Suppose that you measure the flow rate of blood in an artery. You find that your measurements are well-fit be the equation dᏙ /dt =10 - 2 cos(120t)


in units of milliliters per second.


a) What volume of blood flows through the artery in 10 seconds? (include units)


b) What volume of blood flows through the artery in one minute? (include units)

The probability that the word so formed starts with M is 1/6, and the probability that it starts with M and ends with E is 1/30.

i) To find the probability that the word starts with M, we need to consider the total number of possible arrangements of the letters and the number of arrangements that start with M. The word "MOBILE" has 6 letters, so there are 6! = 720 possible arrangements of the letters. To find the number of arrangements that start with M, we can fix the M in the first position and arrange the remaining 5 letters in the remaining positions, which gives us 5! = 120 arrangements. Therefore, the probability that the word starts with M is:

P(starts with M) = number of arrangements that start with M / total number of arrangements

= 120 / 720

= 1/6

ii) To find the probability that the word starts with M and ends with E, we can fix the M in the first position and the E in the last position, and then arrange the remaining 4 letters in the remaining positions. This gives us 4! = 24 arrangements. Therefore, the probability that the word starts with M and ends with E is:

P(starts with M and ends with E) = number of arrangements that start with M and end with E / total number of arrangements

= 24 / 720

= 1/30

Thus, the probability that the word so formed starts with M is 1/6, and the probability that it starts with M and ends with E is 1/30.

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C. The constant of proportionality between the diameter and circumference of a circle is pi.

The constant pi comes from the relationship between the diameter and circumference of a circle.

Constant of Proportionality

A constant of proportionality is a number that describes the ratio between 2 values. No matter the measurements of a circle, the constant of proportionality between a circumference and diameter is always the same. This means that the circumference divided by the diameter ≈ 3.14.

Pi

Pi is an irrational number that can be estimated but never completely solved. The value of pi can be used to complete many different calculations such as the area of a circle, and it is used in many different functions like sin. For this reason, pi is one of the most important constants in math.

Jade rides 1.275 miles before stopping for a water break.

To solve this problem, we need to multiply the length of the trail by the fraction representing the portion of the trail that Jade rides.

First, we need to convert the mixed number 5 1/10 into an improper fraction. We do this by multiplying the whole number (5) by the denominator of the fraction (10) and adding the numerator (1). This gives us 51/10.

Next, we multiply 51/10 by 1/4 to find the fraction of the trail that Jade rides before stopping for a water break:

(51/10) x (1/4) = 51/40

To convert this fraction into a decimal, we divide the numerator by the denominator:

51 ÷ 40 = 1.275

Therefore, Jade rides 1.275 miles before stopping for a water break.

In summary, to find how many miles Jade rides before stopping, we convert the mixed number representing the length of the trail into an improper fraction, multiply it by the fraction representing the portion of the trail that Jade rides, and then convert the resulting fraction into a decimal to get our answer.

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Since the area of a similar figure is proportional to the square of its linear dimensions, if one similar figure has an area that is nine times the area of another, the larger figure must have dimensions that are three times the dimensions of the smaller figure.

This is because the area is the square of the linear dimensions. So, if we increase the linear dimensions by a factor of 3, the area increases by a factor of 3^2 = 9.

Therefore, the answer is 3.

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Answer:

T=64

Step-by-step explanation:

Multiply both sides by 4

t/4=16

t/4×4=16×4 Cancel out the 4

t=64

Over the last couple of years, Arnav's height has increased by 20% so his current height is 1.8 meters.

Arnav's height initially was 1.5 meters. Over the last couple of years, his height increased by 20%. To find the new height, we can use the formula: new height = initial height × (1 + percentage increase).

In this case, the initial height is 1.5 meters and the percentage increase is 20%, which can be expressed as a decimal (0.2). Using the formula, we can calculate Arnav's new height as follows:

New height = 1.5 meters × (1 + 0.2) = 1.5 meters × 1.2 = 1.8 meters.

After the 20% increase in height over the last couple of years, Arnav's current height is 1.8 meters.

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Answer: 1 cups of nuts : 1 1/2 cups of raisins

Step-by-step explanation:

a) The distance the ball rebounds on the fifth bounce is approximately 7.59 ft.

b) The total distance the ball has traveled after the fifth bounce is approximately 52.61 ft.

Let's denote the height of the ball after its nth bounce by h_n. Then we can express the relationship between the height of the ball after each bounce in terms of a recursive formula:

h_0 = 16 (initial height)

h_1 = (3/4) * h_0 (rebound distance after the first fall)

h_2 = (3/4) * h_1 (rebound distance after the second fall)

h_3 = (3/4) * h_2 (rebound distance after the third fall)

h_4 = (3/4) * h_3 (rebound distance after the fourth fall)

h_5 = (3/4) * h_4 (rebound distance after the fifth fall)

a) To find the distance the ball rebounds on the fifth bounce, we need to calculate h_5:

h_5 = (3/4) * h_4

= (3/4) * ((3/4) * ((3/4) * ((3/4) * 16)))

= (3/4)^5 * 16

= 7.59375 ft

Therefore, the ball rebounds approximately 7.59 ft on the fifth bounce.

b) To find the total distance the ball has traveled after the fifth bounce, we need to add up all of the distances traveled during the falls and rebounds:

total distance = distance of first fall + rebound distance after first fall + rebound distance after second fall + rebound distance after third fall + rebound distance after fourth fall + rebound distance after fifth fall

total distance = 16 + (3/4) * 16 + (3/4)^2 * 16 + (3/4)^3 * 16 + (3/4)^4 * 16 + (3/4)^5 * 16

total distance = 16 + 12 + 9 + 6.75 + 5.0625 + 3.7969

total distance = 52.6094 ft

Therefore, the ball travels approximately 52.61 ft after the fifth bounce.

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Full Question:

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

Be sure to show and explain all work using mathematical formulas and terminology. A bouncy ball is dropped from a height of 16ft and always rebounds ¼ of the distance of the previous fall.

a) What distance does it rebound the 5th time?

b) What is the total distance the ball has travelled after this time?

The button's area is approximately 50.27 square centimeters.

To find the area of the button, we need to know the diameter of the button. We can find this by using the formula for circumference of a circle:

C = πd

where C is the circumference and d is the diameter.

Substituting the given value for C:

25.12 cm = πd

Solving for d:

d = 25.12 cm / π

d ≈ 8 cm

Now that we know the diameter, we can use the formula for area of a circle:

A = πr^2

where r is the radius (half the diameter).

Substituting the value for d:

r = d/2 = 4 cm

Substituting this value into the formula:

A = π(4 cm)^2

A ≈ 50.27 cm^2

Therefore, the button's area is approximately 50.27 square centimeters.

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The rate of change of the height of the top of the ladder is -144/h ft/sec when the base of the ladder is 6 ft from the house.

The area of the triangle formed by the ladder, wall, and ground is decreasing at a rate of 163.2 ft^2/sec when the base of the ladder is 6 ft from the house.

The angle between the ladder and the ground is decreasing at a rate of 1/8 rad/sec when the base of the ladder is 6 ft from the house.

The rate of change of the height of the top of the ladder, we need to use the Pythagorean Theorem:

where h is the height of the top of the ladder, d is the distance of the base of the ladder from the house, and L is the length of the ladder.

Taking the derivative with respect to time, t, and using the chain rule, we get:

We are given that d = 6 ft, dd/dt = 24 ft/sec, and L = 10 ft. We need to find dh/dt when d = 6 ft.

Plugging in the values, we get:

2h (dh/dt) + 2(6)(24) = 2(10) (0) (since the ladder is not changing length)

Simplifying, we get:

Dividing by 2h, we get:

The area of the triangle formed by the ladder, wall, and ground is given by:

where b is the distance of the base of the ladder from the wall, and h is the height of the triangle.

Taking the derivative with respect to time, t, and using the product rule, we get:

We are given that db/dt = -24 ft/sec, h = L, and dh/dt = -144/h. We need to find dA/dt when d = 6 ft.

Plugging in the values, we get:

Simplifying, we get:

The rate of change of the angle between the ladder and the ground, we use the trigonometric identity:

Dividing by sec^2(theta), we get:

d(theta)/dt = (-24/h^3) - (2h^2/5)

We can plug in the value of h = (L^2 - d^2)^(1/2) = (100 - 36)^(1/2) = 8 ft when d = 6 ft to get:

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The area of the table is approximately 651 square inches.

What is Area ?

Area is a measure of the size of a two-dimensional shape or surface, such as a rectangle, circle, or triangle. It is expressed in square units, such as square inches, square feet, or square meters.

Let's use the Pythagorean theorem to find the length of the table top:

Substituting the given values, we get:

40*40 = [tex]length^{2}[/tex] + 21*21

Simplifying and solving for length, we get:

[tex]length^{2}[/tex]= 1600 - 441

[tex]length^{2}[/tex] = 961

length = 31 inches (rounded to the nearest inch)

Now that we know the length and width of the table, we can find the area by multiplying them together:

area = length x width

area = 31 x 21

area = 651 square inches (rounded to the nearest inch)

Therefore, the area of the table is approximately 651 square inches.

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If A queen-sized mattress is 20 inches longer than it is wide. A king-sized mattress is 1280 square inches.

In mathematics, a variable is a symbol or letter that represents a value that can change or vary in a given context or problem. The area of the queen-sized mattress is x(x + 20) square inches. The equation to determine the area of the king-sized mattress is (x + 16)(x + 20) = x(x + 20) + 1280

Let x be the width of the queen-sized mattress in inches.

Then the length of the queen-sized mattress is x + 20 inches.

The length of the king-sized mattress is the same as the length of the queen-sized mattress, which is x + 20 inches.

We can use the formula for the area of a rectangle to find the area of each mattress:

Area of queen-sized mattress = length x width = (x + 20) x x = x^2 + 20x

Area of king-sized mattress = length x width = (x + 20) x (x + 16) = x^2 + 36x + 320

Area of king-sized mattress = Area of queen-sized mattress + 1,280

Substituting the expressions we found for the areas, we get:

x^2 + 36x + 320 = x^2 + 20x + 1280

Simplifying and solving for x, we get:

16x = 960

x = 60

So the width of the queen-sized mattress is 60 inches, and its length is 80 inches.

The width of the king-sized mattress is 76 inches, and its length is 80 inches.

The area of the queen-sized mattress is:

60^2 + 20(60) = 4,800 square inches

The area of the king-sized mattress is:

76^2 + 36(76) + 320 = 6,080 square inches

And we can verify that the area of the king-sized mattress is indeed 1,280 square inches more than that of the queen-sized mattress:

6,080 - 4,800 = 1,280

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If Mateo jogged 25 9/10 miles last week and jogged the same course all 7 days, then he jogged an average of (25 9/10) / 7 = 3 11/14 miles per day.

To convert this mixed number to an improper fraction, we can multiply the whole number by the denominator of the fraction and add the numerator, then place the result over the denominator:

3 * 14 + 11 = 53

53/14

So, Mateo jogged an average of 53/14 miles per day last week

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Claude purchased 3 hardback books

What is the meaning of purchase?

Purchase refers to the act of buying or acquiring a product, service, or other item in exchange for money or some other form of payment. Purchases can be made by individuals, businesses, or other organizations, and can be made in a variety of ways, including online, in-store, or through a third-party vendor.

Let's assume that Claude purchased x paperback books and y hardback books.

From the problem statement, we can set up a system of two equations to represent the information given,

x + y = 10 (the total number of books Claude purchased is 10)

3x + 8y = 45 (the total cost of the books Claude purchased is $45)

We can use the first equation to solve for x in terms of y:

x = 10 - y

Substituting this into the second equation,

3(10 - y) + 8y = 45

Simplifying the equation,

30 - 3y + 8y = 45

5y = 15

y = 3

Therefore, Claude purchased 3 hardback books. To find the number of paperback books, we can use the equation we derived earlier:

x = 10 - y = 10 - 3 = 7

So, Claude purchased 7 paperback books and 3 hardback books.

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The first option is correct. There is no significant difference in the purchasing patterns across career types

In order for a dataset to be a valid representation of the total population, it needs to be collected in a way that ensures that it is a fair and accurate sample of the population.

One way to ensure this is through random sampling, where individuals are selected to participate in the study without any bias or preconceived notions about their characteristics. This helps to reduce the potential for selection bias and ensures that the sample is representative of the larger population.

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Answer:

When x is greater than -5, the expression inside the absolute value bars is positive, so we can simply remove the bars.

So the equation y = |x+5| can be rewritten as:

y = x+5 (when x > -5)

To evaluate the expression 37 * arcsin(sin(-3π/8)), follow these steps:

1. First, identify the expression: 37 * arcsin(sin(-3π/8))
2. Calculate the value of sin(-3π/8) using the sine function: sin(-3π/8)
3. Apply the arcsin function to the result from step 2: arcsin(sin(-3π/8))
4. Multiply the result from step 3 by 37: 37 * arcsin(sin(-3π/8))

Let's solve each step:

2. sin(-3π/8) = -0.3826834324 (rounded to 10 decimal places)

3. arcsin(-0.3826834324) = -π/8 (in exact form, since the input is the sine of a known angle)

4. 37 * (-π/8) = -37π/8

So, the expression 37 * arcsin(sin(-3π/8)) evaluates to -37π/8 in exact form.

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The numerical value of the two-sample t-statistic for this test is 0.414 . So, the correct option is A).

To determine if there is a significant difference in the mean bacterial concentration in air between carpeted and uncarpeted rooms, the two-sample t-test can be used.

First, we need to calculate the sample means and standard deviations for each group. The sample mean for the carpeted rooms is 22.0 with a standard deviation of 184, while the sample mean for the uncarpeted rooms is 16.9 with a standard deviation of 175.

Next, we can calculate the t-statistic using the formula

t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^0.5

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Plugging in the values, we get

t = (22.0 - 16.9) / ((184^2/7 + 175^2/7)^0.5) = 0.414

Comparing the calculated t-value with the critical t-value for a two-tailed test with 12 degrees of freedom at a 0.05 significance level, we find that the critical t-value is 2.179. Since the calculated t-value is less than the critical t-value, we fail to reject the null hypothesis and conclude that there is no significant difference in the mean bacterial concentration in air between carpeted and uncarpeted rooms.

So, the correct answer is A).

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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 statement that is true is that the amount of rain in the 24- hour period is 8 inches

From the question, we have the following parameters that can be used in our computation:

In 90 minutes 1/2 inch of rain falls

This means that

Rate = (1/2 inch)/90 minutes

So, we have

Rate = (1/2 inch)/(1.5 hour)

The amount of rain in the 24- hour period is

Amount = Rate * Time

So, we have

Amount = (1/2 inch)/(1.5 hour) * 24 hours

Evaluate

Amount = 8 inches

Hence, the amount of rain in the 24- hour period is 8 inches

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The measure of angle PQR is 90 degrees.

Since PR is the diameter of the circle, it follows that angle POR is a right angle, i.e., it measures 90 degrees.

By the inscribed angle theorem, the measure of angle PQR is half the measure of angle POR. Thus,

angle PQR = 1/2 * angle POR

= 1/2 * 90

= 45 degrees.

However, this is not the final answer since angle PQR is not a stand-alone angle, but rather a part of a right-angled triangle PQR.

Since the three angles in a triangle add up to 180 degrees, and we already know that angle PQR is 45 degrees, it follows that:

angle PRQ + angle PQR + angle QPR = 180 degrees

Since angle PQR = 45 degrees, we have:

angle PRQ + 45 + angle QPR = 180 degrees

Rearranging, we get:

angle PRQ + angle QPR = 135 degrees

Since angles PRQ and QPR are complementary angles (together they form a right angle), their sum is 90 degrees. Therefore,

angle PRQ + angle QPR = 90 degrees

Substituting this into the previous equation, we get:

90 degrees = 135 degrees

This is a contradiction, and hence our assumption that angle PQR measures 45 degrees is false.

Therefore, we conclude that angle PQR must measure 90 degrees, since it is the only angle that can satisfy the given conditions.

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The average price of milk in 2018 was $6.45 per gallon.

The average price of milk in 2021 was $189.15 per gallon.

When x = 0 (which represents the year 2018), the function becomes:

3.55 + 2.90(1 + 0)³

= 3.55 + 2.90(1)³

= 3.55 + 2.90

= 6.45

The average price of milk in 2018 was $6.45 per gallon.

When x = 3 (which represents the year 2021), the function becomes:

3.55 + 2.90(1 + 3)³

= 3.55 + 2.90(4)³

= 3.55 + 2.90(64)

= 3.55 + 185.6

= 189.15

The average price of milk in 2021 was $189.15 per gallon.

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The final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.

Let x represent the original price of the TVs. The store is offering a Black Friday sale of 42% off, which we can represent as a function f(x) = 0.58x (since 100% - 42% = 58%). They are also giving a $100 rebate to all customers, represented by the function g(x) = x - 100.

Now, we want to find f(g(x)), which means applying the function f(x) to the result of the function g(x). So, f(g(x)) = f(x - 100).

To do this, plug in (x - 100) for x in the f(x) function: f(x - 100) = 0.58(x - 100).

This function, f(g(x)), represents the final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.

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The approximate area of the regular decagon, rounded to the nearest hundredth, is 190.78 square units.

To find the area of a regular decagon with an apothem of 6.2 units, we can use the formula:

Area = (1/2) × apothem × perimeter

To find "s", we can use the fact that a regular decagon can be divided into 10 congruent triangles, where each triangle has an interior angle of 144 degrees. We can use trigonometry to find the length of the side "s" using one of these triangles:

Now we can find the perimeter of the decagon:

Finally, we can substitute the apothem and perimeter into the formula to find the area:

Rounding to the nearest hundredth, the area of the regular decagon is approximately 190.78 square units.

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The derivative of the given function with respect to x is:
f'(x) = e^(x^2 + 1) * (1 + 2x^2)

To find the derivative of the given function. Let's first rewrite the function for clarity: f(x) = x * e^(x^2 + 1).

To find the derivative f'(x) with respect to x, we'll apply the product rule since we have a product of two functions: x and e^(x^2 + 1). The product rule states that if you have a function f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).

In this case, g(x) = x and h(x) = e^(x^2 + 1). First, let's find the derivatives g'(x) and h'(x):

g'(x) = d/dx (x) = 1
h'(x) = d/dx (e^(x^2 + 1)) = e^(x^2 + 1) * d/dx (x^2 + 1) = e^(x^2 + 1) * (2x)

Now, we can apply the product rule:

f'(x) = g'(x) * h(x) + g(x) * h'(x) = 1 * e^(x^2 + 1) + x * (e^(x^2 + 1) * 2x)

Simplifying the expression, we get:

f'(x) = e^(x^2 + 1) + 2x^2 * e^(x^2 + 1)

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The explict formula, in slope intercept form is an = n/5

The given sequence is 1/5, 2/5, 3/5.

We can observe that this is an arithmetic sequence, where the first term is 1/5, the common difference is 1/5

To find the explicit formula for an arithmetic sequence, we can use the formula:

an = a1 + (n-1)d

Substituting the values we know for this sequence, we get:

an = 1/5 + (n - 1)*(1/5)

Evaluate

an = n/5

Thus, the nth term of this sequence can be found by substituting the value of n in the formula an = n/5

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Complete question

1/5 2/5 3/5

What is the explicit formula in slope intercept form

The given infinite series diverges.

Let f(x) = -5/x. Then, we can see that f(x) is a continuous, positive, and decreasing function for x ≥ 1. Now, we can apply the integral test to determine whether the series converges or diverges.

∫₅^∞ -5/x dx = -5 ln(x) |₅^∞ = -∞

Since the improper integral diverges, by the integral test, the infinite series also diverges.

To apply the integral test, we need to verify the following conditions:

f(x) is a continuous, positive, and decreasing function for x ≥ 1.

The series Σ aₙ and the integral ∫₁^∞ f(x) dx have the same convergence behavior.

Let f(x) = -5/x. Then, f(x) is a continuous function for x ≥ 1. Furthermore, f(x) is positive and decreasing because its derivative is f'(x) = 5/x² > 0 for x ≥ 1.

We can evaluate the integral ∫₁^∞ f(x) dx as follows:

∫₁^∞ -5/x dx = -5 ln(x) |₁^∞ = -∞

Since the improper integral diverges, the series Σ -5/n also diverges by the integral test.

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