income = $20,000

Depreciation = $2,000

Current assets (increase) = $2,000

Current liabilities (increase) = $1,000

Note: An increase in current assets is a reduction in cash balance while an increase in current liabilities is an increase in cash

Considering the note above,

Cash is provided by operating activities = 20000 + 2000 - 2000 + 1000

                                                                   = $21,000

Answer(s):

d= 4 (D=4) or 4=d (4=D)

Step-by-step explanation: You can solve it in 2 ways.

The [tex]1^{st}[/tex] way:

4d−4 = 5d−8

   +4        +4

   4d = 5d-4

  -5d   -5d

    -1d = -4

     /-1    /-1

      d = 4

The [tex]2^{nd}[/tex]

4d−4 = 5d−8

   +8        +8

   4d+4 = 5d

  -4d       -4d

        4 = 1d

        /1    /1

        4 = d

Hope this helps. :)

The annual rate of interest is 4.80%

Step-by-step explanation:

The formula for compound interest, including principal sum is

[tex]A=P(1+\frac{r}{n})^{nt}[/tex] where:

Suppose that time invest $10,000 in an account that offers are percent annual interest, compounded quarterly if the investment increases to $12,694.34 in five years

∵ P = $10,000

∵ A = $12,694.34

∵ n = 4 ⇒ compounded quarterly

∵ t = 5 years

- Substitute all these values in the formula above

∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{4(5)}[/tex]

∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{20}[/tex]

- Divide both sides by 10,000

∴ [tex]1.269434=(1+\frac{r}{4})^{20}[/tex]

- Insert ㏒ to both sides

∴ [tex]log(1.269434)=log(1+\frac{4}{n})^{20}[/tex]

∴ [tex]log(1.269434)=20log(1+\frac{4}{n})[/tex]

- Divide both sides by 20

∴ [tex]0.00518=log(1+\frac{4}{n})[/tex]

- Remember [tex]log_{a}b=c[/tex] can be written as [tex]a^{c}=b[/tex]

∵ The base of the ㏒ is 10

∴ [tex]10^{0.00518}=(1+\frac{r}{4})[/tex]

∴ [tex]1.011998806=1+\frac{r}{4}[/tex]

- Subtract 1 from both sides

∴ [tex]0.011998806=\frac{r}{4}[/tex]

- Multiply both sides by 4

∴ 0.04799522 = r

∵ r is the rate in decimal

- To find the annual rate of interest R% multiply r by 100%

∴ R% = 0.04799522 × 100% = 4.799522%

∴ R% ≅ 4.80%

The annual rate of interest is 4.80%

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Answer:

Option C

Step-by-step explanation:

As you can see in the graph the shaded region is in 1st quadrant which means that both x and y are greater than or less than equal to zero. From this statement only option A and B get eliminated. But still we will look further...

The equation of the yellow line should be determined in order to know the complete answer. If you don't know how to write equation of line in intercept form then just assume the line's equation to be :

y=mx + c    ; where m and c are constants.

Now you just need to find two points satisfying the line's equation. As you can see in the graph (2,0) and (0,-2) are the points lying on the line. Now put them in the assumed line's equation to determine the constants.

0 = m × 2 + c  .........equation (1)

-2 = m × 0 + c .........equation (2)

solving equation (2) gives c = -2

put the value of c in equation (1) then solve it to get the value of m

0 = m × 2 - 2

m=1

Therefore line's equation is y = x - 2

Since the value of y is greater than and equal to the value of y which we get from the line's equation so

y ≥ x - 2

rearrange the above inequality

x - y ≤ 2

So these three conditions are there

x ≥ 0

y ≥ 0

x - y ≤ 2

which gives the shaded region

Final answer:

Using the principle of inclusion-exclusion, it's calculated that 25% of the teachers surveyed had neither high blood pressure nor heart trouble.

Explanation:

To find the percentage of teachers who had neither high blood pressure nor heart trouble, we can use the principle of inclusion-exclusion in set theory. We begin by adding the number of teachers with each condition, then subtract those counted twice because they have both conditions.

The formula we will use is:

Total surveyed - (High blood pressure + Heart trouble - Both) = Neither condition

Substituting the numbers from the survey, we get:

120 - (70 + 40 - 20) = 120 - (90) = 30

So, 30 teachers had neither high blood pressure nor heart trouble.

To find the percentage, we divide the number of teachers with neither condition by the total surveyed and then multiply by 100:

(30 / 120) * 100 = 25%

Therefore, 25% of the teachers surveyed had neither high blood pressure nor heart trouble.

Answer:

Sometime between the third and fourth year (3.5 years)

Step-by-step explanation:

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

[tex]S=555+33y[/tex]

As for the French class, since we lose 2 students each year the equation is

[tex]F=230-2y[/tex]

We require to know the value of y in the exact moment when S=3F

[tex]555+33y=3(230-2y)[/tex]

Operating

[tex]555+33y=690-6y[/tex]

Reducing

[tex]39y=690-555[/tex]

[tex]39y=135[/tex]

[tex]y=135/39=3.5\ years[/tex]

It means that sometime between the third and fourth year, there will be 3 times as many students taking Spanish as French

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

According to the data provided, we know the initial enrollment in the Spanish class is 555. The trend shows 33 more students will enroll each year. So, being y the time in years, the equation for the number of students in the Spanish class is

Answer:

a) 5 balls

b) 13 balls

Step-by-step explanation:

a) by the pigeon holes theorem where the pigeon is the ball and the holes here are the colors. For her to have at least 3 balls of the same colors, we can consider the worst case where she selected 2 balls of each color, that makes it 4 balls. Then by the 5th ball it should match either of the colors, making it 3 balls of same color.

b) Similarly, in the worst case let's say she selects all 10 red balls, then the next 3 balls must be blue. So she needs to select a total of 13 balls to make sure that she has at least 3 blue balls.

The height of the cloud above the lake is approximately 86.19 meters, calculated using trigonometry relations of angles of elevation and depression from Dustin's point of view standing on a 40-meter high cliff.

To find the height of the cloud above the lake, we use trigonometry. Dustin observes the cloud at an angle of elevation of 30° and the reflection at an angle of depression of 60°. The height of the cliff is 40 meters. Since angles of elevation and depression are measured from the horizontal, the angle between Dustin's line of sight to the cloud and to its reflection is 90° (30° + 60°). The height of the cloud can be found by considering two right-angled triangles that Dustin forms with his line of sight to the cloud and to the reflection.

Triangle ADC represents the direct line of sight to the cloud, with angle CAD being 30°, and side AC the height of the cliff (40 meters).

Triangle BEC represents the line of sight to the reflection of the cloud, with angle CBE being 60°, and side BC the height of the cliff (40 meters).

Using the trigonometric relations:

Since CE and CD form a straight line, the total height of the cloud above the lake is the sum of CD and CE. Using the trigonometry values (Tan 30° = 1/√3, and Tan 60° = √3):

Therefore, the height of the cloud above the lake, DE = CD + CE = 40 * (1/√3 + √3).

After calculation, DE = 40 * (1/√3 + √3) ≈ 40 * 2.1547 ≈ 86.19 meters.

So, the height of the cloud above the lake is approximately 86.19 meters.

Answer:

After 4 months

Step-by-step explanation:

The cost at both the programs consists of a "fixed cost" (reg fee)  & "variable cost" ( per month fee).

Let number of months be "x"

Yo-Yoga:

35 fixed

90 per month

So equation would be: 35 + 90x

Essence Yoga:

75 fixed

80 per month

So equation would be: 75 + 80x

To find number of month when cost would be same, we equate both equations and solve for x:

35 + 90x = 75 + 80x

10x = 40

x = 40/10

x = 4

hence, after 4 months, both cost would be same

The projectile will be back at ground level at t = 0 (initially fired) and t = 8 seconds after being fired.

To find when the height of the projectile is less than 128 feet, we set s < 128 and solve for t using the given position equation:

[tex]\[ -16t^2 + v_0t + s_0 < 128 \][/tex]

Given:

- [tex]\( v_0 = 128 \)[/tex]  feet per second

- [tex]\( s_0 = 0 \)[/tex] (since the object is fired upward from ground level)

Substitute these values into the equation:

[tex]\[ -16t^2 + 128t < 128 \][/tex]

Now, let's solve this inequality for t. First, let's rewrite it in standard quadratic form:

[tex]\[ -16t^2 + 128t - 128 < 0 \][/tex]

Divide all terms by -16 to simplify:

[tex]\[ t^2 - 8t + 8 > 0 \][/tex]

Now, we need to find the values of t for which this inequality holds true.

To solve quadratic inequalities, we can find the roots of the corresponding quadratic equation [tex]\( t^2 - 8t + 8 = 0 \)[/tex], and then determine the intervals where the quadratic expression [tex]\( t^2 - 8t + 8 \)[/tex] is positive.

The roots of the quadratic equation [tex]\( t^2 - 8t + 8 = 0 \)[/tex] can be found using the quadratic formula:

[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

where [tex]\( a = 1 \), \( b = -8 \), and \( c = 8 \).[/tex]

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

[tex]\[ t = \frac{8 \pm \sqrt{64 - 32}}{2} \][/tex]

[tex]\[ t = \frac{8 \pm \sqrt{32}}{2} \][/tex]

[tex]\[ t = \frac{8 \pm 4\sqrt{2}}{2} \][/tex]

[tex]\[ t = 4 \pm 2\sqrt{2} \][/tex]

So, the roots of the equation are [tex]\( t = 4 - 2\sqrt{2} \) and \( t = 4 + 2\sqrt{2} \).[/tex]

Now, we can test the intervals [tex]\( (-\infty, 4 - 2\sqrt{2}) \), \( (4 - 2\sqrt{2}, 4 + 2\sqrt{2}) \), and \( (4 + 2\sqrt{2}, \infty) \)[/tex] to determine where the inequality [tex]\( t^2 - 8t + 8 > 0 \)[/tex] holds true.

However, it seems there might be an error in the equation. Let's reassess the situation. If the projectile is fired straight upward from ground level, it will reach its maximum height and then fall back to the ground. The time at which it returns to the ground can be found by setting \( s = 0 \) in the position equation:

[tex]\[ -16t^2 + 128t = 0 \][/tex]

Factor out -16t:

-16t(t - 8) = 0

This equation will be true when either -16t = 0 or t - 8 = 0.

Solving each equation:

1.  -16t = 0

t = 0

2. t - 8 = 0

t = 8

So, the projectile will be back at ground level at t = 0 (when it's initially fired) and at t = 8 seconds after being fired.

Answer:

It's D

Step-by-step explanation:

Hope this helps :))

Answer:

It will take 5.6 hours

Step-by-step explanation:

If Ken (K) can clear in two hours and bettina (B) in five hours then;

5K=2B; If B=1 K=2/5 = 0.4 (2-0.4=1.6) plus 4 from B are 5.6 or also: If B=4 (Bettina started 1 hour later 5-1=4) K=8/5=1.6 then K+B=1.6+4=5.6 h

Final answer:

Ken and Bettina Reeves will take approximately 51.4 minutes to finish clearing snow together after Bettina has already put in 1 hour of work by herself. This is calculated by combining their work rates and determining how much of the job is left after Bettina's initial work.

Explanation:

The question involves determining how much longer it will take Ken and Bettina Reeves to finish clearing snow together after Bettina has already worked for 1 hour by herself. To solve this, we can find out their collective work rate when they work together, and use it to calculate the remaining time required to clear the snow.

Step-by-Step Solution:

Find Bettina's work rate: Bettina can clear snow in 5 hours, which means her work rate is ⅓ (one-fifth) of the driveway per hour.

Calculate how much Bettina has cleared in 1 hour: She has cleared ⅓ of the snow.

Ken's work rate is ½ (one-half) of the driveway per hour since he can clear the snow by himself in 2 hours.

Calculate their combined work rate: ½ (Ken) + ⅓ (Bettina) = ⅗ (seven-tenths) of the driveway per hour when working together.

Subtract the portion Bettina has already cleared: 1 - ⅓ = ⅔ (four-fifths) of the driveway remains.

Divide the remaining work by their combined work rate: ⅔ ÷ ⅗ = ⅖ (6/7) hours.

Convert the time to minutes: ⅖ hours is approximately 51.4 minutes.

Therefore, working together, Ken and Bettina will take approximately 51.4 minutes to remove the rest of the snow.

Answer: The interest that she will have at the end of 6 years is $2700

Step-by-step explanation:

Chanelle deposits $7,500 and she does not withdraw or deposit money for 6 years. This means that the initial amount and the interest accrued was not altered in 6 years.

She ears 6% in interest. To determine the amount of interest that she would have at the end of 6 years. We will apply the simple interest formula.

I = PRT/100

Where

I = simple interest

P = principal

R = interest rate

T = number of time in years

From the information given

P= $7,500

R = 6

T = 6

I = (7500 × 6 × 6)/100

I = $2700

Answer:

the 3rd one (:

Step-by-step explanation:

Answer:

First One:

x { 6, 9, 12, 15, 18 }
y { -14, -4, 8, 22, 38 }

Step-by-step explanation:

Not sure if the other guy is trolling...
The x values constantly rise by 3, so we find the y value change
The differences are 10, 12, 14, 16
If we do it again we get 2, 2, 2. So since we did the reduction 2 times, this is a quadratic function.
Why isn't "the third one"
Constant rise by 2

Change of 9, 9, 9, 9
This is one reduction, so it is not quadratic.

Answer:

19.43 ft

Step-by-step explanation:

Using SOHCAHTOA,

opposite = x

Hypothenus = 25ft

Sin α = opposite /Hypothenus

Sin 51° = x/25

x = 25(Sin51°)

x = 19.4286

x = 19.43(approximate to 2 d.p)

The accumulation of that number of people in a relatively small area would be consistent with Stokols's definition of​ density. Option D

Definition of density

Density refers to the number of individuals per unit of space. In this scenario, there were 77,000 people present in a stadium with a capacity of 73,000, indicating a high density of individuals in the given area.

Stokols's definition of density aligns with this situation, as it describes the accumulation of individuals in a relatively small area. Crowding, mob behavior, and learned helplessness are not directly related to the concept of density.

During a recent football game, 77,000 people were present in a stadium with a capacity of 73,000. The accumulation of that number of people in a relatively small area would be consistent with Stokols's definition of

 A) crowding.

 B) learned helplessness.

 C) mob behavior.

 D) density.

Answer:

Step-by-step explanation:

Having drawn the line, Kendall must verify that the point P belongs to the line y = 2x-1 and then calculate the distance between A-P  and verify if it is the closest to A or there is another one of the line

Having the point P(3,5) substitue x to verify y

y=2*(3)-1=6-1=5 (3,5)

Now if the angle formed by A and P is 90º it means that it is the closest point, otherwise that point must be found

[tex]d_{AP}=\sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}}=\sqrt{(5-7)^{2}+(3-(-2}))^{2}}=\\\sqrt{(-2)^{2}+(5)^{2}}=\sqrt{29}[/tex]

and we found the distance PQ and QA

; [tex]d_{PQ}=\sqrt{125}[/tex], [tex]d_{QA}=12[/tex]

be the APQ triangle we must find <APQ through the cosine law (graph 2).

Answer:

8x + 29 < 17

Step-by-step explanation:

Let the unknown number be x .

According to the question ,

8 times x when added to 29 will give a sum less than 17 .

So , we can write

8x + 29 < 17.

We can also solve it

so we will get

8x < -12

x<-1.5.

Answer:

g equals 1

g = 1

Isolate the variable by dividing each side by factors that don't contain the variable.

Hope this helps!

Answer:

g=1

Step-by-step explanation:

Given equation is \[−3+5+6g=11−3g\]

Simplifying, \[2 + 6g = 11 - 3g\]

Bringing all terms containing g to the left side of the equation and all the numeric terms to the other side,

\[3g + 6g = 11 - 2\]

=> \[9g = 9\]

=> \[g=\frac{9}{9}\]

=> g = 1

Validating by substituting in the given equation:

Left Hand Side = -3 + 5 + 6 = 8

Right Hand Side = 11 - 3 = 8

Hence the two sides of the equation are equal when g = 1.

Answer:

50 kg water.

Step-by-step explanation:

We have been given that the number of kilograms of water in a human body varies directly as the mass of the body.

We know that two directly proportional quantities are in form [tex]y=kx[/tex], where y varies directly with x and k is constant of variation.

We are told that an 87​-kg person contains 58 kg of water. We can represent this information in an equation as:

[tex]58=k\cdot 87[/tex]

Let us find the constant of variation as:

[tex]\frac{58}{87}=\frac{k\cdot 87}{87}[/tex]

[tex]\frac{29*2}{29*3}=k[/tex]

[tex]\frac{2}{3}=k[/tex]

The equation [tex]y=\frac{2}{3}x[/tex] represents the relation between water (y) in a human body with respect to mass of the body (x).

To find the amount of water in a 75-kg person, we will substitute [tex]x=75[/tex] in our given equation and solve for y.

[tex]y=\frac{2}{3}(75)[/tex]

[tex]y=2(25)[/tex]

[tex]y=50[/tex]

Therefore, there are 50 kg of water in a 75​-kg person.

To find out how many kilograms of water are in a 75 kg person, we can set up a proportion and solve for x. The number of kilograms of water is directly proportional to the mass of the body.

To solve this problem, we can use the concept of direct variation. Direct variation states that two variables are directly proportional to each other. In this case, the number of kilograms of water is directly proportional to the mass of the body.

We are given that an 87 kg person contains 58 kg of water. To find out how many kilograms of water are in a 75 kg person, we can set up a proportion:

(87 kg) / (58 kg) = (75 kg) / (x kg)

Cross multiplying, we get:

87 kg * x kg = 58 kg * 75 kg

Simplifying, we find that x = 39 kg.

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Answer:

8923 years

Step-by-step explanation:

Half life of C-14 = 5730yrs

decay rate= 34%

Halt life (t^1/2) = (ln2) / k

5730 = (ln2) /k

k = (ln2) / 5730

k = 1.209 * 10^-4

For first order reaction in radioactivity,

ln(initial amount) = -kt

ln(34/100) = -(1.209*10^-4)t

-1.0788 = -(1.209*10^-4)t

t = -1.7088/ -1.209*10^-4

t = 8923 years

It would take 8918 years for the tree to decay to 34%.

The half life is the time taken for a substance to decay to half of its value. It is given by:

[tex]N(t) = N_0(\frac{1}{2} )^\frac{t}{t_\frac{1}{2} } \\\\where\ t=period, N(t)=value\ after\ t\ years, N_o=original\ amount, t_\frac{1}{2} =half\ life\\\\Given\ t_\frac{1}{2} =5730,N(t)=0.34N_o, hence:\\\\0.34N_o=N_o(\frac{1}{2} )^\frac{t}{5730} \\\\t=8918\ years[/tex]

It would take 8918 years for the tree to decay to 34%.

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Answer:

[tex]\frac{(2-y)x}{5}\text{ litres}[/tex]

Step-by-step explanation:

Let l litre of 7 grams of salt per litre is combined with x litres of water which contains y grams of salt per litre to yield a solution with 2 grams of salt per litre.

Thus,

Salt in l litre + salt in x litre = salt in resultant mixture

7l + xy = 2(l +x)

7l + xy = 2l + 2x

7l - 2l = 2x - xy

5l = x(2-y)

[tex]\implies l =\frac{x(2-y)}{5}[/tex]

Hence, [tex]\frac{x(2-y)}{5}[/tex] litres of 7 gram of salt per litre is mixed.

Option a) reflection over the north-south axis is the position of the person and the direction the person faces.

If we only consider the person's movement along the East-West axis, this movement was:

Taking east as positive and west as negative, this is equivalent to +20 -25 +15 = 10 or walks east 10 paces

Considering the person's movement along the North-South axis, this movement was:

which cancel out each other.

Hence, The answer is Option a) reflection over the north-south axis is the position of the person and the direction the person faces.

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Answer:

6 problems correct 7 incorrect, and 5 skipped over

Step-by-step explanation:

3+5=8 multiple of 8 that ends in 8:48

she got 6 problems corrects

4x6=48 48-13=35

35/5=7

20-7-8=5

Answer:

The answer to your question is: letter B. 16 days

Step-by-step explanation:

Materials = $3.89 square foot

Day of labor = $121.26

Total $2982.68

Square feet = 252 more than the number of days

Process

                  materials = days of labor + 252

materials = m

days of labor = d

                             m = d + 252  (I)

Total Equation  

                         3.89m + 121.26d = 2982.68

                         3.89(d + 252) + 121.26d = 2982.68        

                         3.89d + 980.28 + 121.26d = 2982.68

                         3.89d + 121.26d = 2982.68 - 980.28

                                       125.15d = 2002.4

                                                 d = 16

Number of days 16.

Answer:

The distance 5 miles North-East of the intersection between the car and the truck increasing at 71.06 miles per hour at that moment.

Step-by-step explanation:

Looking at the attached figures, Fig 1 shows the diagram of the car and the truck.

Using Pythagoras theorem on Fig 1a,

[tex]l^{2} = \sqrt{3^{2} + 4^{2} }[/tex]

[tex]l = \sqrt{9 +16} \\\\l= \sqrt{25} \\\\l = 5 miles[/tex]

The resultant displacement between the car and the truck at that same moment is 5 miles.

From the velocity vector diagram on Fig 2,

The resultant velocity R is given as

[tex]R = \sqrt{45^{2} + 55^{2} }\\\\R = \sqrt{2025 + 3025 }\\\\R = \sqrt{5050 }\\\\R = 71.06mph[/tex]

Therefore, the distance 5 miles North-East of the intersection between the car and the truck increasing at 71.06 miles per hour at that moment.

Answer:

The probability of cured people in who took the remedy is 8/9.

Step-by-step explanation:

Success rate of the cold remedy = 88%

The number of people who took the remedy = 45

Now, 88% of 45 = [tex]\frac{88}{100}  \times 45 = 39.6[/tex]

and 39.6 ≈ 40

So, out of 45 people, the remedy worked on total 40 people.

Now, let E: Event of people being cured by cold remedy

Favorable outcomes = 40

[tex]\textrm{Probability of Event E}  = \frac{\textrm{Total number of favorable outcomes}}{\textrm{Total outcomes}}[/tex]

or, [tex]\textrm{Probability of people getting cured}  = \frac{\textrm{40}}{\textrm{45}}[/tex]  = [tex]\frac{8}{9}[/tex]

Hence, the probability of cured people in who took the remedy is 8/9.

Final answer:

To find the probability of at least four patients out of 25 actually having the flu, the binomial probability formula is used. For the expected number of flu cases, the probability (4%) is multiplied by the number of patients (25).

Explanation:

The question pertains to the calculation of probabilities regarding how many patients actually have the flu given a certain success rate of 4% for flu diagnosis among those reporting symptoms. Specifically, we are asked to find the probability that at least four out of the next 25 patients calling in actually have the flu. To solve this, we would use the binomial probability formula:

P(X ≥ k) = 1 - ∑ [ P(X = i) where i goes from 0 to k-1 ]

In this case, X represents the number of patients who actually have the flu, k is the number we are interested in (at least four), and P(X = i) is the probability that exactly i patients have the flu. To express the distribution of X, we use a binomial distribution as we are dealing with a fixed number of independent trials, each with two possible outcomes (having the flu or not).

To determine the expected number of patients with the flu, we would multiply the probability of an individual having the flu (4%) by the number of patients calling in, which is 25.

Expected number of patients with the flu = 25 * 0.04 = 1

The depth of the water in the cylindrical tank is increasing at a rate of 0.1 m/min. This result is gotten by using the concept of related rates in calculus and differentiating volume with respect to time. The given rate of water inflow and dimensions of the cylindrical tank are substituted into the resultant formula.

This question is dealing with rates of change, specifically in the context of volume and height within a cylindrical tank. It's a problem in calculus, more specifically related to related rates.

To start, let's understand that the volume V of a cylinder is given by the formula V = πr²h, where r is the radius, h is the height, and π is a constant (~3.14159). We can substitute r with 4m (half of the diameter of 8m) and differentiate both sides with respect to time. Differentiating both sides of the equation with respect to t (time) gives dV/dt =πr² dh/dt (since r is constant with respect to time, but h changes).

Given that the rate at which water flows into the tank, dV/dt, is 5m³/min we can substitute this into the formula. This gives us 5 = π(4)²dh/dt. Simplifying the equation gives us dh/dt = 5 / (π(4)²), which approximately equals 0.1 m/min when rounded to three decimal places.

So, the depth of the water in the tank is increasing at a rate of 0.1 m/min.

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The rate at which the depth of water is rising in the cylindrical tank with a diameter of 8 m and water flowing in at 5 m^3/min is approximately 0.199 m/min.

To find the rate at which the depth of the water is rising, we can use the formula for the volume of a cylinder:

where:

Differentiating both sides of the equation with respect to time \( t \) gives us:

Given that the diameter is 8 m, the radius \( r \) is 4 m. We are also given that the rate of change of volume dV/dt is 5 m³/min. Plugging in these values, we can solve for \( dh/dt \), the rate at which the depth of the water is rising:

Calculating this gives:

[tex]\[ dh/dt \approx 0.199 \text{ m/min} \][/tex]

So, the rate at which the depth of the water is rising is approximately 0.199 m/min.