Please help I need all of this in alphabetical order

Answer:

61 degrees

Step-by-step explanation:

Triangle interior measures add up to 180 degrees.

61 + 58 + x = 180

119 + x = 180

x = 61

hope this helps :) !!!

The inverse for each relation:

1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)} - {(-2, 1), (3, 2), (-3, 3), (2, 4)}

2. {(4,2),(5,1),(6,0),(7,‐1)} - {(2, 4), (1, 5), (0, 6), (-1, 7)}

3. Inverse equation: y=(-1/8)x+3/8

4. Inverse equation: y=3/2x+15/2

5. Inverse equation: y=2x-20

6. Inverse equation: y=[tex]x^{(1/2)}+3[/tex]

7. Since fog(x) = gof(x) = x, f and g are inverse functions.

8. Since fog(x) = gof(x) = x, f and g are inverse functions.

1. To find the inverse of the relation, we need to swap the positions of x and y for each point and then solve for y.

{(1, -2), (2, 3), (3, -3), (4, 2)}

Inverse: {(-2, 1), (3, 2), (-3, 3), (2, 4)}

2. Again, we swap x and y and solve for y.

{(4, 2), (5, 1), (6, 0), (7, -1)}

Inverse: {(2, 4), (1, 5), (0, 6), (-1, 7)}

3. To find the inverse equation for y=-8x+3, we swap x and y and solve for y.

x=-8y+3

x-3=-8y

y=(x-3)/-8

Inverse equation: y=(-1/8)x+3/8

4. To find the inverse equation for y=2/3x-5, we swap x and y and solve for y.

x=2/3y-5

x+5=2/3y

y=3/2(x+5)

Inverse equation: y=3/2x+15/2

5. To find the inverse equation for y=1/2x+10, we swap x and y and solve for y.

x=1/2y+10

x-10=1/2y

y=2(x-10)

Inverse equation: y=2x-20

6. To find the inverse equation for y=(x-3)², we swap x and y and solve for y.

x=(y-3)²

[tex]x^{(1/2)}=y-3[/tex]

[tex]y=x^{(1/2)}+3[/tex]

Inverse equation: [tex]y=x^{(1/2)}+3[/tex]

7. To verify that f(x)=5x+2 and g(x)=(x-2)/5 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.

fog(x) = f(g(x)) = f((x-2)/5) = 5((x-2)/5) + 2 = x

gof(x) = g(f(x)) = g(5x+2) = ((5x+2)-2)/5 = x/5

Since fog(x) = gof(x) = x, f and g are inverse functions.

8. To verify that f(x)=1/2x-7 and g(x)=2x+14 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.

fog(x) = f(g(x)) = f(2x+14) = 1/2(2x+14) - 7 = x

gof(x) = g(f(x)) = g(1/2x-7) = 2(1/2x-7) + 14 = x

Since fog(x) = gof(x) = x, f and g are inverse functions.

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(Explanation below)

x=2

x = 2 is the solution of the equation

Two or more expressions with an Equal sign is called as Equation.

The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]

We have to find the value of x

Subtracting 3 from both sides:

[tex](3X-5)^(^1^/^4^) = 1[/tex]

Raising both sides to the fourth power:

3X - 5 = 1^4

3X - 5 = 1

Adding 5 to both sides:

3X = 6

Dividing by 3:

X = 2

Therefore, x = 2 is the solution of the equation

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77.6% of patients in the antibiotic treatment group experienced improvement in symptoms, while 80.2% of patients in the placebo group experienced improvement. The control group had a slightly higher percentage of improvement. The placebo effect could have contributed to the difference in improvement rates.

The percent of patients in the treatment group who experienced improvement in symptoms is 77.6% ((66/85) x 100). The percent of patients in the control group who experienced improvement in symptoms is 80.2% ((65/81) x 100).

The control group had a higher percentage of patients experience improvement in symptoms (80.2%) compared to the treatment group (77.6%).

One possible explanation for the observed difference between the percentages of patients in the antibiotic and placebo treatment groups that experience improvement in symptoms of sinusitis is that the placebo effect may have played a role.

The placebo effect is a phenomenon in which patients who receive a treatment that is not expected to have a therapeutic effect experience an improvement in their symptoms due to their belief in the treatment.

Therefore, the symptomatic treatments provided in the placebo group may have led to an improvement in symptoms, even though they did not receive an antibiotic.

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Each breath mint costs $0.78. The correct answer is C.

To solve this problem, we can use the concept of a system of linear equations. Let x be the cost of one breath mint and y be the total amount of money Shari and Jamal had.

We know that Shari bought 3 breath mints and received $2.76 change, so her equation will be:
3x + 2.76 = y

Jamal bought 5 breath mints and received $1.20 change, so his equation will be:
5x + 1.20 = y

Now we have a system of two equations with two variables:
3x + 2.76 = y
5x + 1.20 = y

We can solve for x by setting the two equations equal to each other:
3x + 2.76 = 5x + 1.20

Now, solve for x:
2x = 1.56
x = 0.78

So, each breath mint costs $0.78. The correct answer is C.

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To show that Maria's claim is incorrect, we need to find a fraction that is located between 1/5 and 1/7 on a number line but does not have a denominator of 6.

One way to do this is to find the least common multiple (LCM) of 5 and 7, which is 35, and then find a fraction with a denominator of 35 that falls between 1/5 and 1/7.

To do this, we can find the equivalent fractions of 1/5 and 1/7 with a denominator of 35:

1/5 = 7/35

1/7 = 5/35

Now we need to find a fraction between 7/35 and 5/35. One such fraction is:

6/35

This fraction is located between 7/35 and 5/35 on the number line, but its denominator is 35, not 6. Therefore, Maria's claim is incorrect.

Another way to show that Maria's claim is incorrect is to find a counterexample by simply listing all the fractions between 1/5 and 1/7 and showing that not all of them have a denominator of 6. For example:

1/6, 1/7, 1/8, 1/9, 1/10, ..., 1/34, 1/35

As we can see, not all of these fractions have a denominator of 6, so Maria's claim is incorrect.

Answer:

13/70

Step-by-step explanation:

In order to  show that Maria's claim is incorrect, we need to find a fraction that is located between 1/5 and 1/7 on a number line,  but does not have a denominator of 6.

Let's  find the  common multiple (CM) of 5 and 7, which is 70, or 35. But this case try 70  and then find a fraction with a denominator of 70 that falls between 1/5 and 1/7.

equivalent fractions of 1/5 and 1/7 with a denominator of 70
1/7 < x < 1/5 , will be equivalent to 1/7 ( 10/10 ) < x < 1/5 ( (14/14)
10/70 < x < 14/70..
x is the fraction   between 10/70  and 14 /70.  Unknown  fraction is:

13/70

This fraction is located between 10/70  and 14/70  on the number line, but its denominator is 70 , not 6. Therefore, Maria's claim is incorrect.

Note that the first coordinate of M is -1. (Option D)

Given :- coordinates of F = (-4,-2) = (x1,y1)

coordinates  of G = (2,-2) = (x2,y2)

coordinates of P = (2,-8) = (x3,y3)

and distance between point M and P is half of the distance between FG

To find :- first coordinate of point M

solution :- let the coordinate of M be (x4,y4)

as we know that distance between of the opposite point of parallel line segments are equal

so, second coordinate of M = -8

now by distance formula

FG = √(x2-x1)² + (y2-y1²)

= √[2-(-4)]² + [-2-(-2)]²

= √(2+4)² + (2-2)²

= √(6)² + (0)²

=√36

F G = 6

so, distance between point M and P = 1/2 × F G

= 1/2 × 6

= 3units

again, by distance formula

MP = √(x3-x⁴)² + (y3-y4)²

3 = √(2-x⁴)² + [-8-(-8)]²

squaring on both side

(3)² = (√(2-x⁴)² + [-8-(-8)]²)²

9 = (2-x⁴)² + [-8-(-8)]²

9 = (2-x⁴)²+(0)²

9 = (2-x⁴)²

√9 = 2-x⁴

3 = 2-x⁴

x⁴ = 2-3

x⁴ = -1

Hence the first coordinate of M is -1

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Point M is located in the third quadrant.

The distance between point M and point P is half the distance between point F and point G.

• Segment MP is parallel to segment FG. What is the first coordinate of point M?

The area of the regular hexagon is 509.2 square units (to the nearest tenth).

The formula for the area of a regular polygon is:

[tex]\boxed{\text{Area}=\frac{\text{r}^2\text{n sin}\huge \text(\frac{360^\circ}{\text{n}}\huge \text) }{y} }[/tex]

where:

From inspection of the given regular polygon:

Substitute the values into the formula and solve for area:

[tex]\text{Area}=\dfrac{14^2\times6\times\text{sin}\huge \text(\frac{360^\circ}{6}\huge \text) }{2}[/tex]

        [tex]=\dfrac{196\times6\times\text{sin} (60^\circ)}{2}[/tex]

        [tex]=\dfrac{1176\times\frac{\sqrt{3} }{2} }{2}[/tex]

        [tex]=\dfrac{588\sqrt{3} }{2}[/tex]

        [tex]=294\sqrt{3}[/tex]

        [tex]=509.2 \ \text{square units (nearest tenth)}[/tex]

Therefore, the area of the regular hexagon is 509.2 square units (to the nearest tenth).

To solve the equation, we need to first simplify both sides:

(4x - 6)/5 + 1 = (x + 1)/5 - 2/5

Multiplying both sides by 5 to eliminate the denominator:

4x - 6 + 5 = x + 1 - 2

Simplifying further:

4x - 1 = x - 1

Subtracting x from both sides:

3x - 1 = -1

Adding 1 to both sides:

3x = 0

Dividing both sides by 3:

x = 0

Therefore, the solution to the equation is x = 0.

Answer:  x=28

Step-by-step explanation:

Given:      <A=68

Find:     x

Reasoning:  

<B = 2x+x

<B= 3x

<C=x     they say the sides across from <C is same as other side so the

             angles are the same

Solution:

All angles of a triangle =180

<A + <B + <C =180    >substitute

68 + 3x + x =180      > combine like terms

68 + 4x = 180           > subtract 68 from both sides

4x=112                       >divide both sides by 4

x=28

the resulting 90% confidence interval for the average cost for healthcare of a senior citizen receiving Medicare  is [$12,000, $14,000].

The estimated average cost of providing healthcare for a senior citizen receiving Medicare is $13,000 per year, and the margin of error for this estimate is $1,000 with a 90% confidence level.

To find the confidence interval, we need to add and subtract the margin of error from the estimated mean.

Lower Limit = Estimated Mean - Margin of Error

Lower Limit = 13,000 - 1,000

Lower Limit = 12,000

Upper Limit = Estimated Mean + Margin of Error

Upper Limit = 13,000 + 1,000

Upper Limit = 14,000

Therefore, the resulting 90% confidence interval for the average cost for healthcare of a senior citizen receiving Medicare is [$12,000, $14,000]. This means we are 90% confident that the true mean cost of providing healthcare for a senior citizen receiving Medicare is between $12,000 and $14,000 per year.

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The probability that both marbles drawn will be green is 16.4%.

The probability of drawing a green marble on the first draw is 8/19.

Since there are no replacements, the probability of drawing another green marble on the second draw is 7/18 (since there are now only 18 marbles left in the bag, including 7 green marbles).

Therefore, the probability of drawing two green marbles in a row is:

(8/19) × (7/18)

= 56/342

To convert this to a percentage, we can divide 56 by 342 and multiply by 100:

(56/342) × 100 = 16.37%

Therefore, the probability that both marbles drawn will be green is 16.4%.

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If Ofra tried to solve an equation 3x = 4.5, The statement "Ofra correctly solved the equation" is correct. So, correct option is C.

We can see this by substituting x = 1.5 into the original equation 3x = 4.5:

3(1.5) = 4.5

Simplifying the left-hand side, we get:

4.5 = 4.5

This is a true statement, which means that x = 1.5 is a valid solution to the equation 3x = 4.5.

Therefore, Ofra did not make any mistakes in solving the equation. She correctly set up the equation 3x = 4.5 by multiplying both sides by 3 to isolate x, and then calculated the value of x to be 1.5, which is the correct solution.

Option (c) is the correct answer.

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Complete question is:

Ofra tried to solve an equation.

3x = 4.5, Setting up x = 1.5 Calculating

Where did Ofra make her first mistake?

Choose 1 answer:

a) Setting up

b)  Calculating

c) Ofra correctly solved the equation.

Answer:

1,215,000 cubic centimeters

Step-by-step explanation:

1. Find the volume of the cylinder

v = π r (squared) x h

v = 3.14 x 50 (squared) x 100

v = 3.14 x 2,500 x 100

v = 3.14 x 250,00

v = 785,000 cubic centimeters

2. Find the volume of the rectangular prism

v = l x w x h

v = 100 x 200 x 100

v = 2,000,000 cubic centimeters

3. Subtract

2,000,000 - 785,000 = 1,215,000 cubic centimeters

The average depth of the Indian Ocean is 3.7 time greater than that of Arctic Ocean. Therefore, the correct option is A.

To find how many times as great the average depth of the Indian Ocean is compared to the Arctic Ocean, we need to divide the average depth of the Indian Ocean by the average depth of the Arctic Ocean.

1: Divide the average depth of the Indian Ocean by the average depth of the Arctic Ocean:

3900 meters (Indian Ocean) / 1050 meters (Arctic Ocean) = 3.7142857

2: Round the result to the nearest tenth:

3.7

So, the average depth of the Indian Ocean is approximately 3.7 times greater than the average depth of the Arctic Ocean. The correct answer is A. 3.7.

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2(2.5) + 5(23) = 150

3(2.5) + 4(23) = 144.5

Both equations are satisfied, so our solution is correct.

To solve the problem, let's first assign some variables. Let x be the cost of one pine tree and y be the cost of one hydrangea bush. We can then use these variables to set up a system of equations:

2x + 5y = 150 (equation 1)

3x + 4y = 144.5 (equation 2)

We can solve this system of equations using various methods. Here, we will use the substitution method.

From equation 1, we can solve for x in terms of y:

2x = 150 - 5y

x = (150 - 5y)/2

We can then substitute this expression for x into equation 2:

3((150 - 5y)/2) + 4y = 144.5

Multiplying both sides by 2 to eliminate the fraction:

3(150 - 5y) + 8y = 289

Expanding and simplifying:

450 - 15y + 8y = 289

-7y = -161

y = 23

We can now substitute this value for y into either equation 1 or 2 to solve for x:

2x + 5(23) = 150

2x = 5

x = 2.5

Therefore, one pine tree costs $2.50 and one hydrangea bush costs $23.

To check our work, we can substitute these values into both equations:

2(2.5) + 5(23) = 150

3(2.5) + 4(23) = 144.5

Both equations are satisfied, so our solution is correct.

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The coordinates of point M come out to be 4.8, 4.4

This case is solved by using the section formula which states that

The coordinate of point P that divides the line segment AB in the ratio of m:n where the coordinate of A is [tex]x_1,y_1[/tex] and the B is [tex]x_2,y_2[/tex] is described as

[tex]\frac{mx_2+nx_1}{m+n}[/tex],[tex]\frac{my_2+ny_1}{m+n}[/tex]

The line to be divided = PQ

Coordinates of P = (6,2)

Coordinates of Q = (3,8)

Ratio = 2:3

Thus, the coordinates of M = [tex]\frac{2*3+3*6}{2+3}[/tex],[tex]\frac{8*2+2*3}{2+3}[/tex]

= 24/5 , 22/5

= 4.8, 4.4

Point M with coordinates (4.8,4.4) lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3

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90 + 63 = 153

this is because the pink square means that degree is 90°

To find the value of tan(θ), we first need to calculate the values of sine and cosine for the given point (4, 3) terminal side. We can use the Pythagorean theorem to find the length of the hypotenuse (r):

r = √((4)^2 + (3)^2) = √(16 + 9) = √25 = 5

Now, we can find sin(θ) and cos(θ) at the terminal side:

sin(θ) = opposite/hypotenuse = 3/5
cos(θ) = adjacent/hypotenuse = 4/5

Then, we can calculate tan(θ):

tan(θ) = sin(θ) / cos(θ) = (3/5) / (4/5) = 3/4

Now we need to find tan(2θ). We can use the double-angle formula for tangent:

tan(2θ) = (2 * tan(θ)) / (1 - tan^2(θ))

Substitute the value of tan(θ):

tan(2θ) = (2 * (3/4)) / (1 - (3/4)^2) = (3/2) / (1 - 9/16) = (3/2) / (7/16)

Now, we'll multiply by the reciprocal to solve for tan(2θ):

tan(2θ) = (3/2) * (16/7) = 24/7

So, tan2θ = 24/7. Your answer is: 24/7

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The statement on finding the areas of a trapezoid and a kite are True.

To determine the area of a trapezoid, it can be broken down into separate geometrical shapes. One possible breakdown would include a rectangle with two adjacent right triangles or an isosceles triangle with one right triangle configuration. By calculating each smaller compartment's size and summing them together, one can obtain the total area for the trapezoid.

Similarly, in order to find the surface area of a kite shape, drawing a diagonal creates two adjoining triangles that are easily computed individually then summed.

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Options for this question :
True

False

As k goes to infinity, the expression (k / (k+1)) approaches 1. Therefore, the limit becomes: lim (k -> infinity) 10 * (1^k) = 10
Since the limit is greater than 1, the Ratio Test indicates that the series is divergent.

To determine if the given series is convergent or divergent, we can use the Ratio Test. The series is given by:

Σ(5^(k-1) * 2^(k+1) / k^k) from k=1 to infinity

First, let's find the ratio of consecutive terms, a_(k+1)/a_k:

a_(k+1)/a_k = [(5^k * 2^(k+2)) / (k+1)^(k+1)] * [k^k / (5^(k-1) * 2^(k+1))]

Now, let's simplify the expression:

a_(k+1)/a_k = (5 * 2) * (k^k / (k+1)^(k+1))

Now, let's take the limit as k goes to infinity:

lim (k -> infinity) a_(k+1)/a_k = lim (k -> infinity) 10 * (k^k / (k+1)^(k+1))

We can rewrite the expression as:

lim (k -> infinity) 10 * ((k / (k+1))^k)

As k goes to infinity, the expression (k / (k+1)) approaches 1. Therefore, the limit becomes:

lim (k -> infinity) 10 * (1^k) = 10

Since the limit is greater than 1, the Ratio Test indicates that the series is divergent.

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The random variable in the first situation is the number of hits for the players of a baseball team and in the second situation is the distance traveled by the tee shots in a golf game.

1) The random variable in this distribution is the number of hits for the players of a baseball team. This is a discrete random variable because hits are counted as whole numbers and cannot take on non-integer values.

2) The random variable in this distribution is the distance traveled by the tee shots in a golf game. This is a continuous random variable because the distances traveled can take on any value within a certain range, including non-integer values. The exact distance traveled by a tee shot can be measured to any degree of precision, and there are infinitely many possible distances within the range of possible outcomes. Therefore, it is a continuous random variable.

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Hunter assumed he would only get 64 problems correct on his test, but he actually got 78 correct, So his percent error is 18%.

To calculate Hunter's percent error, we'll use the given formula:

Percent error = ((Prediction - Actual) / Actual) x 100

Prediction = 64 (the number of problems Hunter assumed he would get correct)
Actual = 78 (the number of problems he actually got correct)

Now, plug in the values:

Percent error = ((64 - 78) / 78) x 100
Percent error = (-14 / 78) x 100
Percent error ≈ -17.95%

Since percent error is typically expressed as a positive value, we can round to the nearest percent and report it as:

Percent error ≈ 18%

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Answer:

60%

Step-by-step explanation:

Take 96 and divide it by 160.

(easier if done on a calculator.)

For example: Find A/B as a percentage: take "A" and divide it by "B"

The correct expression for the volume of the prism is:

V = (1/2)(x)(x + 7)(x + 1)

This expression is derived from the formula for the volume of a prism, which is V = Bh, where B is the area of the base and h is the height of the prism. For a right triangle base, the area is equal to half the product of the base and height, or (1/2)(b)(l). Substituting the given values, we get:

B = (1/2)(x)(x + 7)

h = x + 1

Multiplying B and h together and simplifying, we get:

V = (1/2)(x)(x + 7)(x + 1)

Therefore, this is the correct expression for the volume of the given prism.

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a) P(x) = R(x) - C(x) = 6x - (0.001x^2 + 18x + 40) = -0.001x^2 - 12x - 40

b) R(200) = 6(200) = 1200
  C(200) = 0.001(200)^2 + 18(200) + 40 = 4000
  P(200) = R(200) - C(200) = 1200 - 4000 = -2800

c) R'(x) = 6
  C'(x) = 0.002x + 18
  P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18) = -0.002x - 12

d) R'(200) = 6
  C'(200) = 0.002(200) + 18 = 18.4
  P'(200) = -0.002(200) - 12 = -12.4

Here are the answers to each part:

a) P(x) is the profit function, which is calculated as the difference between the revenue function and the cost function: P(x) = R(x) - C(x). In this case, P(x) = 6x - (0.001x^2 + 18x + 40).

b) To find R(200), C(200), and P(200), plug x = 200 into each function:
R(200) = 6(200) = 1200
C(200) = 0.001(200^2) + 18(200) + 40 = 7600
P(200) = 1200 - 7600 = -6400

c) To find R'(x), C'(x), and P'(x), we need to find the derivative of each function with respect to x:
R'(x) = d(6x)/dx = 6
C'(x) = d(0.001x^2 + 18x + 40)/dx = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18)

d) To find R'(200), C'(200), and P'(200), plug x = 200 into each derivative function:
R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = 6 - 18.4 = -12.4

I hope this helps! Let me know if you have any further questions.

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The mean travel time is 60 minutes, the standard deviation is approximately 6.08 minutes, and the probability that Alice will not be late for her class is 0.50 or 50%.

To find the mean travel time, we need to add up the mean times for each stage of Alice's journey. Let's calculate it step by step:

Step 1: Alice's walking time from home to the bus stop:

Mean walking time = 13 minutes

Step 2: Waiting time for the bus to arrive:

Mean waiting time = 5 minutes

Step 3: Bus journey time from the bus loop:

Mean bus journey time = 24 minutes

Step 4: Walking time from the bus loop to the lecture theatre:

Mean walking time = 18 minutes

Now, let's calculate the total mean travel time:

Mean travel time = Mean walking time + Mean waiting time + Mean bus journey time + Mean walking time

= 13 + 5 + 24 + 18

= 60 minutes

So, the mean travel time is 60 minutes.

To find the standard deviation of Alice's travel time, we need to calculate the variance for each stage and then sum them up. Finally, we take the square root to get the standard deviation. Let's calculate it step by step:

Step 1: Alice's walking time from home to the bus stop:

The variance of walking time = 4 minutes squared

Step 2: Waiting time for the bus to arrive:

The standard deviation of waiting time = 2 minutes

Step 3: Bus journey time from the bus loop:

The standard deviation of bus journey time = 5 minutes

Step 4: Walking time from the bus loop to the lecture theatre:

The variance of walking time = 4 minutes squared

Now, let's calculate the total variance of travel time:

Variance of travel time = Variance of walking time + Variance of waiting time + Variance of bus journey time + Variance of walking time

= 4 + 4 + 25 + 4

= 37 minutes squared

Finally, the standard deviation of travel time is the square root of the variance:

The standard deviation of travel time = [tex]\sqrt(37)[/tex]

≈ 6.08 minutes (rounded to 2 decimal places)

So, the standard deviation of Alice's travel time is approximately 6.08 minutes.

To find the probability that Alice will not be late for her class, we need to calculate the z-score for the desired arrival time and then find the corresponding probability from the standard normal distribution table. Let's calculate it step by step:

Step 1: Calculate the total travel time from home to the lecture theatre:

Total travel time = Mean travel time = 60 minutes

Step 2: Calculate the difference between the desired arrival time and the total travel time:

Time difference = 8 am - 7 am = 1 hour = 60 minutes

Step 3: Calculate the z-score using the formula:

z = (Time difference - Mean travel time) / Standard deviation of travel time

z = [tex]\frac{(60 - 60) }{ 6.08}[/tex]

z = 0

Step 4: Find the probability corresponding to the z-score from the standard normal distribution table.

Since the z-score is 0, the probability is 0.50 (or 50%).

Therefore, the probability (to 2 decimal places) that Alice will not be late for her class is 0.50 or 50%.

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The equation is (x^2 + y^2 - 38x + 6y = -369).

The center of the lake is 19 miles east and 3 miles south of the town hall, which means the coordinates of the center are (19,-3). The radius of the lake is 0.5 miles.

Using the standard equation of a circle, we have:

(x - h)^2 + (y - k)^2 = r^2

where (h,k) is the center of the circle and r is the radius.

Substituting the given values, we get: (x - 19)^2 + (y + 3)^2 = 0.5^2

Expanding the left side, we get: x^2 - 38x + 361 + y^2 + 6y + 9 = 0.25

Simplifying and rearranging terms, we get:

x^2 + y^2 - 38x + 6y + 369.25 = 0.25

Subtracting 369 from both sides, we get:

x^2 + y^2 - 38x + 6y = -369

Therefore, the equation that represents the lake on the map is:

(x - 19)^2 + (y + 3)^2 = 0.5^2, which can be simplified to (x^2 + y^2 - 38x + 6y = -369).

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