A neighborhood watch association surveyed 40 neighbors about their feelings of safety in the neighborhood. They will survey an additional 80 neighbors. Based on the information, predict how many of the 80 neighbors will feel safe?

Earl works a total of 3 days in a week. From Monday through Friday, he works in the bookstore on 1 day and in the athletic center on 2 days. On Saturday and Sunday, he cooks food on 50% of the days, which would be a total of 1 day. Therefore, he works a total of 3 days in a week.

To calculate the percentage of Monday through Friday that Earl works, we need to first calculate the total number of days in a week, which is 7. Then, we need to subtract  the weekend days, which are Saturday and Sunday, leaving us with 5 days.

Finally, we can calculate the percentage by dividing the number of days Earl works from Monday through Friday (which is 1) by the total number of weekdays (which is 5), and multiplying by 100. So, Earl works 20% of Monday through Friday.

In summary, Earl works 3 days in a week, 1 day in the bookstore and 2 days in the athletic center. He also cooks food on 1 day during the weekend. Earl works 20% of Monday through Friday.

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Answer:

13×X + 35= 100 (equation)

X=(100-35)/13 = 5 hours

Answer:

13x+35=100

5 hours

Step-by-step explanation:

New shoes= $100

Hours he makes= $13

Hours he worked= unknown so its "x"

Now it's 5

He already has= $35

13 x 5 = 65) + 35 = 100

If an 'I' statement is true, then the truth value of its corresponding E statement is equals to the true. So, option(a) is right answer of this problem.

Truth value : The truth value of a statement is either true or false, depending on whether the logic makes sense or not. Let us assume two statements, p : Ram goes to school daily

q : Ram will get good marks

We use implication for determining value. In logic, implication is relationship between different propositions in which the second proposition is a logical consequence of the first.

first case : If Ram does not go to school daily then he will get good marks. That is p is false and q is true, ¬ p --> q ⇒ true ( using implication rule)

second case : If Ram does not go to school daily then he will not get good marks. That is p is false and q is false , ¬ p --> ¬ q ⇒ true ( using implication rule)

Hence, required value of statement is true.

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a.  If the CD paid 12. 3% interest for the first 7 years, he balance be after the first 7 years will be $492.89.

b.  If the CD paid 6% interest for the next 10 years, the balance be after the first 17 years would be $784.98.

c.  If the CD paid 4. 1% interest for the remaining 13 years, the balance be after 30 years would be $1,265.59.

d. The difference between the appraised value of the guitar and the amount the original $150 would have earned in the CD is $2,784.41.

A. The annual interest rate for a CD that pays 12.3% interest compounded daily is 12.3%/365 ≈ 0.0337% per day. The balance after 7 years can be calculated using the formula:

Balance = $150 x (1 + 0.000337)^((365 x 7) / 365) ≈ $492.89

Rounding to the nearest cent, the balance after 7 years is $492.89.

B. After 7 years, the remaining term of the CD is 30 - 7 = 23 years. The annual interest rate for a CD that pays 6% interest compounded daily is 6%/365 ≈ 0.0164% per day. The balance after 17 years can be calculated using the formula:

Balance = $492.89 x (1 + 0.000164)^((365 x 10) / 365) ≈ $784.98

Rounding to the nearest cent, the balance after 17 years is $784.98.

C. After 17 years, the remaining term of the CD is 30 - 17 = 13 years. The annual interest rate for a CD that pays 4.1% interest compounded daily is 4.1%/365 ≈ 0.0112% per day. The balance after 30 years can be calculated using the formula:

Balance = $784.98 x (1 + 0.000112)^((365 x 13) / 365) ≈ $1,265.59

Rounding to the nearest cent, the balance after 30 years is $1,265.59.

D. The difference between the appraised value of the guitar and the amount the original $150 would have earned in the CD is:

$4,200 - $1,265.59 - $150 ≈ $2,784.41

Rounding to the nearest cent, the difference is $2,784.41.

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The line crosses the y-axis at -9 and has a slope of 4, meaning it increases 4 units in the y-direction for every 1 unit increase in the x-direction.

A graph is a visual representation of data, typically involving the use of points, lines, and curves to show how different values are related to each other. Graphs can be used to display a wide range of information, from mathematical functions and scientific data to business trends and social networks.

Here,

To graph y = 4x - 9, we can start by plotting the y-intercept, which is -9 on the y-axis. Then, we can use the slope of 4 to find other points on the line.

Slope = 4 can be written as rise/run = 4/1. This means that for every increase of 1 in the x-direction, the y-value increases by 4.

Starting from the y-intercept (-9,0), we can go up 4 units and to the right 1 unit to get the point (1,-5). We can continue this pattern and plot more points on the line, or we can simply draw a straight line through the y-intercept and (1,-5) to represent the equation y = 4x - 9.

Here's what the graph looks like:

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The domain of the rational function is: option (2)  {x ∈ ℝ | x ≠ -4, 3, 4}

A rational number is any number that can be expressed as the ratio or fraction of two integers, where the denominator is not zero. In other words, a rational number is a number that can be written in the form of p/q, where p and q are integers, and q is not equal to zero.

The domain of a rational function is the set of all real numbers for which the function is defined, and the denominator is not equal to zero. So, we need to find the values of x for which the denominator of the given rational function is not zero.

The denominator of the given rational function is:

x³ - 3x² - 16x + 48

We can factor this polynomial using synthetic division or polynomial long division:

x³- 3x² - 16x + 48 = (x - 4)(x - 3)(x + 4)

So, the denominator of the rational function is not defined when:

x - 4 = 0 or x - 3 = 0 or x + 4 = 0

Solving these equations, we get:

x = 4 or x = 3 or x = -4

Therefore, the domain of the given rational function is:

{x ∈ ℝ| x ≠ –4, 3, 4}

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Step-by-step explanation:

x = sqrt (y-9)     square both sides

x^2 = y-9           add 9 to both sides

y = x^2 + 9        <====== this parabola has a POSITIVE x^2 coefficient ( +1)...

                                      so it is bowl shaped and opens UPWARD

The colony will need to triple in number after about 7.58 days.

Since the number of bacteria doubles every 5 days, we can write the relationship between the number of bacteria and time as an exponential function:

N(t) = N0 x 2^(t/5)

where N0 is the initial number of bacteria and t is the time in days.

To find out how long it will take for the colony to triple in number, we need to solve the equation:

N(t) = 3N0

Substituting the expression for N(t) from above, we get:

N0 x 2^(t/5) = 3N0

Dividing both sides by N0, we get:

2^(t/5) = 3

Taking the logarithm of both sides (with base 2) gives:

t/5 = log2(3)

Multiplying both sides by 5, we get:

t = 5 x log2(3)

Using a calculator or a computer program to evaluate the logarithm, we get:

t ≈ 7.58

Therefore, the colony will need to triple in number after about 7.58 days.

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The potters are qualified for the home loan as their total monthly payment is $831.02, which is less than $1000.00.

The total cost of the cottage along with the annual insurance and taxes is $118,000 + $710 + $2800 = $121,510.

The down payment made by the potters is $14,000. Therefore, the amount to be financed through a mortgage is $121,510 - $14,000 = $107,510.

Using the formula for the monthly payment of a mortgage, which is given by:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]

where P is the principal (amount to be financed), i is the monthly interest rate, and n is the total number of monthly payments.

For a 5%, 15-year mortgage, the monthly interest rate is 0.05/12 = 0.0041667, and the total number of monthly payments is 15 x 12 = 180.

Plugging in the values, we get:

M = $107,510 [ 0.0041667 (1 + 0.0041667)^180 ] / [ (1 + 0.0041667)^180 - 1 ]

M = $831.02

Therefore, the total monthly payment for the mortgage and the annual insurance and taxes is $831.02 + $59.17 + $233.33 = $1123.52, which is more than the maximum allowed payment of $1000.00. Hence, the potters are qualified for the home loan.

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A nonspecific defense, also known as innate immunity, is the first line of defense that the body uses against pathogens, such as bacteria, viruses, and fungi.

This defense is considered nonspecific because it is not targeted towards a particular pathogen but rather provides a general defense mechanism that helps the body to prevent and fight infections. Examples of nonspecific defenses include physical barriers such as skin and mucous membranes, chemical barriers such as stomach acid and lysozyme, and cellular defenses such as phagocytic cells and natural killer cells.

2. The body's second line of defense is also part of the innate immune system and involves the activation of nonspecific leukocytes, including neutrophils, monocytes/macrophages, and natural killer cells. This defense takes effect when pathogens manage to breach the physical and chemical barriers of the body and enter into the tissues. These leukocytes then recognize and attack the invading pathogens through various mechanisms such as phagocytosis, release of cytotoxic substances, and activation of the complement system.

3. Nonspecific leukocytes play critical roles in the body's second line of defense by recognizing and attacking foreign pathogens. Neutrophils are the most abundant leukocytes in the body and are the first responders to infection. They are recruited to the site of infection and destroy pathogens through phagocytosis and release of antimicrobial substances. Monocytes/macrophages also play a crucial role in phagocytosis and release cytokines that activate other immune cells. Natural killer cells are responsible for recognizing and killing virus-infected cells and tumor cells.

Lastly, question 4. When Jera's skin became red and warm around the cut, it was a sign of inflammation, which is a normal response of the immune system to infection. Inflammation is characterized by redness, heat, swelling, and pain, and is caused by the release of chemicals such as histamine and cytokines in response to infection. The purpose of inflammation is to increase blood flow to the site of infection, bringing immune cells and nutrients to aid in the fight against the invading pathogens.

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Two triangles have perimeters that can be expressed as 12x + 12 and 8x - 2,  with a difference of 4x + 14, and have perimeters of 47 and 28 when x = 3.

a. The perimeter of each triangle can be found by adding up the lengths of all three sides:

Triangle 1: (4x+2) + (7x+7) + (x+3) = 12x + 12

Triangle 2: (2x-5) + (x+7) + (5x-4) = 8x - 2

b. To find the difference in perimeter between the larger triangle and the smaller triangle, we can subtract the smaller perimeter from the larger perimeter:

(12x + 12) - (8x - 2) = 4x + 14

c. To find the perimeter for each triangle when x = 3, we can substitute x = 3 into the expressions found in part (a):

Triangle 1: (4(3)+2) + (7(3)+7) + (3+3) = 47

Triangle 2: (2(3)-5) + (3+7) + (5(3)-4) = 28

Therefore, the perimeter of Triangle 1 is 47 units and the perimeter of Triangle 2 is 28 units when x = 3.

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Answer:

point form - (2,2)

x=2 y=2

For the first equation, we can use the product rule. Let f(x) = 2x+ and g(x) = 5x+2. Then, f'(x) = 2 and g'(x) = 5. Using the product rule formula, we get:

k'(x) = f(x)g'(x) + g(x)f'(x)
k'(x) = (2x+)(5) + (5x+2)(2)
k'(x) = 10x+ + 4x+10

For the profit function, P(x) = -2x^2 + 6x + 7, we can use the derivative to find the marginal profit. The derivative of P(x) is:

P'(x) = -4x + 6

To find the marginal profit when 4 units are produced and sold, we plug in x = 4 into P'(x):

P'(4) = -4(4) + 6
P'(4) = -10

Therefore, the marginal profit when 4 units are produced and sold is -10 hundred dollars or -$1,000. This means that for each additional unit produced and sold beyond the initial 4 units, the profit will decrease by $1,000.
Hi! It seems that your question has some missing information and typos, but I'll try my best to help you with the given data. The main task here is to find the marginal profit when 4 units are produced and sold, and interpret the answer.

The profit function P(x) appears to be incomplete. However, since we're asked to find the marginal profit, we need to take the derivative of the profit function with respect to x. Marginal profit is the derivative of the profit function with respect to the number of units sold (x). We can use the chain rule, product rule, or quotient rule to find the derivative, depending on the given profit function.

Once we have the derivative, we can evaluate it at x = 4 to find the marginal profit when 4 units are produced and sold. The result will indicate how much additional profit (in hundreds of dollars) the company will gain for producing and selling one more unit.

Please provide the complete profit function P(x), and I will be glad to help you find the marginal profit and interpret the answer.

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Tina walks to the library for her studies three times a week. During each visit, she walks 3/10 mile to the library and then walks 4/10 mile back home. Therefore, Tina walks a total of 1.4 miles each week for her library studies (3 times a week x (3/10 mile to library + 4/10 mile back home) = 1.4 miles).

If Tina wants to win a prize for walking, she would need to walk more than 1.4 miles per week. The amount of additional distance she needs to walk depends on the requirements for the prize.

For example, if the prize requires her to walk 2 miles per week, Tina would need to walk an additional 0.6 miles (2 miles - 1.4 miles) to meet the goal. This could be achieved by adding an extra walk to her routine or extending the distance of her existing walks.

It is important to note that walking is a great form of exercise and can have many benefits for overall health and well-being. By incorporating regular walks into her routine, Tina can improve her physical fitness and potentially achieve her goal of winning a prize.

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The word sentence for the given number line represents the X is greater than zero. The inequality is the X is not less than or equal to zero.

In the given number line the values are from negative infinity to positive infinity. Though the number line is having all values, there is some part that is shaded only on certain parts of the number line. The part is from number 1 to positive infinity marking zero as a circle.

The circle represents the value starts counting from the next step. In the given number line circle is shaded at number zero and the value started from the next value which is number 1 onwards to positive infinity.

So, the given number line represents X is greater than zero and X is not less than or equal to zero.

In mathematical words,

X < 0 and X≠0

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Given question is not having enough information to solve, for better understanding attaching the number line image below.

-43x < 79 - 16

To solve the inequality −43x + 16 < 79, we need to isolate the variable x.

Let's begin by subtracting 16 from both sides of the inequality:

−43x + 16 - 16 < 79 - 16

Simplifying the equation, we have:

−43x < 63

Next, we divide both sides of the inequality by -43. However, when we divide by a negative number, the direction of the inequality sign will be flipped:

x > 63 / -43

Simplifying further, we have:

x > -1.465

Therefore, the solution to the inequality is x > -1.465.

In interval notation, we can represent the solution as (-1.465, ∞), indicating that x is greater than -1.465 and extends indefinitely towards positive infinity.

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The answer of the given question based on the right triangular prism is , the surface area of the prism is,

(D) 2.6 + 12 + 8 + 8 = 30.

Since, A right triangular prism is a three-dimensional solid that has two congruent, parallel triangular faces (also known as bases) and three rectangular faces that connect the bases at right angles. The perpendicular distance between the two bases is called the height of the prism.

To find the surface area of the prism, we need to calculate the area of each of its faces and then add them up.

The prism has two congruent triangular faces and three rectangular faces.

The area of each triangular face is equal to one-half the product of the base and height, which gives:

1/2 x 3 x 12 = 18

The area of each rectangular face is equal to the product of its length and width, which gives:

8 x 3 = 24 (two of these faces)

12 x 8 = 96 (one of these faces)

Therefore, total surface area of  prism is given by:

2 x 18 + 2 x 24 + 96 = 60 + 96 = 156

So, the correct expression that represents the surface area of the prism is option,

(D) 2.6 + 12 + 8 + 8 = 30.

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The complete factorization of the expression is  (m+n)(x-y).

The complete factorization of the expression is determined as follows;

To factorize the expression (m+n)(2x-y)-x(m+n), we can first factor out the common factor (m+n):

(m+n)(2x-y)-x(m+n) = (m+n)(2x-y-x)

Next, we will factorize completely as follows;

2x - x - y = x - y

(m+n)(2x-y-x)  = (m+n)(x-y)

Therefore, the fully factorized form of the expression is (m+n)(x-y).

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The drive-thru is able to estimate their wait time more consistently will be A. Burger Quick, because it has a smaller IQR.

The interquartile range (IQR) is a measure of variability that represents the range of values in the middle 50% of scores in a distribution. It is used when the data is skewed and the median is a better measure of central tendency than the mean.

In this scenario, Burger Quick has an IQR of 14.5, while Fast Chicken has an IQR of 4.5.

The IQR is calculated by subtracting the first quartile (Q1) from the third quartile (Q3).

For Burger Quick, Q1 is 9.5 and Q3 is 24, so the IQR is 24 - 9.5 = 14.5. For

Fast Chicken, Q1 is 10 and Q3 is 14.5, so the IQR is 14.5 - 10 = 4.5.

Therefore, Burger Quick has a larger IQR than Fast Chicken, indicating that its data is more consistent and less spread out. This means that Burger Quick is better able to estimate its wait time consistently.

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The length, slope and midpoints of the segment in the drawing, obtained using the distance and midpoint formula are;

The length of [tex]\overline{AB}[/tex] = √(29)

The midpoint of [tex]\overline{AB}[/tex] is (3, -0.5)

The slope of segment [tex]\overline{CD}[/tex] = 2/5

The midpoint of segment [tex]\overline{CD}[/tex] = (-1.5, 2)

The slope of a segment on the coordinate plane is the ratio of the rise to the run of the segment.

The coordinates of the required points in the figure are; A(2, 2), B(4, -3), C(1, 3), and D(-4, 1)

The distance formula that can be used in finding the distance between points on the coordinate plane can be presented as follows;

d = √((x₂ - x₁)² + (y₂ - y₁)²)

The distance formula indicates that the length of [tex]\overline{AB}[/tex] can be found as follows;

[tex]\overline{AB}[/tex] = √((4 - 2)² + (-3 - 2)²) = √(29)

The midpoint formula indicates tha midpoint of the segment [tex]\overline {AB}[/tex] can be found as follows

The midpoint of [tex]\overline{AB}[/tex] = ((2 + 4)/2,  (2 + -3)/2) = (3, -0.5)

The slope of [tex]\overline{CD}[/tex] = ((3 - 1)/(1 - (-4)) = 2/5

The midpoint of [tex]\overline{CD}[/tex] = ((1 + (-4))/2, (3 + 1)/2 = (-1.5, 2)

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The measure of a central angle of a regular polygon with 7 sides is approximately 51.4 degrees.

For a regular polygon with 7 sides, we can substitute n = 7 into the formula:

central angle = 360 degrees / 7

central angle ≈ 51.4 degrees

A regular polygon is a polygon with equal sides and equal angles. The measure of each interior angle of a regular polygon with n sides is given by the formula:

interior angle = (n-2) x 180 degrees / n

For example, for a regular polygon with 7 sides:

interior angle = (7-2) x 180 degrees / 7

interior angle = 5 x 180 degrees / 7

interior angle ≈ 128.6 degrees

Since a central angle of a regular polygon is an angle formed by two consecutive radii from the center of the polygon, the measure of a central angle is equal to the measure of the exterior angle.

The measure of an exterior angle of a regular polygon with n sides is given by the formula:

exterior angle = 360 degrees / n

For a regular polygon with 7 sides, we can use the formula above to find the measure of each exterior angle:

exterior angle = 360 degrees / 7

exterior angle ≈ 51.4 degrees

Therefore, the measure of a central angle of a regular polygon with 7 sides is approximately 51.4 degrees, rounded to the nearest tenth of a degree as requested.

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7) Roy buys 5 whole pizzas.

8) There are 36 gifts altogether.

9) There is 5 cows and 8 chickens.

10) He travel 210 km.

We have the four parts of question:

Now, We have the information:

7) A whole pizza costs P 190.00 and P 40. 00 for every additional topping.

If he spent P 1070 for pizza with 3 sets of additional toppings.

So, The formation of equation is:

=> (1070 - 3 × 40) ÷ 190

=> 5

8) Total gifts are : 4

and Inside each large gift are 2 medium-sized gifts and inside each

medium-sized gift are 3 small gifts.

So, the formation of equation;

=> 4 × 2 = 8

=> 8 × 3 = 24

The total is:

8 + 24 + 4 = 36

9) There are 13 heads and 36 feet .

We know:

Cows have (C) 4 legs

Chickens have (c)  2 legs

So, The equation will be:

C + c=13   ....eq.(1)

Multiply by 4, we get:

4C+2c=36

Then,

4C+4c=52

4C+2c=36

--------------- (-)

2c=16

c = 8 chickens

We put the value of c in eq.(1), we get  

C=13 - 8

C = 5 cows

Hence, There is 5 cows and 8 chickens.

10) Alexander travelled at 6:00 a.m.

and, average speed is 70 km per hour.

So, He travelled 140 km in 2 hours.

After two hours of driving, he stopped for 30 minutes for a rest.

He continued driving and reach his hometown at 9:30 a.m.

So, therefore he continues driving at 8:30

then he arrived at his destination at 9:30

So after his break for driving he travels 1 more hour

So in total he travelled 3 hours to arrive at his destination

3 × 70 = 210.

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The options that are statistical are:

how many pets do you have?

what time do you go to sleep?

how many siblings do you have?

The options that are non-statistical are:

what kind of dog does bruno have?

what is clara's brother's name?

what time did javed go to sleep?

A statistical question is defined as one that will obtain the data that will vary from one particular response to another. However, a non-statistical question is defined as one that will obtain data that is basically exact and then has only one response.

A question that will not provide a variety of different answers is referred to as not a statistical question. For example, we can say that 'how many siblings do I have?' is not referred to as statistical. The answer will definitely have just one response, and not many.

A non-statistical question in math is defined as a question that will not provide a variety of answers. Finally, non-statistical questions provide us with exact answers that do not change.

Thus, the options that are statistical are:

how many pets do you have?

what time do you go to sleep?

how many siblings do you have?

The options that are non-statistical are:

what kind of dog does bruno have?

what is clara's brother's name?

what time did javed go to sleep?

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The p-value of the test is 0.017 > 0.01, which means that there is not convincing evidence that the mean time behind the whell for all U.S drivers is less than 51 minutes.

We are told that:

Time spent by U.S. drivers on average each day = 51 minutes behind the wheel

Thus, the null hypothesis is:

H₀: μ = 51

The alternate hypothesis is:

Hₐ: μ < 51

The formula for the test statistic is:

z = (x' - μ)/(σ/√n)

where:

x' is sample mean

n is sample size

μ is population mean

σ is standard deviation

Thus:

z = (46.4 - 51)/(18.8/√75)

z = -2.12

From online p-value from z-score calculator, we have:

The p-value of the test is referred to as the probability of finding a sample mean that is less than 46.4, which is the p-value of z when X = 46.4.

Looking at the z-table, z = -2.12 has a p-value of 0.017.

Decision:

The p-value of the hypotheses test is seen to be0.017 > 0.01.

Thus, this tells us that we don't have convincing evidence that the mean time spent behind the wheel for all U.S drivers is less than 51 minutes.

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Answer:

3/5

Step-by-step explanation:

If it takes 15 minutes to complete one lap around the park, then the runner completes 1/15 of a lap in 1 minute.

To find out what fraction of a lap the runner completes in 9 minutes, we can multiply the fraction completed in 1 minute by the number of minutes elapsed:

Fraction completed in 9 minutes = (1/15) x 9 = 9/15 = 3/5

Therefore, the runner has completed 3/5 of a lap in 9 minutes

Sally's estimated maximum heart rate (MHR) is 199 beats per minute.

To estimate Sally's maximum heart rate (MHR), we will use the following formula:

MHR = 220 - age

1. First, note Sally's age, which is 21 years old.
2. Then, apply the formula: MHR = 220 - 21.
3. Calculate the result: MHR = 199 beats per minute.

Sally's estimated maximum heart rate (MHR) is 199 beats per minute.

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Answer:

divide by 100

Step-by-step explanation:

hope it helps

Convert 1 Kilograms to Meters (kg to m) with our conversion calculator and conversion tables. To convert 1 kg to m use direct conversion formula below.

Convert 1 Kilograms to Meters (kg to m) with our conversion calculator and conversion tables. To convert 1 kg to m use direct conversion formula below.1 kg = 1 m.

Convert 1 Kilograms to Meters (kg to m) with our conversion calculator and conversion tables. To convert 1 kg to m use direct conversion formula below.1 kg = 1 m.You also can convert 1 Kilograms to other Weight (popular) units.

[tex]Direct \: \: conversion \: \: formula: 1 Kilograms * 1 = 1 Meters[/tex]

The measure of angle A is 20 degrees, the measure of angle B is 65 degrees, and the measure of angle C is 95 degrees.

To find the measure of the angles in triangle ABC, we first need to determine the total ratio of measures.

The total ratio is 4 + 13 + 19 = 36.

Next, we can use the ratios to find the measure of each angle.

Let x be the measure of the smallest angle in triangle ABC.

Then the measures of the angles are:

Angle A = 4x
Angle B = 13x
Angle C = 19x

We know that the sum of the angles in a triangle is 180 degrees, so we can set up the equation:

4x + 13x + 19x = 180

Simplifying, we get:

36x = 180

Dividing both sides by 36, we get:

x = 5

Therefore, the measures of the angles in triangle ABC are:

Angle A = 4x = 4(5) = 20 degrees
Angle B = 13x = 13(5) = 65 degrees
Angle C = 19x = 19(5) = 95 degrees

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We can prove that 25k² + 15k is an even integer whenever 5k - 3 is an integer by considering two cases: when k is even and when k is odd.

Let's assume that 5k - 3 is an integer. Then, we can write k as k = (5k - 3 + 3)/5 = (5k - 3)/5 + 3/5. Since (5k - 3)/5 is an integer, we can write it as (5k - 3)/5 = n, where n is an integer. Thus, we have k = n + 3/5.

Now, we can substitute this expression for k into 25k² + 15k as follows:

25k² + 15k = 25(n + 3/5)² + 15(n + 3/5)

Expanding the square, we get:

25(n² + 6n/5 + 9/25) + 15n + 9 = 25n² + 45n/5 + 34/5

Simplifying, we get:

25k² + 15k = 5(5n² + 9n) + 34/5

Since 5n² + 9n is an integer, we can write it as m, where m is an integer. Thus, we have:

25k² + 15k = 5m + 34/5

Now, we can consider two cases:

Case 1: k is even. In this case, k can be written as k = 2p, where p is an integer. Substituting this expression into 5k - 3, we get:

5k - 3 = 5(2p) - 3 = 10p - 3

Since 10p is even, we can conclude that 10p - 3 is odd. Therefore, m must be odd, since 5m + 34/5 is even. Thus, 25k² + 15k is even, since it can be written as 5m + 34/5, where 5m is even and 34/5 is even.

Case 2: k is odd. In this case, k can be written as k = 2p + 1, where p is an integer. Substituting this expression into 5k - 3, we get:

5k - 3 = 5(2p + 1) - 3 = 10p + 2

Since 10p is even, we can conclude that 10p + 2 is even. Therefore, m must be even, since 5m + 34/5 is even. Thus, 25k² + 15k is even, since it can be written as 5m + 34/5, where 5m is even and 34/5 is even.

In both cases, we have shown that 25k² + 15k is an even integer whenever 5k - 3 is an integer. Therefore, the statement is proved.

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