Use the word bank and fill in the blanks. (Use each word only once) Word Bank: natural number, whole number, integer, rational, irrational

a) The coyote was out exercising for 55 minutes since it jogged for 45 minutes and did a cool-down walk for 10 minutes.

b) For his entire trip, the coyote moving in a straight line continually had a displacement of 5.75 miles.

Displacement refers to the change in the position of an object.

Displacement is also defined as the length of the shortest distance from the initial to the final position of a point under motion.

The time spent jogging = 45 minutes

The average jogging speed = 7 miles/hour

The distance covered during jogging = 5.25 miles (7 x 45/60)

The time the coyote spent walking = 10 minutes

The average walking speed = 3 miles/hour

The distance covered walking = 0.5 miles (3 x 10/60)

a) The total time spent exercising = 55 minutes (45 + 10).

b) The displacement (total distance) of the coyote in 55 minutes = 5.75 miles (5.25 + 0.5).

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The function of the power series f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

And a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

The common ratio is the distance between each number in a geometric series. The proportion of a number or two consecutive numbers. The common ratio, which is the same for all numbers or common, is the number divided by the number that comes before it in the sequence.

To find the power series for f(x), we first need to find the power series for g(x) = 6/(1+x):

g(x) = 6/(1+x) = 6(1 - x + x² - x³ +...) (geometric series with common ratio -x)

Next, we integrate term by term:

∫ g(x) dx = ∫ 6(1-x+x²-x³+...) dx

= 6(x - x²/2 + x³/3 - x⁴/4 + ...) + C , where C is a constant.

Since we're only interested in finding the coefficients of the power series, we can ignore the constant term.

Thus, the power series for f(x) is:

f(x) = 6ln(1+x) = 6(x - x²/2 + x³/3 - x⁴/4 + ...) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n

where a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

Therefore, f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

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1.

y= -5x + 4

slope intercept form is y = mx + b (m) being the slope and (b) being the y intercept

The appropriate formula for calculating the dose of diphenhydramine for a pediatric patient is weight (kg) x (age (years)/150) x 33 mg = dose (mg). In this case, the dose would be (9/150) x 33 mg = 2.2 mg, rounded to the nearest tenth of a milligram.

The most appropriate formula to calculate the dose for a pediatric patient is the Clark's rule formula. This formula is used to calculate the dose of a medication for a child based on their weight and the recommended adult dose. The formula is as follows:
Child's dose = (weight of child in pounds / 150) x adult dose
Using this formula, we can calculate the appropriate dose for the 9-year-old patient:
Child's dose = (weight of child in pounds / 150) x 33 mg
Without knowing the weight of the child, we cannot calculate the exact dose. However, once the weight is known, the formula can be used to calculate the appropriate dose and it can be rounded to the nearest tenth of a milligram as necessary.

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The correct statement regarding the approximate probability of calling 3 households in Virginia in a row that have motorcycles is given as follows:

The student can use a random number generator and assign numbers 1 and 2 to represent a household with a motorcycle and numbers 3 through 25 to represent a household without a motorcycle.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Since the probability is of 8%, the experiment is constructed as follows:

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Answer:

Yes, the triangle is right-angled since one of the angles is 90°

Step-by-step explanation:

The sum of the three angles of a triangle must add up to 180°

Here the three angles are 3x – 7, 4x – 1, and 5x + 20.

Adding them up gives

3x – 7 + 4x – 1 + 5x + 20 = 180

Grouping like terms:
3x + 4x + 5x - 7 - 1 + 20 = 180

12x + 12 = 180

12x = 180 - 12 = 168

x = 168/12 = 14

Substituting this value of x into each of the expressions:
3x – 7  = 3(14) - 7 = 42 - 7 = 35°

4x – 1 = 4(14) - 1 = 56 - 1 = 55°

5x + 20 = 5(14) + 20 = 70 + 20 = 90°

Since one of the angles is 90° it is indeed a right angled triangle

Option D: The number of patrons who prefer roller coasters will be 2 times the number of patrons who prefer food.​

The activities are given as follows:

The number of patrons for each activity is given as follows:

Hence the proportions are:

For the next 300 patrons, the estimates are given as follows:

Hence option d is correct.

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To find the inverse equation of f(x) = 3x + 5, we first replace f(x) with y:

y = 3x + 5

Next, we switch x and y and solve for y:

x = 3y + 5

x - 5 = 3y

y = (x - 5)/3

Therefore, the inverse of f(x) is f^-1(x) = (x - 5)/3.

To find the inverse equation of g(x) = x^2 - 6, we replace g(x) with y:

y = x^2 - 6

Next, we switch x and y and solve for y:

x = y^2 - 6

x + 6 = y^2

y = ±sqrt(x + 6)

Since we want a function, we take the positive square root:

y = sqrt(x + 6)

Therefore, the inverse of g(x) is g^-1(x) = sqrt(x + 6).

The relationship between a function and its inverse is such that if (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f^-1. In other words, the roles of x and y are reversed in the inverse function. The domain of the function becomes the range of the inverse, and the range of the function becomes the domain of the inverse.

In general, not all functions have inverses. For a function to have an inverse, it must be one-to-one (i.e., each x-value in the domain maps to a unique y-value in the range). If a function is not one-to-one, then it may be possible to restrict the domain to a smaller interval so that the restricted function does have an inverse. When a function has an inverse, the domain and range of the inverse follow a pattern as stated above.

Answer:

8183.27

Step-by-step explanation:

Answer:

8183.3

Step-by-step explanation:

500(1+0.15)20=8183.27 round it to become 8183.3

The vertex form of the equation for function g is g(x) = a(x + 1)² - 38, where a is the same as in function f.

A quadratic function is one of the forms f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

The vertex form of the equation of a quadratic function is given by:

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex. To find the vertex form of the equation for function g, we need to determine the vertex of function f and then adjust it to 2 units below and 3 units to the left.

To find the vertex of function f, we can use the formula:

h = -b / 2a

where a and b are the coefficients of the quadratic term and the linear term, respectively.

For function f(x) = x² - 4x - 32, we have a = 1 and b = -4, so the vertex has an x-coordinate of:

h = -b / 2a = -(-4) / 2(1) = 2

To find the y-coordinate of the vertex, we can substitute this value of h into the function:

f(2) = 2² - 4(2) - 32 = -36

So the vertex of function f is (2, -36).

To adjust the vertex of the function f 2 units below and 3 units to the left, we need to subtract 2 from the y-coordinate and add 3 to the x-coordinate. Therefore, the vertex of function g is:

(h, k) = (2 - 3, -36 - 2) = (-1, -38)

Therefore, the vertex form of the equation for function g is g(x) = a(x + 1)² - 38, where a is the same as in function f.

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Answer:8

Step-by-step explanation: I got 8 because when you multiply 4/5 and 10 together you put 10 over one then multiply across.

After that you will get 40/5 and that is 8.

Answer: practice 2.  8 inches   practice 3. 220[tex]ft^2[/tex]  

practice 4. 78° , 78° , 102° , 102 ° Practice 5. 12.5

Step-by-step explanation:

43. g(6) = 5602.33

44. f(5) = 61.

The exponential function is defined by the equation:g(x) = 31​(7)x − 2So, we need to evaluate the function g(x) at x = 6. Then, g(6) is given by;g(6) = 31​(7)6 − 2= 31​(7)4= 16807/3Thus, g(6) = 5602.33 (rounded to two decimal places).Similarly, the exponential function is defined by the equation:f(x) = 4(2)x − 1 − 2We need to evaluate the function f(x) at x = 5. Then, f(5) is given by;f(5) = 4(2)5 − 1 − 2= 4(16) − 1 − 2= 64 − 1 − 2= 61Thus, f(5) = 61.

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The mathematical expression for the given phrase using x as the variable is:

9x + (x+1)

We can simplify this expression by combining like terms:

9x + x + 1 = 10x + 1

Therefore, the expression is 10x + 1.

Answer: Bones break because so much force is applied onto them at a singular moment, more than can be handled. Most commonly from falls where you hit your bone in a specific spot.

Bones can break, also known as a fracture, due to a variety of reasons. Here are some common causes:

Trauma: A sudden force or impact, such as a fall, can cause a bone to break.

Overuse: Repeated stress on a bone over time can cause a stress fracture, which is a small crack in the bone.

Medical Conditions: Certain medical conditions, such as osteoporosis, cancer, or infections, can weaken bones and make them more susceptible to fractures.

Vitamin and Mineral Deficiencies: Insufficient levels of calcium, vitamin D, and other minerals necessary for strong bones can increase the risk of fractures.

The severity of a fracture can vary depending on the force of impact and the strength of the bone. Some fractures may only cause minor pain and swelling, while others may require surgery and a prolonged healing process.

It is important to seek medical attention if you suspect a fracture as prompt diagnosis and treatment can help prevent complications and promote healing. Treatment options for a fracture may include immobilization, casting, surgery, or physical therapy, depending on the severity and location of the break.

For a line perpendicular to y = - 5 x - 2 and passing through the point ( 10 , 5 ), the equation in slope-intercept form is y = (1/5)x + 3.

To write an equation for a line perpendicular to y = -5x - 2 and passing through the point (10, 5), we first need to find the slope of the new line. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. The slope of the original line is -5, so the slope of the new line is 1/5.

Next, we can use the point-slope form of an equation to write the equation for the new line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Plugging in the values we have, we get:

y - 5 = (1/5)(x - 10)

Finally, we can rearrange this equation to put it in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. To do this, we'll distribute the 1/5 and then add 5 to both sides of the equation:

y - 5 = (1/5)x - 2
y = (1/5)x + 3

So the equation for the new line is y = (1/5)x + 3. This is our final answer.

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[tex]\sqrt{26^{2} - 24^{2}} = \sqrt{(26-24)(26+24)} = \sqrt{2(50)} = \sqrt{100} = 10 \\[/tex]

[tex]x = \sqrt{10^{2} - 6^{2}} = \sqrt{100 -36} = \sqrt{64} = 8 \\[/tex]

[tex]\implies \bf x = 8[/tex]

Answer:

8

Step-by-step explanation:

Begin with the bigger right angle triangle

Hypotenuse is 26, second side is 24, third side is unknown (a)

26² - 24² = a²

676 - 576 =a²

100 = a²

Therefore a is 10

Now use the value of a which is 10 to solve the smaller triangle

Hypotenuse is 10, one side is 6, third side x is unknown

10²- 6²= x²

100 - 36 =x²

64 =x²

X is 8

Answer:

approximately 10.3 days to reach 500 bacteria

Step-by-step explanation:

Let's use the formula for exponential growth to write an equation that represents the situation: N = N0 * e^(rt)

where N is the number of bacteria, N0 is the initial number of bacteria, e is Euler's number (approximately 2.718), r is the growth rate, and t is the time in days.

We know that after 4 days, the number of bacteria is 71, so we can plug these values into the equation to solve for the growth rate: 71 = N0 * e^(4r)

Similarly, after 7 days (4 + 3), the number of bacteria is 185: 185 = N0 * e^(7r)

Now we have two equations with two unknowns (N0 and r). We can divide the second equation by the first equation to eliminate N0: 185/71 = e^(3r)

Taking the natural logarithm of both sides, we get: ln(185/71) = 3r

Solving for r, we get: r = ln(185/71) / 3 ≈ 0.558

Now we can use the first equation and the growth rate we just found to solve for N0:

71 = N0 * e^(4 * 0.558)

N0 ≈ 11.7

So the initial number of bacteria was approximately 11.7.

To find out how long it will take to reach 500 bacteria, we can plug in the values we know into the equation and solve for t: 500 = 11.7 * e^(0.558t)

Dividing both sides by 11.7, we get: e^(0.558t) ≈ 42.74

Taking the natural logarithm of both sides, we get: 0.558t ≈ ln(42.74)

Solving for t, we get: t ≈ ln(42.74) / 0.558 ≈ 10.3 days

Therefore, it will take approximately 10.3 days for the sandwich to be fully covered with bacteria.

Answer:

the first one:  1:39

the second one: 7:12

the third one: 10:24

Step-by-step explanation:

the short hand points to the hour and the long hand points to the minutes. when the long hand is at 1, that's 5 minutes and when it's at 2, that's 10 minutes and so on and so forth. and each of the little dash marks in between the numbers is 1 minute.

i hope this was correct, i'm not great with time

Question 1:

Question 2:

Question 3:

The sample size needed to ensure that the probability that the sample mean differs from the population mean by more or less then 2.0 hours is less than at a 90% confidence level is 127.

The sample size needed to ensure that the probability can be calculated using the formula:
n = (zα/2)2 2 / (E2)

Where n is the sample size, is the population standard deviation (12.4 hours in this case), E is the margin of error (2.0 hours in this case) and zα/2 is the critical value from a z-table corresponding to the 90% confidence level (1.645 in this case).

Thus, n = (1.645)2 (12.4)2 / (2.0)2 = 126.8

Therefore, the sample size needed to ensure that the probability that the sample mean differs from the population mean by more or less then 2.0 hours is less than at a 90% confidence level is 127.

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Answer:

x = - 2, y = - 8

Step-by-step explanation:

13x - 6y = 22 → (1)

x = y + 6 → (2)

substitute x = y + 6 into (1)

13(y + 6) - 6y = 22

13y + 78 - 6y = 22

7y + 78 = 22 ( subtract 78 from both sides )

7y = - 56 ( divide both sides by 7 )

y = - 8

substitute y = - 8 into (2)

x = y + 6 = - 8 + 6 = - 2

then x = - 2 and y = - 8

The probability of exactly two heads in tossing three fair coins is 3/8.

To find the probability of exactly two heads, we need to count the number of outcomes in the sample space that have exactly two heads and divide that by the total number of outcomes in the sample space.
In the sample space {hhh, hht, hth, htt, thh, tht, tth, ttt}, there are three outcomes that have exactly two heads: hht, hth, and thh.
So the probability of exactly two heads is 3/8.

In mathematical terms, this can be represented as:
P(exactly two heads) = number of outcomes with exactly two heads / total number of outcomes
P(exactly two heads) = 3 / 8

Therefore, the probability of exactly two heads in tossing three fair coins is 3/8.

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As we proved that there exists a δ > 0 such that the function f(x) > 0 for all x ∈ (p − δ, p + δ).

To prove this, we will use the definition of continuity of a function. According to this definition, for a function to be continuous at a point p, the limit of the function as x approaches p must be equal to the value of the function at p.

In this case, we know that f(p) is greater than 0. So, if we can find a δ such that for all x within δ units of p, f(x) is also greater than 0, then we can show that f is continuous at p.

Let's begin by assuming that there is no such δ, i.e., for any δ > 0, there exists an x within δ units of p such that f(x) is less than or equal to 0. This means that we can find a sequence of points xn that approach p such that f(xn) is less than or equal to 0 for all n.

Since f is continuous, we know that lim xn → p f(xn) = f(p) (by the definition of continuity). But we also know that f(p) is greater than 0, so lim xn → p f(xn) cannot be less than or equal to 0. This is a contradiction, which means our assumption that there is no δ such that f(x) is greater than 0 for all x within δ units of p must be false.

Therefore, there must exist a δ > 0 such that f(x) is greater than 0 for all x within δ units of p, which proves our original statement.

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Complete Question:

Let f : (a, b) → R be continuous such that for some p ∈ (a, b), f(p) > 0. Show that there exists a δ > 0 such that f(x) > 0 for all x ∈ (p − δ, p + δ).

Answer:

Step-by-step explanation:

9994ddaefeaf

Answer:(1)/(4)

Step-by-step explanation:n>1/4

The volume of the cone that will fill the cone halfway is [tex]198.72 cm3[/tex] by the given base radius of 9/2cm and 14cm deep.

The  formula for the volume of the cone is

[tex]V = (1/3)πr^2h[/tex]

where h is the height of the cone, r is its base's radius and is a mathematical constant roughly equal to 3.14159.

We must first determine the height of the cone that is filled halfway in to calculate the volume of the cone that will fill it halfway.

Half of the cone's entire height, or 14 cm divided by 2, or 7 cm, corresponds to the height of the half-filled cone.

The cone's base has a radius of 9/2 cm.

We can calculate the volume of the cone that will fill the cone halfway using the formula for a cone's volume as follows:

[tex]V = (1/3)πr^2h[/tex]

[tex]V = (1/3)π(9/2)^2(7) (7)[/tex]

[tex]V ≈ 198.72 cm^3[/tex] (rounded to 2 decimal places) (rounded to 2 decimal places)

Therefore, [tex]198.72 cm3[/tex] is the volume of the cone that will fill it halfway.

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Answer: A, D, and F

Step-by-step explanation:

SOH CAH TOA

Sin = Opposite/Hypotenuse

Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent

Answer:

tanθ=6/8, sinθ=6/10, cosθ= 8/10

Step-by-step explanation:

the ratio of tan is opposite over adjacent from the angle's perspective, and 6 is opposite whereas 8 is adjacent so tan(x)=6/8

the ratio of sin is opposite over hypotenuse from the angle's perspective, where 6 is opposite from it whereas the hypotenuse will always be the side opposite to 90 so it's 10 so sin(x)=6/10

the ratio of cos is adjacent over the hypotenuse from the angle's perspective, where 8 is adjacent to it whereas 10 is the hypotenuse so the ratio will be cos(x)=8/10

To show that the triangles are equivalent, we need to show that each triangle can be transformed into the other by a combination of translations, rotations, and reflections.

Let's assume that △ γ1 γ2 γ3 is an equilateral triangle, and γ1 γ2 + γ1 γ3 +γ2 γ3 = γ1^2 + γ2^2 + γ3^2.

First, we can translate the triangle so that γ1 is at the origin (0,0) in the complex plane. Then, we can rotate the triangle so that γ2 is on the positive real axis. This rotation can be achieved by multiplying each complex number by a suitable complex number of the form e^(iθ), where θ is the angle that γ2 makes with the positive real axis.

After this transformation, we have:

γ1 = 0

γ2 = r

γ3 = x + iy

where r is a positive real number and x, y are real numbers.

Using the fact that the triangle is equilateral, we have:

|γ2 - γ3| = |γ1 - γ3|

|r - x - iy| = |x + iy|

Squaring both sides and simplifying, we get:

r^2 + x^2 + y^2 - 2rx = x^2 + y^2

r^2 - 2rx = 0

This gives us x = r/2, and substituting this into the equation γ1 γ2 + γ1 γ3 +γ2 γ3 = γ1^2 + γ2^2 + γ3^2 gives:

r^2 + r(x + iy) + r(x - iy) = 3r^2

Simplifying this equation, we get:

r^2 = x^2 + y^2

which is equivalent to the equation we obtained earlier.

Therefore, we have shown that the two conditions are equivalent, and the triangles are equivalent under translations, rotations, and reflections.

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The radian measure of an angle coterminal to (15π)/4 is (7π)/4. The correct answer is A

To find an angle that is coterminal with (15π)/4 between 0 and 2π, we can subtract or add any multiple of 2π until we get an angle between 0 and 2π.

First, we can simplify (15π)/4 as follows:

(15π)/4 = (4π + π)/4 = π + (π/4)

Now, we can subtract 2π until we get an angle between 0 and 2π:

π + (π/4) - 2π = -π/4

Since -π/4 is negative, we can add 2π to get an angle between 0 and 2π:

-π/4 + 2π = (8π - π)/4 = (7π)/4

Therefore, an angle coterminal with (15π)/4 between 0 and 2π is (7π)/4. The correct answer is A

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